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EIGENVALUE ESTIMATES AND SAMPLING

FOR TIME-FREQUENCY LOCALIZATION OPERATORS

BY

SCOTT IZU, B.S., M.S.

A dissertation submitted to the Graduate School

in partial fulfillment of the requirements

for the degree

Doctor of Philosophy

Major Subject: Mathematics

New Mexico State University

Las Cruces New Mexico

July 2009

“Eigenvalue Estimates and Sampling for Time-Frequency Localization Opera-

tors,” a dissertation prepared by Scott Izu in partial fulfillment of the require-

ments for the degree, Doctor of Philosophy, has been approved and accepted by

the following:

Linda LaceyInterim Dean of the Graduate School

Joe LakeyChair of the Examining Committee

July 2009

Committee in charge:

Dr. Joe Lakey, Chair

Dr. Caroline Sweezy

Dr. Tiziana Giorgi

Dr. Chuck Creusere

ii

DEDICATION

I dedicate this work to my wonderful wife Suleica and my father Allen.

iii

ACKNOWLEDGMENTS

I would like to thank my wife, Suleica, for the countless hours of support and

encouragement. She has helped me through many stuggles and been supportive

through it all.

I would like to thank my father, Allen, for the mentoring and guidance through

difficult times. He always had something positive to say and helped me realize

that I was actually growing through each struggle.

I would like to thank my advisor, Joe Lakey. He was willing to change the

traditional roles a student and an advisor play, adapting to my way of communica-

tion and learning. On several occasions, he was willing to debate topics, providing

clarity to my views. He was willing to venture into different research projects, re-

sulting in many valuble experiences and an excellent introduction to the world of

research. He has spent a great deal of time and effort in helping me to succeed

and I appreciate all the work that he has done.

I would like to thank the following for their encouragement and devotion to de-

veloping young minds: Tonia Izu, George Schuttinger, Rupinder Sekhon, Donald

Sarason and Stewart McKechnie.

Finally, I would like to thank the following organizations for their support: the

Ford Foundation, the SAGE Scholars Program, the National Science Foundation

and the Accenture Foundation.

iv

VITA

February 21, 1980 Born in San Jose, California

1998-2000 A.A., De Anza Community CollegeCupertino, California

2001-2003 B.S., University of CaliforniaBerkeley, California

2003-2005 M.S., New Mexico State UniversityLas Cruces, New Mexico

2006-2007 Teaching Assistant, Department of Mathematics,New Mexico State University.

Honors and Awards

Ford Foundation Fellow, 2004-2008.

AGEP Fellow, 2003-2006.

Anna Schrufer Kist Scholarship, 2005.

NSF Summer Research Grant, 2004.

NSF Honorable Mention, 2004.

Highest Honors, 2003, UC Berkeley.

Highest Distinction, 2003, UC Berkeley.

Percy Lionel Davis Award for Excellence in Mathematics, 2003, UC Berkeley.

SAGE Scholar, 2002-2003.

Accenture Foundation Minority Scholarship, 2002-2003.

Accessibility Award, 2003, USDA.

A+ Certification, 2001.

v

PUBLICATIONS

Joseph Lakey and Scott Izu. Time-Frequency Localization and Sampling of

Multiband Signals. Acta Applicandae Mathematicae.

Joseph Lakey, Mike Coombs, Scott Izu and Chris Weaver. On Models for

Coordination of Activity and It’s Disruption. ARO DAAD19-02-1-0211.

FIELD OF STUDY

Major Field: Harmonic Analysis

vi

ABSTRACT

EIGENVALUE ESTIMATES AND SAMPLING

FOR TIME-FREQUENCY LOCALIZATION OPERATORS

BY

SCOTT IZU, B.S., M.S.

Doctor of Philosophy

New Mexico State University

Las Cruces, New Mexico, 2009

Dr. Joe Lakey

In this work, we build on the classical time-frequency analysis tools developed

by Landau, Slepian and Pollack. We focus on prolate spheroidal wave functions

as a tool to analyze time-frequency localized signals. Recent developments in pe-

riodic nonuniform sampling by Venkataramani and Bresler allow us to develop

several sampling formulas and discrete methods for applying time-frequency lo-

calization concepts to multiband signals. Using concepts introduced by Landau,

we explore the behavior of the eigenvalues for time-frequency localization oper-

ators. By extending the work of Khare and George, we are able to give several

vii

alternatives for calculating these eigenvalues. We also include an error analysis

for these calculations.

Our presentation includes a brief history and overview of several concepts

related to time-frequency analysis and sampling. We develop distribution theory

and functional theoretic foundations in order to provide a rigorous basis for our use

of the Poisson summation formula, impulse sampling, synthesis/analysis operators

and Bessel sequences.

Our overview emphasizes possible applications in digital signal processing and

includes several examples. We also include a small code library to demonstrate

how to modularize code based on theory.

viii

CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 THE FOURIER TRANSFORM . . . . . . . . . . . . . . . . . . . 5

2.1 Operator Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Tempered Distributions . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 The Poisson Summation Formula . . . . . . . . . . . . . . . . . . 14

2.4 Periodic Tempered Distributions and Support . . . . . . . . . . . 19

2.5 Fourier Series I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Fourier Series II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Discrete Time Fourier Transform I . . . . . . . . . . . . . . . . . 36

2.8 Discrete Fourier Transform I . . . . . . . . . . . . . . . . . . . . . 38

2.9 Discrete Fourier Transform II . . . . . . . . . . . . . . . . . . . . 47

2.10 Discrete Fourier Transform III . . . . . . . . . . . . . . . . . . . . 55

3 GENERALIZED BASES . . . . . . . . . . . . . . . . . . . . . . . 62

3.1 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Bessel Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Complete Bessel Sequences . . . . . . . . . . . . . . . . . . . . . . 74

3.4 ω-Linearly Independent Bessel Sequences . . . . . . . . . . . . . . 75

ix

3.5 Frame Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.6 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.7 Riesz Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.8 Riesz Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.9 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.9.1 Additions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.9.2 Psuedo Inverse Definition . . . . . . . . . . . . . . . . . . 86

3.9.3 Banach Spaces and Pseudo Inverses . . . . . . . . . . . . . 86

3.9.4 Frame Sequences and Frames . . . . . . . . . . . . . . . . 88

3.9.5 Riesz Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.9.6 Operators associated with Frames . . . . . . . . . . . . . . 89

3.9.7 Duality Principles . . . . . . . . . . . . . . . . . . . . . . . 90

3.10 Wavelet Based Noise Cancellation Algorithm . . . . . . . . . . . . 90

4 SAMPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.1 Poisson Summation Formula for L2 . . . . . . . . . . . . . . . . . 98

4.2 Basic Shannon Sampling . . . . . . . . . . . . . . . . . . . . . . . 100

4.3 Shannon Sampling with a Tiling Set . . . . . . . . . . . . . . . . 101

4.4 General Shannon Sampling Formulas . . . . . . . . . . . . . . . . 104

4.5 Periodic Nonuniform Sampling with Only Complete Aliasing . . . 106

4.6 Periodic Nonuniform Sampling . . . . . . . . . . . . . . . . . . . . 110

4.7 More on ΩK-Tiling Sets . . . . . . . . . . . . . . . . . . . . . . . 114

x

5 TIME-FREQUENCY LOCALIZATION OPERATORS . . . . . . 117

5.1 Prolate Spheriodal Wave Functions . . . . . . . . . . . . . . . . . 117

5.2 Related Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.3 PSWF Sampling Formula . . . . . . . . . . . . . . . . . . . . . . 124

5.4 Eigenvalues Greater Than 1/2 . . . . . . . . . . . . . . . . . . . . 127

5.5 When No Significant Eigenvalues Exist . . . . . . . . . . . . . . . 133

5.6 Eigenvalue Calculations . . . . . . . . . . . . . . . . . . . . . . . 137

5.7 Matrix Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 140

6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.1 Computation Scripts . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.2 Fourier Transform Digital Signal Representations . . . . . . . . . 164

7.3 Fourier Transform Library . . . . . . . . . . . . . . . . . . . . . . 168

7.4 Wavelet Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.5 Eigenvalue Calculations . . . . . . . . . . . . . . . . . . . . . . . 180

7.6 Plots and Configuration . . . . . . . . . . . . . . . . . . . . . . . 182

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

xi

LIST OF TABLES

1 Commutation Properties . . . . . . . . . . . . . . . . . . . . . . . 11

2 Fourier Series I Equations . . . . . . . . . . . . . . . . . . . . . . 23

3 Fourier Series II Equations . . . . . . . . . . . . . . . . . . . . . . 30

4 Fourier Series III Equations . . . . . . . . . . . . . . . . . . . . . 30

5 Discrete Time Fourier Transform I Equations . . . . . . . . . . . 36

6 Discrete Time Fourier Transform II Equations . . . . . . . . . . . 37

7 Discrete Fourier Transform I Equations . . . . . . . . . . . . . . . 40

8 Discrete Fourier Transform II Equations . . . . . . . . . . . . . . 49

9 Discrete Fourier Transform III Equations . . . . . . . . . . . . . . 56

10 Characterizing Bessel Sequences . . . . . . . . . . . . . . . . . . . 84

11 Banach and Hilbert Space Notation . . . . . . . . . . . . . . . . . 87

12 Uniform Sampling Example . . . . . . . . . . . . . . . . . . . . . 113

13 Spectral Splicing . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

14 Related Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 123

15 Eigenvalues for Example 5.20 . . . . . . . . . . . . . . . . . . . . 138

16 First Set of Parameters for Example 5.20 . . . . . . . . . . . . . . 139

17 Second Set of Parameters for Example 5.20 . . . . . . . . . . . . . 139

xii

LIST OF FIGURES

1 Plot of ρ from Example 2.2 . . . . . . . . . . . . . . . . . . . . . . 8

2 Plot of ρ from Example 2.4 . . . . . . . . . . . . . . . . . . . . . . 9

3 Shifts of ψ in Example 2.19 . . . . . . . . . . . . . . . . . . . . . 17

4 Plot of ψ from Example 2.19 . . . . . . . . . . . . . . . . . . . . . 18


6 Plot of a from Example 2.25 . . . . . . . . . . . . . . . . . . . . . 24

7 Plot of A from Example 2.25 . . . . . . . . . . . . . . . . . . . . . 24

8 Plot of b from Example 2.25 . . . . . . . . . . . . . . . . . . . . . 25

9 Plot of B from Example 2.25 . . . . . . . . . . . . . . . . . . . . . 25

10 Alternate definition for a in Example 2.25 . . . . . . . . . . . . . 26

11 Alternate definition for A in Example 2.25 . . . . . . . . . . . . . 26



14 Plot of f from Example 2.27 . . . . . . . . . . . . . . . . . . . . . 33


16 Approximation for f in Example 2.27 . . . . . . . . . . . . . . . . 34

17 Approximation for f in Example 2.27 . . . . . . . . . . . . . . . . 34

18 Plot of g0g1(t) from Example 2.28 . . . . . . . . . . . . . . . . . . 35

19 Plot of g0g1 from Example 2.28 . . . . . . . . . . . . . . . . . . . 35

xiii

20 Plot of b from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 42

21 Plot of B from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 42

22 Plot of c from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 43

23 Plot of C from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 43

24 Plot of d from Example 2.30 . . . . . . . . . . . . . . . . . . . . . 44

25 Plot of D from Example 2.30 . . . . . . . . . . . . . . . . . . . . 44

26 Approximation for c from Example 2.30 . . . . . . . . . . . . . . 45

27 Approximation for C from Example 2.30 . . . . . . . . . . . . . . 45

28 Approximation for c from Example 2.30 . . . . . . . . . . . . . . 46

29 Approximation for C from Example 2.30 . . . . . . . . . . . . . . 46

30 Plot of g0 from Example 2.32 . . . . . . . . . . . . . . . . . . . . 51

31 Plot of g0 from Example 2.32 . . . . . . . . . . . . . . . . . . . . 51



34 Approximation for g0 from Example 2.32 . . . . . . . . . . . . . . 53






40 Plot of samples of f from Example 2.34 . . . . . . . . . . . . . . . 59

xiv

41 Plot of samples of f from Example 2.34 . . . . . . . . . . . . . . . 59

42 Summary of Fourier Transform Examples . . . . . . . . . . . . . . 60

43 Summary of Fourier Transform Examples . . . . . . . . . . . . . . 61

44 3 Stage Wavelet Transform Analysis Filter Bank . . . . . . . . . . 92

45 3 Stage Wavelet Packet Analysis Filter Bank . . . . . . . . . . . . 95

46 LAPP Lightning Pulse Before Denoising . . . . . . . . . . . . . . 96

47 LAPP Lightning Pulse After Denoising . . . . . . . . . . . . . . . 96

48 Wavelet Packet Matrix . . . . . . . . . . . . . . . . . . . . . . . . 97

49 Wavelet Packet Coefficients . . . . . . . . . . . . . . . . . . . . . 97

50 Support for f from Example 4.3 . . . . . . . . . . . . . . . . . . . 101



53 Support for f from Example 4.10 . . . . . . . . . . . . . . . . . . 113

xv

1 INTRODUCTION

The purpose of this dissertation is to present methodologies and concepts

for dealing with analog signals in a digital setting. There are many applications

for such methodologies including cell phone communication, video cameras and

satellite sensors. Signals are used in our everyday lives and over recent years we

have seen a large transition from analog to digital. Generally speaking, analog

signals have an uncountable domain and range, while digital signals have a finite

domain and range. This advantage along with many others has pushed the desire

to develop digital technologies even though most real signals tend to be analog.

In the late 40’s at Bell National Laboratories, Shannon presented his famous

sampling theorem. The result was that band-limited signals may be completely

represented by countably infinite and evenly spaced samples. Thus, analog band-

limited signals may be viewed as having a countably infinite domain and uncount-

ably infinite range. The Shannon sampling theorem lies at the heart of many dig-

ital technologies used today and is a first step toward bridging the gap between

analog and digital signals. In practice, applications use samples which are finite in

number and quantized in value. This reduces the domain from countably infinite

to finite and range from uncountably infinite to finite. Theoretically, this gives a

further reduction of the space of analog signals which may be represented in the

digital setting. Realistically, this introduces errors into the digital representation.

1

For band-limited functions, Shannon also introduced a sampling formula which

demonstrates how to reconstruct an analog signal from its samples. This process

involves interpolating the samples using sinc functions. Since Shannon, many

others have attempted to generalize his sampling formula to reduce the errors

involved with using samples which are finite in number and quantized in value.

We start our discussion in Chapter 2 with the development of basic time-

frequency concepts surrounding the Fourier Transform. We demonstrate how the

domain of a function may effectively be reduced from uncountably infinite to

countably infinite as one moves from the Fourier Transform to the Fourier Series.

This is the basic concept involved in the Shannon sampling formula. We point

out that the Discrete Time Fourier Transform represents the time-frequency dual

of the Fourier Series. We also demonstrate how the domain of a function may be

further reduced from countably infinite to finite as we move from the Fourier Series

to the Discrete Fourier Transform. This presentation of the Fourier Transform is

similar to that of [13] but there are several differences including the development

of the underlying distribution theory.

Under certain circumstances, the Discrete Time Fourier Transform may be

used to numerically compute the Fourier Transform of a function. This concept

will be used in some of our examples. Under other circumstances, the Discrete

Fourier Transform may be used to numerically compute the Fourier Transform of a

function. That is, under certain circumstances, a finite sequence and its Discrete

2

Fourier Transform may both be interpolated to reconstruct a continuous func-

tion and its Fourier Transform. We give some new methodologies and examples

regarding this concept.

The Shannon sampling formula may be generalized to reduce the error asso-

ciated with using finitely many samples instead of a countably infinite number

of samples. One common method of generalizing the Shannon sampling formula

comes from generalizing the underlying orthonormal basis. In Chapter 3, we intro-

duce several generalizations of an orthonormal basis including Bessel sequences,

frame sequences, frames, Riesz sequences and Riesz bases. Most of the concepts

presented are based on the work of Casazza and Christensen. However, our presen-

tation differs greatly from these authors. At the end of Chapter 3, we will discuss

further insights gained from our presentation and common misconceptions found

in the literature.

In Chapter 4, we present several generalizations of the Shannon sampling for-

mula involving functions band-limited to multiple intervals. Some are based on

applying operators to an orthornomal basis. Others are based on frames and

are typically referred to as periodic nonuniform sampling formulas (see [24]). We

include a formula which bridges the gap between typical Shannon sampling and

periodic nonuniform sampling formulas. In order to bridge this gap, we switch

notation from that which is found in [24] and provide a slight generalization to

their periodic nonuniform sampling formula.

3

All of the sampling formulas presented involve reconstruction of signals with

compact frequency support. Since applications use samples which are finite in

number, it seems that the key to developing good reconstruction formulas lies in

time-frequency localization. Chapter 5 introduces the time-frequency localization

operators. Possible reconstruction formulas involving time-frequency localization

operators have already been developed and the associated error may be expressed

in terms of eigenvalues. New estimates and methods are introduced for calculating

eigenvalues of the time-frequency localization operators.

4

2 THE FOURIER TRANSFORM

In this Chapter, we discuss four settings of the Fourier Transform: on the real

line R, on the unit circle T, on the integers Z and on the modular integers ZN .

The authors of [13] refer to the Fourier Transform in these four settings as the

Fourier Transform (FT), Fourier Series (FS), Discrete Time Fourier Transform

(DTFT) and Discrete Fourier Transform (DFT) respectively.

We start with the definitions and commutation properties of standard oper-

ators used in Fourier Analysis. These operators include the Fourier Transform,

Dilation, Translation, Modulation, Reverse, Time-Limiting and Band-Limiting

operators. We then develop the distribution theory for tempered distributions

which will play an important role as we discuss the Fourier Transform in each

setting. We include the Poisson Summation formula for tempered distributions

which serves as the foundation for many sampling formulas.

We then follow the authors of [13] in an attempt to relate the four settings of

the Fourier Transform. For example, we give the FT representations for FS and

DTFT signals. We also give FT, FS and DTFT representations for different types

of DFT signals. However, there are several differences between our presentation

and that of [13]. First of all, we derive our equations rigorously using distribu-

tion theory. Naturally, many of the equations presented here generalize those of

[13]. Second, we express the time and frequency sample periods explicitly with

5

parameters T and Ω. Finally, we provide additional equations so that the duality

between time and frequency may be fully represented.

One of the main concepts is that impulse sampling FT and FS signals leads

to DTFT and DFT signals, respectively. We will show that periodization in

the time domain is equivalent to impulse sampling in the frequency domain, so

that by duality, periodizing FT and DTFT signals leads to FS and DFT signals,

respectively. While the authors of [13] consider impulse sampling DTFT and DFT

signals, we view this as subsampling and do not present the equations. In a similar

fashion, we ignore periodizing FS and DFT signals.

Our view that any FS signal is the periodization of an FT signal allows us

to obtain the periodization equation for an FT signal and the FT representation

of an FS signal simultaneously. This view also leads to the concept that there

are Fourier Series equations for compactly supported functions which parallel

the Fourier Series equations for periodic functions. These are presented as the

Campbell sampling equations. Similarly, the Discrete Fourier Transform involves

finite discrete sequences which may be viewed as either periodic or finite.

The familiar view of a finite discrete sequence as representing the weights of a

periodic impulse train is discussed in the first DFT Section. A periodic impulse

train satisfies the following four properties: i) it is a tempered distribution and

has a well-defined Fourier Transform, ii) its time weights may be interpolated to

reconstruct the impulse train, iii) its frequency weights may be interpolated to re-

6

construct the Fourier Transform and iv) its time and frequency weights are related

by the Discrete Fourier Transform. In a sense, these properties imply that Fourier

Analysis on periodic impulse trains may be performed on a computer. There is

one major problem. Periodic impulse trains do not model real signals very well.

We end the chapter with a third DFT Section which gives an example of a class

of Schwartz functions which satisfy the following properties: i) the signal and its

Fourier Transform are Schwartz functions, ii) the time samples may be interpo-

lated to reconstruct the signal, iii) the frequency samples may be interpolated to

reconstruct the Fourier Transform and iv) the time and frequency samples are

related by the Discrete Fourier Transform. To the best of our knowledge, this is

the first such example found in the literature.

2.1 Operator Definitions

We start by defining the Fourier Transform and various operators on L2(R).

We then list commutation properties of these operators. See [20] and [22] for more

information on Fourier analysis on the Schwartz space and over groups.

Definition 2.1. The Schwartz space, denoted by S, consists of all infinitely dif-

ferentiable functions which are rapidly decreasing. We say φ is rapidly decreasing

if for every n,m ∈ N there exists a constant C such that

supx∈R|(1 + |x|2)nφ(m)(x)| ≤ C

7

With respect to the Schwartz space, we say φj → φ in S(R) if for every n,m ∈ N,

xnφ(m)j → xnφ(m) uniformly in R.

Example 2.2. The following is an example of a Schwartz function which has

support in [−1, 1]. This function is graphed in Figure 1.

ρ(t) =

e|t|2

|t|2−1 if |t| < 1

0 if |t| ≥ 1

Figure 1: Plot of ρ from Example 2.2

Definition 2.3. The Fourier Transform operator, denoted by F , takes functions

from the time domain to the frequency domain and is defined by the following

equation for any φ ∈ S(R).

Fφ(x) =

∫ ∞−∞

φ(t)e−2πixtdt

8

The Fourier Transform as defined is a continuous bijective linear operator from

S(R) to S(R). Using a density argument, this definition extends uniquely to a

unitary operator of L2(R).

We may use f and f to denote the Fourier Transform and Inverse Fourier

Transform of f , respectively. Sometimes we will want to be clear whether the

discussion involves the time domain or the frequency domain. If this is the case,

we will use t to represent the time variable and ω to represent the frequency

variable. The following two equations represent the Fourier Transform and its

inverse for f ∈ S(R).

f(ω) =∫∞−∞ f(t)e−2πiωtdt

f(t) =∫∞−∞ f(ω)e2πiωtdw

Example 2.4. Let ρ be as defined in Example 2.2. Then, the Fourier Transform

of ρ is a Schwartz function and is shown in Figure 2.

Figure 2: Plot of ρ from Example 2.4

9

Definition 2.5. For real parameters α > 0, β and γ, the Dilation (Dα), Transla-

tion (Tβ), Modulation (Mγ) and Reverse (R) operators are defined as follows.

Dαf(x) =√αf(αx)

Tβf(x) = f(x− β)

Mγf(x) = e−2πiγxf(x)

Rf(x) = f(−x)

Many authors take FD2π to be the definition of the Fourier Transform. That

is, their definition of the Fourier Transform includes a factor of 1√2π

in front of the

integral while the factor of 2π is missing from the exponential in the integral. Using

Definition 2.3, the Reverse operator is equal to the Fourier Transform operator

squared. In addition, we have a simple equation for the Inverse Fourier Transform:

F−1 = FR = RF

Definition 2.6. For sets S and Σ, the Time Limiting (QS) and Band Limiting

(PΣ) operators are defined as follows.

QSf = f1S

PΣ = F−1QΣF

The Time and Band Limiting Operators QS and PΣ are projection opera-

tors (being self-adjoint and idempotent). The ranges of these operators, denoted

10

R(QS) and R(PΣ), represent the space of functions time and band limited to

the sets S and Σ, respectively. Table 1 lists the commutation properties of the

operators defined above. See [25] for further discussion.

Table 1: Commutation Properties

FDα = D 1αF FQS = P−SF

FTβ = MβF DαQS = Q 1αSDα

FMγ = T−γF TβQS = QS+βTβ

FR = RF MγQS = QSMγ

DαTβ = T βαDα RQS = Q−SR

DαMγ = MαγDα FPΣ = QΣF

DαR = RDα DαPΣ = PαΣDα

TβMγ = e2πiβγMγTβ TβPΣ = PΣTβ

TβR = RT−β MγPΣ = PΣ−γMγ

MγR = RM−γ RPΣ = P−ΣR

2.2 Tempered Distributions

We introduce S ′(R), the space of tempered distributions. Then, we extend

the Fourier Transform, translation, modulation, multiplication and convolution to

include S ′(R). See [14] for more information.

11

Definition 2.7. The space of tempered distributions, denoted S ′(R) is the set

of all linear functionals which are continuous over S(R). We say f is continuous

over S(R) if f(φj)→ 0 in C whenever φj → 0 in S(R).

Definition 2.8. The Fourier Transform of a tempered distribution f ∈ S ′(R) is

defined by the following equation for any φ ∈ S(R).

f(φ) = f(φ)

This definition extends the definition of the Fourier Transform over L2(R) to

a unique weakly continuous bijective linear operator from S ′(R) to S ′(R). We

now extend translation, modulation, multiplication and convolution to include

tempered distributions.

Definition 2.9. For real parameters β and γ, the translation and modulation of

a tempered distribution f ∈ S ′(R) are both tempered distributions defined by the

following equations for any φ ∈ S(R).

Tβf(φ) = f(T−βφ)

Mγf(φ) = f(Mγφ)

Definition 2.10. The space OM(R) consists of all infinitely differentiable func-

tions which are slowly increasing. We say φ is slowly increasing if for every m ∈ N

there exists constants n, C such that

supx∈R|φ(m)(x)| ≤ C(1 + |x|2)n

12

Definition 2.11. The product of a tempered distribution f ∈ S ′(R) and a slowly

increasing function g ∈ OM(R) is a tempered distribution defined by the following

equation for any φ ∈ S(R).

(gf)(φ) = f(gφ)

Definition 2.12. The convolution of two tempered distributions f, g ∈ S ′(R)

where g ∈ OM(R) is a tempered distribution defined by the following equation for

any φ ∈ S(R).

g ∗ f(φ) = f(g(T−tφ))

Theorem 2.13. Suppose f, g ∈ S ′(R) where g ∈ OM(R). Then, we have

g ∗ f = gf

Proof. We observe how these distributions act on a test function using the defi-

nitions of multiplication and convolution. Let φ ∈ S(R). Then, we have

gf(φ) = f(gφ)

= f(gφ)

= f(g(Mtφ))

= f(g(T−tφ))

= g ∗ f(φ)

= g ∗ f(φ)

13

There are a few comments to make before leaving this section. First of all,

Theorem 2.13 provides an equivalent way of defining convolution between two

tempered distributions. Second, the commutation property FTβ = MβF still

holds in S ′(R). Finally, we have the following inclusion (see [14]).

S(R) ⊂ OM(R) ⊂ S ′(R)

2.3 The Poisson Summation Formula

We start with the Poisson Summation Formula for Schwartz functions. This is

used to develop the Poisson Summation Formula for distributions whose Fourier

Transform is slowly increasing. We end the section with an application to p-

periodic partitions of unity.

Theorem 2.14. Suppose φ ∈ S(R). Then, for any p > 0, φ satisfies the Poisson

Summation Formula pointwise. That is,

∞∑m=−∞

Tmpφ = 1p

∞∑m=−∞

φ(mp

)M−mp1R

Proof. Since φ is a Schwartz function there exists a constant C such that

|φ(t−mp)| ≤ C1+|t−mp|2

Thus, the partial sums for the summation on the left uniformly converge to a

continous function whose Fourier Series coefficients are φ(mp

). Thus, the summa-

tion on the right converges to the summation on the left in L2([0, p)).

14

Since φ is a Schwartz function, the sequence of coefficients φ(mp

) is in `1.

Thus, the partial sums for the summation on the right uniformly converge to a

continuous function and the Poisson Summation formula holds pointwise.

Theorem 2.15. Suppose g is a tempered distribution such that g ∈ OM(R).

Then, for any p > 0, g satisfies the Poisson Summation formula in the sense of

distributions. That is,

∞∑m=−∞

Tmpg = 1p

∞∑m=−∞

g(mp

)M−mp1R

Proof. We observe how these distributions act on a test function φ ∈ S(R). Using

the commutation properties listed in Table 1, FMmpφ = T−mpφ. The product gφ

is a Schwartz function and satisfies the Poisson Summation Formula as shown in

Theorem 2.14. Evaluating the Poisson Summation Formula at t = 0 yields,

∞∑m=−∞

g(T−mpφ) =∞∑

m=−∞g(Mmpφ)

=∞∑

m=−∞

gφ(mp)

= 1p

∞∑m=−∞

gφ(mp

)

Corollary 2.16. Suppose g is a tempered distribution such that g ∈ OM(R).

Then, for any p > 0, g satisfies the following two equations.

∞∑m=−∞

Tmpg = g ∗∞∑

m=−∞Tmpδ

F( ∞∑m=−∞

Tmpg)

= g(

1p

∞∑m=−∞

Tmpδ)

15

Proof. We observe how these distributions act on a test function φ ∈ S(R) using

the equalities in the proof of Theorem 2.15.

∞∑m=−∞

Tmpg(φ) =∞∑

m=−∞g(T−mpφ)

=∞∑

m=−∞Tmpδ(g(T−tφ))

=(g ∗

∞∑m=−∞

Tmpδ)

(φ)

∞∑m=−∞

Tmpg(φ) = 1p

∞∑m=−∞

gφ(mp

)

=(g 1p

∞∑m=−∞

Tmpδ)

(φ)

Definition 2.17. Suppose g is a tempered distribution such that g ∈ OM(R) and

p > 0. Then, g is a p-partition of unity if it satisfies the following equality in the

sense of tempered distributions.

∞∑m=−∞

Tmpg = 1

Corollary 2.18. Suppose g is a tempered distribution such that g ∈ OM(R) and

p > 0. Then, g is a p-partition of unity if and only if

g(m/p) =

p if m = 0

0 if m 6= 0,m ∈ Z

Proof. From Corollary 2.16, g is a p-partition of unity if and only if

g(

1p

∞∑m=−∞

Tmpδ)

= δ

16

In this section, we have discussed three key concepts related to the Fourier

Series: the Poisson Summation Formula, impulse trains and p-partitions of unity.

We end this section with an example of a p-partition of unity.

Example 2.19. Let ρ be as defined in Example 2.2. Then, the following defines

an example of a Schwartz function which is also a 1-partition of unity and has

support on the interval [−1, 1] (see Figures 3 - 5).

ψ(t) =

0 if t ≤ −1

ρ(t)ρ(t)+ρ(t+1)

if − 1 < t <= 0

ρ(t)ρ(t−1)+ρ(t)

if 0 < t < 1

0 if 1 ≤ t

Figure 3: Shifts of ψ in Example 2.19

Notice that the integer shifts of ψ sum to 1.

17

Figure 4: Plot of ψ from Example 2.19


Notice that ψ(n) = 0 for all n ∈ Z/0. The Schwartz function ψ was estimated

using the following DTFT equation.

ψ(ω) ≈ 1N

∞∑n=−∞

ψ( nN

)e−2πinωN

18

2.4 Periodic Tempered Distributions and Support

We start defining what it means for a tempered distribution to be periodic.

We then provide a theorem stating that a tempered distribution is periodic if and

only if it may be written as the convolution of an impulse train with a distribution

whose Fourier Transform is slowly increasing. We end with a discussion of support

for distributions.

Definition 2.20. A tempered distribution f ∈ S ′(R) is p-periodic if f = Tpf .

Theorem 2.21. Suppose ψ ∈ S(R) is any p-partition of unity and f ∈ S ′(R) is

a p-periodic tempered distribution. Then,

f =∞∑

m=−∞Tpm(ψf)

Proof. We observe how this distribution acts on a test function φ ∈ S(R).

f(φ) = f(( ∞∑

m=−∞Tpmψ

)φ)

=∞∑

m=−∞f((Tpmψ)φ)

=∞∑

m=−∞f(ψT−pmφ)

=∞∑

m=−∞Tpm(ψf)(φ)

Corollary 2.16 and Theorem 2.21 provide two equivalent ways of defining pe-

riodicity of tempered distributions. That is, we may say f is periodic if f may be

19

expressed as the convolution of an impulse train with a distribution whose Fourier

Transform lies in OM. The authors of [14] define periodic for the space of distri-

butions rather than the space of tempered distributions. However, no generality

is lost here, since any periodic distribution extends uniquely to a tempered distri-

bution. That is, if f is a p-periodic distribution and ψ ∈ S(R) is any compactly

supported p-partition of unity, f defines a periodic tempered distribution by the

following equation for all φ ∈ S(R).

f(φ) = f(ψ∞∑

m=−∞

T−mpφ)

Definition 2.22. A tempered distribution f ∈ S ′(R) is said to have support in a

set S if f(φ) = 0 whenever φ ∈ S(R) and φ1S = 0.

Lemma 2.23. Suppose f ∈ S ′(R) has support in S. If g ∈ OM and satisfies the

equation g1S = 1S, then f = gf .

Proof. Suppose φ ∈ S(R). Then, f(φ)− gf(φ) = f(φ− gφ) = 0.

2.5 Fourier Series I

In this section, we follow [13] developing equations for periodic distributions.

We introduce the notation fI to denote a distribution which is being impulse

sampled in the frequency domain. The distribution fI will be defined using a

Schwartz function ψ but will not depend on the choice of ψ. Examples of fI will

be given following the proof and major results are summarized in Table 2.

20

Theorem 2.24. Suppose f is a T -periodic distribution. Let ψ ∈ S(R) be any

T -partition of unity. Let fI be any tempered distribution such that fI ∈ OM(R)

and satisfies the following property for all m ∈ Z.

fI(mT

) = f(MmTψ) (1)

Then, f and fI satisfy the following equations.

f(t) = 1T

∞∑m=−∞

fI(mT

)e2πimtT (2)

f(t) =∞∑

m=−∞

fI(t−mT ) (3)

f(ω) = 1T

∞∑m=−∞

fI(mT

)δ(ω − mT

) (4)

Proof. While we have not assumed that ψ is compactly supported, we will use

a compactly supported T -partition of unity for the proof. Let φ ∈ S(R) be any

compactly supported T -partition of unity so that φf is compactly supported.

Then, from Corollary 2.18 and Theorem 2.15, for m ∈ Z, we have

φf(mT

) = 1T

∞∑l=−∞

φf( lT

)ψ(m−lT

)

= 1T

∞∑l=−∞

φf( lT

)∫∞−∞ ψ(t)e−

2πi(m−l)tT dt

=∫∞−∞

1T

∞∑l=−∞

φf( lT

)e2πiltT Mm

Tψ(t)dt

=∫∞−∞

( ∞∑l=−∞

TlTφf)Mm

Tψ(t)dt

= f(MmTψ)

= fI(mT

)

21

Further, φf is compactly supported so φf ∈ OM(R) (see [14]). Thus, from

Theorems 2.21 and 2.15 we have

f =∞∑

m=−∞TmT (φf)

= 1T

∞∑m=−∞

M−mTφf(m

T)

= 1T

∞∑m=−∞

M−mTfI(

mT

)

=∞∑

m=−∞TmTfI

This is Equation 3. Equation 4 comes from applying Theorem 2.16 to fI .

This proof implies that fI can be chosen independently of ψ since the terms

f(MmTψ) do not depend on the choice of ψ. An example of fI which is compactly

supported and satisfies Equation 1 is fI = φf where φ ∈ S(R) is any compactly

supported T -partition of unity. A more complicated example of fI which is not

compactly supported and satisfies Equation 1 is fI = φ1f + φ2 where φ1 ∈ S(R)

is any compactly supported T -partition of unity and φ2 is any Schwartz function

whose samples are zero. Notice that under these assumptions the periodization

of φ1f + φ2 is equal to the periodization of φ1f .

There are two special cases where fI ∈ OM(R) that we should mention. If

fI is a Schwartz function, then fI ∈ S(R) and Equation 3 holds pointwise. If

fI ∈ L2(R) with compact support in some set S, then Equation 3 holds in L2(S).

One might say the space of periodic tempered distributions is the largest space

where the Fourier Series holds since the Fourier Series will not even converge in

22

the sense of distributions if the coefficients fI(mT

) do not have slow growth. On the

other hand, if these coefficients are in `2, they may be given using the partition

of unity ψ = 1TI0 . That is, Equation 1 becomes the following.

fI(mT

) =∫TI0

f(t)e−2πimtT dt

Table 2: Fourier Series I Equations

Fourier Series Fourier Transform

f(t) = 1T

∞∑m=−∞

fI(mT

)e2πimtT f(t) =

∫∞−∞ f(ω)e2πiωtdω

fI(mT

) = f(MmTψ)

f(t) =∞∑

m=−∞fI(t−mT )

f(ω) = 1T

∞∑m=−∞

fI(mT

)δ(ω − mT

)

We assume that f is a T -periodic distribution, ψ ∈ S(R) is any T -partition of

unity, fI is any tempered distribution such that fI ∈ OM(R) and fI satisfies

Equation 1 of Theorem 2.24.

Example 2.25. Let ρ as in Example 2.2 and let T = 12. Then, the following

defines functions satisfying Equations 2 - 4 (see Figures 6-11).

a(t) =5−√

3i

2ρ(6t+ 2) + 7ρ(6t) +

5 +√

3i

2ρ(6t− 2)

b(t) = 7∞∑

n=−∞

ρ(6t−3n)+5−√

3i

2

∞∑n=−∞

ρ(6t−1−3n)+5 +√

3i

2

∞∑n=−∞

ρ(6t−2−3n)

23

Figure 6: Plot of a from Example 2.25

The Schwartz function a may be periodized to obtain b.

Figure 7: Plot of A from Example 2.25

The Schwartz function A = a may be impulse sampled to obtain B.

24

Figure 8: Plot of b from Example 2.25

The tempered distribution b is periodic.

Figure 9: Plot of B from Example 2.25

The tempered distribution B = b is an impulse train.

25

Figure 10: Alternate definition for a in Example 2.25

The above tempered distribution may be periodized to obtain b.

Figure 11: Alternate definition for A in Example 2.25

The above tempered distribution may be impulse sampled to obtain B.

26

In Example 2.25, we first created a using a generic function based on the Mat-

lab parameters vector, T and Ω (see the Appendix). The idea is that several

other examples may be created using the same function, but varying the param-

eters under the restrictions mentioned within code comment blocks. After a was

created, A, b and B were created using the DTFT, periodizing and impulse sam-

pling algorithms, respectively. In fact, most of the examples created in this section

came from manipulating vector. In addition, one should note how the theory is

transparent from the Matlab code.

2.6 Fourier Series II

In this section, we develop equations for compactly supported distributions.

Theorem 2.26. Let S be a T -tiling set. Suppose f is a distribution whose support

lies in S0 where S0 is a subset of S and dist(S0, Sc) > 0. Let ψ be any Schwartz

function satisfying the following property.

ψ(t) =

1 if t ∈ S0

0 if t−mT ∈ S0 for some m ∈ Z \ 0

Then, f satisfies the following equations.

f(t) = 1T

∞∑m=−∞

f(mT

)e2πimtT ψ(t) (5)

f(ω) = 1T

∞∑m=−∞

f(mT

)ψ(ω − mT

) (6)

f(mT

) = f(MmTψ) for all m ∈ Z (7)

27

Proof. From Definition 2.22 and Lemma 2.23, we know that ψTmTf is identical

to f for m = 0 and identical to 0 for any other m ∈ Z. Furthermore, f ∈ OM(R)

since f is compactly supported (see [14]). Therefore, by Theorem 2.15, we have

f = ψf

=∞∑

m=−∞ψTmTf

= ψ∞∑

m=−∞TmTf

= ψ 1T

∞∑m=−∞

M−mTf(m

T)

Taking the Fourier Transform yields Equation 6 which is known as the Camp-

bell Sampling equation (see [14]). The Campbell Sampling equation turns out to

hold pointwise since ψ ∈ S(R).

Let φ ∈ S(R) represent a compactly supported function which satisfies the

same constraint as ψ. Then, Equation 5 is satisfied if ψ is replaced by φ. Further,

since φ has compact support, we may choose a Schwartz function g such that

g(t) = e−2πimtT whenever φ(t) 6= 0. From Lemma 2.23 and Corollary 2.18,

f(MmTψ) = f(g)

=∫∞−∞

1T

∞∑l=−∞

f( lT

)M− lTφ(t)e

−2πimtT dt

= 1T

∞∑l=−∞

f( lT

)M− lTφ(m

T)

= 1T

∞∑l=−∞

f( lT

)T lTφ(m

T)

= f(mT

)

28

If the coefficients f(mT

) are in `2, the equations above may be given using the

partition of unity ψ = 1S. That is, if f ∈ L2(S) where S is a tiling set, then f

satisfies the following equations.

f(mT

) =∫Sf(t)e−

2πimtT dt

f(t) = 1T

∞∑m=−∞

f(mT

)e2πimtT 1S(t)

f(ω) = 1T

∞∑m=−∞

f(mT

)1S(ω − mT

)

Notice that S need not be relatively compact. If S = TI0, Equation 6 simplifies

to the following known as the Shannon Sampling formula.

f(ω) =∞∑

m=−∞f(m

T)sinc(Tw −m)

The main equations derived in this section are listed in Table 3 and generalize

to a larger class of distributions. For example, suppose ψ1 and ψ2 are Schwartz

functions such that ψ1ψ2 is a T -partition of unity. Then, certain distributions f

and fI satisfy the equations in Table 4. In the case that ψ1 = ψ2, TmTψ1 is an

orthonormal system of translates (see [25]).

We will finish this section with two examples. Both of these examples will

satisfy the equations given in Table 3. However, only the first example will satisfy

the assumptions given in Theorem 2.26. Again, the Matlab code used to generate

these examples is found in the Appendix.

29

Table 3: Fourier Series II Equations


f(t) = 1T

∞∑m=−∞

f(mT

)e2πimtT ψ(t)

f(mT

) = f(MmTψ) f(ω) =

∫∞−∞ f(t)e−2πiωtdt

f(ω) = 1T

∞∑m=−∞

f(mT

)ψ(ω − mT

)

We assume that f is compactly supported on some set S0 and ψ is any Schwartz

function which is identical to 1 on S0 and identical to 0 on S0 + mT for any

non-zero integer m.

Table 4: Fourier Series III Equations


f(t) = 1T

∞∑m=−∞

fI(mT

)e2πimtT ψ1(t)

f(mT

) = f(MmTψ2) f(ω) =


f(ω) = 1T

∞∑m=−∞

fI(mT

)ψ1(ω − mT

)

∞∑m=−∞

fI(t−mT ) = (ψ2f) ∗∞∑

m=−∞δ(t−mT )

1T

∞∑m=−∞

fI(mT

)δ(ω − mT

) = (ψ2 ∗ f)∞∑

m=−∞δ(ω − m

T)

These equations generalize those from Table 3.

30

Example 2.27. Let ρ be as defined in Example 2.2 and let T = 12. Define S

to be the T -tiling set [−12,−1

4) ∪ [1

4, 1

2) and S0 = [− 7

16,− 5

16) ∪ [ 5

16, 7

16). Then, the

following assignments give an example of functions satisfying Equations 5 - 7.

Plots of these functions are given in Figures 12-17.

ψ0(t) =

0 if t ≤ −1

ρ(2t+1)3ρ(2t+1)+3ρ(2t+2)

if − 1 < t < −12

13

if − 12≤ t ≤ 1

2

ρ(2t−1)3ρ(2t−2)+3ρ(2t−1)

if 12< t < 1

0 if 1 ≤ t

ψ(t) = 3ψ0(8t+ 3) + 3ψ0(8t− 3)

f(t) = ρ(16t+ 6) + .02δ(8t− 3)

Example 2.28. Let T = 12

and let ψ0 be the T -partition of unity defined in

Example 2.27. Then, the following assignments give an example of functions sat-

isfying the Fourier Series II Equations. Notice g1 does not satisfy the assumptions

of Theorem 2.26. Plots of these functions are given in Figures 18-19.

g0(t) = 4 + e4πit + 2e8πit

g1(t) = ψ0(t)

f(t) = g0g1(t)

31


The Schwartz function ψ is identical to 1 over S0 = [− 716,− 5

16) ∪ [ 5

16, 7

16).


The Schwartz function ψ was estimated using the following DTFT equation.

ψ(ω) ≈ 1N

∞∑n=−∞

ψ( nN

)e−2πinωN

32

Figure 14: Plot of f from Example 2.27

The tempered distribution f is supported in S0 = [− 716,− 5

16) ∪ [ 5

16, 7

16) and is the

sum of Schwartz and impulse functions.


The tempered distribution f is the sum of Schwartz and periodic functions.

33

Figure 16: Approximation for f in Example 2.27

The approximation f came from interpolating samples of f (see Equation 5).

Figure 17: Approximation for f in Example 2.27

The approximation˜f came from interpolating samples of f (see Equation 6).

34

Figure 18: Plot of g0g1(t) from Example 2.28

The tempered distribution g0g1 satisfies the Campbell Sampling Equation.

Figure 19: Plot of g0g1 from Example 2.28

The tempered distribution g0g1 may be reconstructed from its samples.

35

2.7 Discrete Time Fourier Transform I

In this section, we discuss the Discrete Time Fourier Transform which is the

dual of the Fourier Series. That is, we may repeat our discussion on Fourier Series

but interchange f , n, T and ψ with f , m, Ω and ψ respectively. The result would

be the DTFT equations listed in Tables 5 and 6. Here, we use the notation f I to

denote a distribution which is being impulse sampled in the time domain. Notice

that the organization of the equations within Tables 5 and 6 has been modified.

Table 5: Discrete Time Fourier Transform I Equations

Fourier Transform

f(ω) =∫∞−∞ f(t)e−2πiωtdt

Discrete Time Fourier Transform

f I( nΩ

) = f(M− nΩψ)

f(ω) = 1Ω

∞∑n=−∞

f I( nΩ

)e−2πiωn

Ω

f(t) = 1Ω

∞∑n=−∞

f I( nΩ

)δ(t− nΩ

)

f(ω) =∞∑

n=−∞f I(ω − nΩ)

We assume that f is an Ω-periodic distribution, ψ ∈ S(R) is any Ω-partition

of unity, f I ∈ OM(R) and f I satisfies f I( nΩ

) = f(M− nΩψ). The superscript I

represents the fact that f I is being impulse sampled in time.

36

Table 6: Discrete Time Fourier Transform II Equations

Fourier Transform

f(t) =∫∞−∞ f(ω)e2πiωtdω

Discrete Time Fourier Transform

f( nΩ

) = f(M− nΩψ)

f(ω) = 1Ω

∞∑n=−∞

f( nΩ

)e−2πiωn

Ω ψ(ω)

f(t) = 1Ω

∞∑n=−∞

f( nΩ

)ψ(t− nΩ

)

We assume that f is a tempered distribution, f is compactly supported on some

set Σ0 and ψ is any Schwartz function which is equal to 1 on Σ0 and equal to 0

on Σ0 + nΩ for any non-zero integer n.

Table 5 represents the case where f is Ω-periodic and should be compared with

Table 2. Table 6 represents the case where f is compactly supported and should

be compared with Table 3.

In several of our examples, we have used the DTFT equation to compute

the Fourier Transform of a Schwartz function. This is due to the fact that the

Riemann sum approximations converge uniformly to φ as N approaches ∞. We

have replaced N with Ω to emphasize the fact that we have used a Riemann sum

to approximate the value of the Fourier Transform integral equation.

37

2.8 Discrete Fourier Transform I

In this section, we develop equations for periodic impulse trains. We make

the assumption that TΩ ∈ N which will allow us to take the TΩ point Discrete

Fourier Transform. Again, the I in fI and f I will represent the fact that fI and

f I are being impulse sampled in the frequency and time domains as shown in

Equations 12 and 15. The distributions fI and f I will be defined using Schwartz

functions ψ1 and ψ2 but will not depend on the choice of ψ1 and ψ2. Finally,

examples of fI and f I will be given in the discussion following the proof.

Theorem 2.29. Let TΩ ∈ N. Suppose f is a T -periodic impulse train where the

spacing between impulses is 1Ω

. Let ψ1 ∈ S(R) be any T -partition of unity and

ψ2 be any Ω-partition of unity. Let fI ,fI be any tempered distributions such that

fI , fI ∈ OM(R) and satisfy the following properties for all m,n ∈ Z.

fI(mT

) = f(MmTψ1) (8)

f I( nΩ

) = f(M− nΩψ2) (9)

Then, f , fI and f I satisfy the following equations.

f(t) = 1T

∞∑m=−∞

fI(mT

)e2πimtT (10)

f(t) =∞∑

m=−∞fI(t−mT ) (11)

f(ω) = 1T

∞∑m=−∞

fI(mT

)δ(ω − mT

) (12)

38

f(ω) = 1Ω

∞∑n=−∞

f I( nΩ

)e−2πiωn

Ω (13)

f(ω) =∞∑

n=−∞f I(ω − nΩ) (14)

f(t) = 1Ω

∞∑n=−∞

f I( nΩ

)δ(t− nΩ

) (15)

f I( nΩ

) = 1T

TΩ−1∑m=0

fI(mT

)e2πimnTΩ for all n ∈ Z (16)

fI(mT

) = 1Ω

TΩ−1∑n=0

f I( nΩ

)e−2πimnTΩ for all m ∈ Z (17)

Proof. Using the results of Section 2.5, Equations 10, 11 and 12 hold. Since f is

an impulse train and the impulses of f are spaced by 1Ω

, we have MΩf = f . Thus,

for any n ∈ Z, we have

fI(m+nTΩ

T) = f(Mm+nTΩ

Tψ1)

= MnΩf(MmTψ1)

= fI(mT

)

Using Equations 9 and 12, for any n ∈ Z, we have

f I( nΩ

) = f(M− nΩψ2)

= 1T

∞∑m=−∞

fI(mT

)e2πimnTΩ ψ2(m

T)

= 1T

TΩ−1∑m=0

fI(mT

)e2πimnTΩ

∞∑l=−∞

ψ2(m+lTΩT

)

= 1T

TΩ−1∑m=0

fI(mT

)e2πimnTΩ

Equation 12 implies that f is an Ω-periodic distribution. Using the results of

Section 2.7, Equations 13, 14 and 15 hold. We may repeat the argument above to

show that Equation 17 follows from Equations 8 and 15.

39

The definition of fI and f I do not depend on the choices of ψ1 and ψ2. Fur-

thermore, examples include fI = fφ1 and f I = fφ2 where φ1, φ2 ∈ S(R) are any

compactly supported T and Ω-partitions of unity, respectively.

In this section, we discussed periodic distributions with periodic FS coeffi-

cients. Table 7 lists a summary of the equations from this section.

Table 7: Discrete Fourier Transform I Equations


f(t) = 1T

∞∑m=−∞

fI(mT

)e2πimtT f(t) =


fI(mT

) = f(MmTψ1) f(ω) =


Discrete Fourier Transform Discrete Time Fourier Transform

f I( nΩ

) = 1T

TΩ−1∑m=0

fI(mT

)e2πimnTΩ f I( n

Ω) = f(M− n

Ωψ2)

fI(mT

) = 1Ω

TΩ−1∑n=0

f I( nΩ

)e−2πimnTΩ f(ω) = 1

Ω

∞∑n=−∞

f I( nΩ

)e−2πiωn

Ω

f(t) =∞∑

m=−∞fI(t−mT )

f(t) = 1Ω

∞∑n=−∞

f I( nΩ

)δ(t− nΩ

)

f(ω) =∞∑

n=−∞f I(ω − nΩ)

f(ω) = 1T

∞∑m=−∞

fI(mT

)δ(ω − mT

)

We assume TΩ ∈ N, f is a T -periodic impulse train with impulses spaced by

1Ω

, ψ1, ψ2 ∈ S(R) are any T -and Ω-partitions of unity, fI ,fI are any tempered

distributions such that fI , fI ∈ OM(R) and fI , f

I satisfy Equations 8 and 9.

40

Example 2.30. Let ρ and ψ be as in Example 2.2 and 2.19. Let T = 12

and

Ω = 6. Define ψ1(t) = ψ( tT

) = ψ(2t) and ψ2(ω) = ψ(ωΩ

) = ψ(ω6). Then, the

following assignments give an example of functions satisfying Equations 8 - 17.

b(t) = 7∞∑

n=−∞ρ(6t− 3n) + 5−

√3i

2

∞∑n=−∞

ρ(6t− 1− 3n) + 5+√

3i2

∞∑n=−∞

ρ(6t− 2− 3n)

c(t) = 76

∞∑n=−∞

δ(t− n2) + 5−

√3i

2

∞∑n=−∞

δ(t− 16− n

2) + 5+

√3i

2

∞∑n=−∞

δ(t− 13− n

2)

D(ω) = 2∞∑

n=−∞ρ( .5ω−3n

.5|n|) + .5

∞∑n=−∞

ρ( .5ω−1−3n.5|n|

) + 1∞∑

n=−∞ρ( .5ω−2−3n

.5|n|)

Plots for these functions are shown in Figures 20 - 29. Notice that we also

have the following equations which give the weights of the delta functions in the

delta trains c and C.

1Ωb( n

Ω) =

76

if n mod 3 = 0

5−√

3i12

if n mod 3 = 1

5+√

3i12

if n mod 3 = 2

, 1TD(m

T) =

4 if m mod 3 = 0

1 if m mod 3 = 1

2 if m mod 3 = 2

The tempered distribution D is an infinite sum of Schwartz functions φn

where the infinite sum of the Schwartz functions φn converges uniformly. For

Figure 24, D was approximated using a finite sum of DTFT approximations for

the Schwartz functions in the summation above.

Notice b may be replaced with g0 from Example 2.28. In addition, since both

are continuous periodic tempered distributions, their Fourier Transform may be

computed by time limiting to one period, taking the Fourier Transform and then

impulse sampling (see Section 2.5).

41

Figure 20: Plot of b from Example 2.30

The tempered distribution b may be impulse sampled to obtain c.

Figure 21: Plot of B from Example 2.30

The tempered distribution B may be periodized to obtain C.

42

Figure 22: Plot of c from Example 2.30

The tempered distribution c is periodic delta train.

Figure 23: Plot of C from Example 2.30

The tempered distribution C = c is a periodic delta train.

43

Figure 24: Plot of d from Example 2.30

The tempered distribution d may be periodized to obtain c.

Figure 25: Plot of D from Example 2.30

The tempered distribution D = d may be impulse sampled to obtain C.

44

Figure 26: Approximation for c from Example 2.30

The approximation c came from interpolating samples of D (see Equation 10).

Figure 27: Approximation for C from Example 2.30

The approximation C came from interpolating samples of b (see Equation 13).

45

Figure 28: Approximation for c from Example 2.30

The approximation c came from periodizing d (see Equation 11).

Figure 29: Approximation for C from Example 2.30

The approximation C came from periodizing B (see Equation 14).

46

2.9 Discrete Fourier Transform II

In this section, we develop several equations for periodic distributions whose

Fourier Series coefficients are supported by a discrete tiling set.

Theorem 2.31. Let TΩ ∈ N and let Σ be any Ω-tiling set. Let ψ1 ∈ S(R) be any

T -partition of unity and let ψ2 be any Schwartz function satisfying the following

property for all m ∈ Z.

ψ2

(mT

)=

1 if m

T∈ Σ

0 if mT6∈ Σ

Suppose f is a T -periodic distribution such that f(MmTψ1) = 0 whenever m ∈ Z

but mT6∈ Σ. Let fI be any tempered distribution such that fI ∈ OM(R) and satisfies

the following property for all m ∈ Z.

fI(mT

) = f(MmTψ1) (18)

Then, f and fI satisfy the following equations.

f(t) = 1T

∑mT∈Σ

fI(mT

)e2πimtT (19)

f(t) =∞∑

m=−∞fI(t−mT ) (20)

f(ω) = 1T

∑mT∈Σ

fI(mT

)δ(ω − mT

) (21)

f( nΩ

) = f(M− nΩψ2) (22)

47

f(ω) = 1Ω

∞∑n=−∞

f( nΩ

)e−2πiωn

Ω ψ2(ω) (23)

f(t) = 1Ω

∞∑n=−∞

f( nΩ

)ψ2(t− nΩ

) (24)

f( nΩ

) = 1T

∑mT∈Σ

fI(mT

)e2πimnTΩ for all n ∈ Z (25)

fI(mT

) = 1Ω

TΩ−1∑n=0

f( nΩ

)e−2πimnTΩ for all m

T∈ Σ (26)

Proof. Using the results of Section 2.5, f satisfies Equations 19, 20 and 21. Thus,

Equation 21 implies that f has compact support in Σ and by Section 2.7, f satisfies

Equations 22, 23 and 24. By evaluating Equation 19, we see that the samples of

f satisfy Equations 25 and 26.

The equations developed in this section are shown in Table 8. Notice that

this section describes a periodic distribution f which has compactly supported FS

coefficients and also descibes a compactly supported distribution f which has pe-

riodic DTFT coefficients. Thus, by duality, this section also describes a compactly

supported distribution f which has periodic FS coefficients.

On the other hand, the previous section described a periodic distribution f

which had periodic FS coefficients. The remaining description, which will be

given in the next section, involves a compactly supported distribution f with

compactly supported FS coefficients. Notice that tiling sets have been used to

describe compactly supported distributions while discrete tiling sets have been

used to describe compactly supported FS coefficients.

48

Table 8: Discrete Fourier Transform II Equations


f(t) = 1T

∑mT∈Σ

fI(mT

)e2πimtT f(t) =


fI(mT

) = f(MmTψ1)


f( nΩ

) = 1T

∑mT∈Σ

fI(mT

)e2πimnTΩ f( n

Ω) = f(M− n

Ωψ2)

fI(mT

) = 1Ω

TΩ−1∑n=0

f( nΩ

)e−2πimnTΩ f(ω) = 1

Ω

∞∑n=−∞

f( nΩ

)e−2πiωn

Ω ψ2(ω)

f(t) =∞∑

m=−∞fI(t−mT )

f(t) = 1Ω

∞∑n=−∞

f( nΩ

)ψ2(t− nΩ

)

f(ω) = 1T

∑mT∈Σ

fI(mT

)δ(ω − mT

)

We assume TΩ ∈ N, Σ is an Ω-tiling set, f is a T -periodic distribution, f is

compactly supported on Σ0 = mT

: m ∈ Z, mT∈ Σ, ψ1 ∈ S(R) is any T -partition

of unity and ψ2 is any Schwartz function which is equal to 1 on Σ0 and equal to

0 on Σ0 + nΩ for any non-zero integer n. We also assume fI is any tempered

distribution such that fI ∈ OM(R) and fI satifies Equation 18.

49

Example 2.32. Let ρ and ψ0 be as in Example 2.2 and 2.28. Let T = 12

and

Ω = 6. Define ψ1(t) = ψ( tT

) = ψ(2t) and Σ = [−4,−2) ∪ [6, 8) ∪ [10, 12). Then,

the following assignments give an example of functions that satisfy the equations

of Theorem 2.31.

ψ2(ω) = ρ(ω+42

) + ρ(ω−62

) + ρ(ω−102

)

g0(t) = e−8πit + 4e12πit + 2e20πit

g1(t) = ψ0(t)

fI(t) = g0g1(t)

Plots for these functions are shown in Figures 30 - 37. Notice that we also have

the following equations which give the weighted samples of g0 and the weights of

the delta functions in g0.

1Ωg0( n

Ω) =

76

if n mod 3 = 0

5−√

3i12

if n mod 3 = 1

5+√

3i12

if n mod 3 = 2

, 1Tg0g1(m

T) =

4 if m = 3

1 if m = −2

2 if m = 5

The tempered distribution g0 is a T -periodic tempered distribution. Therefore,

its Fourier Transform may be computed by multiplying by a T -partition of unity,

taking the Fourier Transform and then impulse sampling. Since g1 is a T -partition

of unity, we know that g0g1 may be impulse sampled to obtain g0. In addition,

g0g1 is just one example of many slowly increasing functions whose samples satisfy

Equation 18. Here, g0g1 was calculated using the DTFT to approximate the FT.

50

Figure 30: Plot of g0 from Example 2.32

The tempered distribution g0 is periodic with compactly supported FS coefficients.

Figure 31: Plot of g0 from Example 2.32

The tempered distribution g0 has compact support and has DTFT coefficients

which are periodic with period T .

51


The tempered distribution g0g1 may be periodized to obtain g0.


The tempered distribution g0g1 may be impulse sampled to obtain g0.

52

Figure 34: Approximation for g0 from Example 2.32

The approximation g0 came from interpolating samples of g0 (see Equation 24).


The approximation ˜g0 came from interpolating samples of g0 (see Equation 23).

53


The approximation g0 came from interpolating samples of g0g1 (see Equation 19).


The approximation g0 came from periodizing g0g1 (see Equation 20).

54

2.10 Discrete Fourier Transform III

In this section, we develop several equations for compactly supported distri-

butions whose Fourier Series coefficients are supported by a discrete tiling set.

Theorem 2.33. Let TΩ ∈ N. Let S be any T -tiling set and Σ any Ω-tiling set.

Let ψ ∈ S (R) be any T -partition of unity satisfying the following equation.

ψ(n

Ω

)=

1 if n

Ω∈ S

0 if nΩ6∈ S

Suppose f is a tempered distribution such that f is a linear combination of the

functions in the set TmTψ : m

T∈ Σ. Then, f satisfies the following equations.

f(ω) =∫∞−∞ f(t)e−2πiωtdt (27)

f(t) =∫∞−∞ f(ω)e2πiωtdω (28)

f(ω) = 1T

∑mT∈Σ

f(mT

)ψ(ω − mT

) (29)

f(t) = 1T

∑mT∈Σ

f(mT

)e2πimtT ψ(t) (30)

f( nΩ

) = 1T

∑mT∈Σ

f(mT

)e2πimnTΩ for all n

Ω∈ S (31)

f(mT

) = 1Ω

∑nΩ∈Sf( n

Ω)e−

2πimnTΩ for all m

T∈ Σ (32)

f(t) = 1Ω

∑nΩ∈Sf( n

Ω)[

1T

∑mT∈Σ

e2πimT

(t− nΩ

)ψ(t)]

(33)

f(ω) = 1Ω

∑nΩ∈Sf( n

Ω)[

1T

∑mT∈Σ

e−2πimnTΩ ψ(ω − m

T)]

(34)

55

Proof. Schwartz functions satisfy Equations 27 and 28. By Corollary 2.18, the

coefficients for the linear combination are 1Tf(m

T) and Equation 29 holds. Finally,

Table 1 implies Equation 30 and properties of ψ imply Equations 31 and 32.

Equations 30 and 34 are generalized FS and DTFT equations. Also, ψ may

be replaced by 1S. The equations developed in this section are shown in Table 9.

Table 9: Discrete Fourier Transform III Equations


f(t) = 1T

∑mT∈Σ

f(mT

)e2πimtT ψ(t) f(t) =


f(mT

) =∫∞−∞ f(t)e−

2πimtT dt f(ω) =



f( nΩ

) = 1T

∑mT∈Σ

f(mT

)e2πimnTΩ f( n

Ω) =

∫∞−∞ f(ω)e

2πiωnΩ dω

f(mT

) = 1Ω

∑nΩ∈Sf( n

Ω)e−2πimnTΩ f(ω) = 1

Ω

∑nΩ∈Sf( n

Ω)[

1T

∑mT∈Σ

e−2πimnTΩ ψ(ω − m

T)]

f(t) = 1Ω

∑nΩ∈Sf( n

Ω)[

1T

∑mT∈Σ

e2πimT

(t− nΩ

)ψ(t)]

f(ω) = 1T

∑mT∈Σ

f(mT

)ψ(ω − mT

)

We assume that TΩ ∈ N, S is a T -tiling set, Σ is an Ω-tiling set, ψ ∈ S(R) is

a T -partition of unity which is 1 on the set S0 = nΩ

: n ∈ Z, nΩ∈ Σ, ψ is 0

on S0 + mT for any m ∈ Z and f is a tempered distribution such that f is in

spanTmTψ : m

T∈ Σ.

56

Example 2.34. Let ρ and ψ be as in Example 2.2 and 2.19. Let T = 12

and Ω = 6.

Define S = [−13,−1

6) ∪ [0, 1

6) ∪ [1

3, 1

2) and Σ = [−4,−2) ∪ [6, 8) ∪ [10, 12). Then,

the following assignments give an example of functions that satisfy the equations

of Theorem 2.33.

g0(t) = 4 + e4πit + 2e8πit

g2(t) = ψ(6t+ 2) + ψ(6t) + ψ(6t− 2)

f(t) = g0g2(t)

Plots for these functions are shown in Figures 38 - 41. Notice that we also have

the following equations which give the weighted samples of f and the weights of

the delta functions in f .

1Ωf( n

Ω) =

76

if n = 0

5−√

3i12

if n = −2

5+√

3i12

if n = 2

, 1Tf(m

T) =

4 if m = 3

1 if m = −2

2 if m = 5

Figures 42 and 43 summarize most of the examples from this section. These

include a periodic distribution b (FS I), a periodic impulse train c (DFT I), the

product g0g1 of a periodic distribution and a partition of unity g1 (FS II), a

periodic distribution g0 whose FS coefficients are supported by a discrete tiling

set (DFT II) and finally, the product g0g2 of a periodic distribution g0 whose FS

coefficients are supported by a discrete tiling set and a partition of unity g2 whose

samples are supported by a discrete tiling set. (DFT III).

57


The tempered distribution f is compactly supported and has FS coefficients which

are also compactly supported. Furthermore, the samples of f are related to the

samples of f by Equations 31 and 32.


58

Figure 40: Plot of samples of f from Example 2.34

The samples of f are supported in a T -tiling set, while the samples of f are

supported in an Ω-tiling set. These samples are related by the DFT.

Figure 41: Plot of samples of f from Example 2.34

59

Figure 42: Summary of Fourier Transform Examples

60

Figure 43: Summary of Fourier Transform Examples

61

3 GENERALIZED BASES

In this chapter we characterize several sequences which generalize orthonormal

bases. We begin with the development of several functional analysis tools. First,

we discuss when a linear operator is bounded. For a bounded linear operator T ,

we also discuss its reduction T and its pseudo inverse T−1. Some of these ideas,

such as the open mapping theorem, may be found in [21] and [23].

Starting with Section 3.2, we assume `2 is the space of square integrable func-

tions and H is a Hilbert space. Under this assumption, there is a 1-1 correspon-

dence between isometries T : `2 → H and orthonormal bases gn ⊂ H. That is,

if T : `2 → H is an isometry, gn = T (δn) defines an orthonormal basis, where δn

is the standard orthonormal basis of `2. Conversely, if gn ⊂ H is an orthonormal

basis, then T (δn) = gn defines an isometry. Recall that a bounded linear operator

can be completely described by its action on an orthonormal basis. In short, we

may say that there is a 1-1 correspondence between isometries T : `2 → H and

orthonormal bases gn ⊂ H through the binding T (δn) = gn.

A similar correspondence holds between other types of bounded linear opera-

tors and the following types of sequences: Bessel sequences, complete Bessel se-

quences, ω-linearly independent Bessel sequences, frame sequences, frames, Riesz

sequences and Riesz bases. Remarkably, for each of these sequences, we can de-

scribe this correspondence with respect to a bounded linear operator T : `2 → H

62

having a combination of the following three properties: T is injective, R(T ) is

closed and T ∗ is injective. This concept was developed independently of the work

in [1], which gives similar characterizations.

We also give characterizations of Bessel sequences, frame sequences, frames,

Riesz sequences and Riesz basis in terms of the four basic operators associated

with Riesz bases: synthesis operator, analysis operator, Gram matrix and frame

operator. Many of the results involving these four operators may be found in the

following works of Casazza and Christensen: [4, 5, 6, 7, 8, 9, 10].

Our presentation differs in significant ways from that in the literature. We have

pulled together pieces from many sources and given several additions. Therefore,

we include the final section of this chapter to compare our work with the literature.

3.1 Functional Analysis

For the following discussion, H1 and H2 will represent Hilbert spaces and

T : H1 → H2 will represent a linear operator. The nullspace and range of T will

be denoted N (T ) and R(T ), respectively.

Theorem 3.1 discusses necessary and sufficient conditions for T to be bounded

(i.e., continuous). We define the reduction T of T by restricting the domain and

codomain of T . Theorem 3.5 discusses necessary and sufficient conditions forR(T )

to be closed. In a sense, this is exactly the case for which the pseudo inverse T−1

exists. Finally, Corollary 3.7 discusses necessary and sufficient conditions for T to

63

be surjective. Throughout the section, we will be mentioning the duality between

T and T ∗. That is, we will often replace T with T ∗ to obtain a result. Some of the

ideas in this section may be extended to the case where H1 and H2 are Banach

spaces. In general, any equivalence which does not involve the composition of the

operators T and T ∗ may be extended to an equivalence for Banach spaces (see

Table 11 and Section 3.9).

Theorem 3.1. Suppose H1 and H2 are Hilbert spaces and T : H1 → H2 is a

linear operator. Then, the following statements are equivalent.

1. There exists B such that ‖Tf‖ ≤ B ‖f‖ for every f ∈ H1

2. There exists B such that ‖T ∗f‖ ≤ B ‖f‖ for every f ∈ H2

3. There exists B such that T ∗T ≤ B2I on H1

4. There exists B such that TT ∗ ≤ B2I on H2

If any of these statements hold, we may use the following value for B.

B2 = ‖T‖2 = ‖T ∗‖2 = ‖T ∗T‖ = ‖TT ∗‖

Proof. We outline the proof. We will show 1 ⇒ 2 and 1 ⇒ 3. The argument for

the opposite directions is similar. Finally, 2⇔ 4 follows from the duality.

64

1⇒ 2. Suppose 1 holds and let f2 ∈ H2. Then,

‖T ∗f2‖ = sup‖f1‖≤1,f1∈H2

|〈T ∗f2, f1〉|

= sup‖f1‖≤1,f1∈H2

|〈T ∗ f2

‖f2‖ , f1〉| ‖f2‖

= sup‖f1‖≤1,f1∈H2

|〈Tf1,f2

‖f2‖〉| ‖f2‖

≤ sup‖f1‖≤1,f1∈H2

‖Tf1‖ ‖f2‖

≤ sup‖f1‖≤1,f1∈H2

B ‖f1‖ ‖f2‖

= B ‖f2‖

1⇒ 3. Suppose 1 holds and let f ∈ H1. Then, T ∗T ≤ B2I on H1 since

〈T ∗Tf, f〉 = 〈Tf, Tf〉

= ||Tf ||2

≤ B2||f ||2

= 〈B2If, f〉

The following Lemmas concern the range of a bounded linear operator.

Lemma 3.2. Suppose H1, H2 are Hilbert spaces, T : H1 → H2 is a bounded

linear operator and h ∈ R(T ). Then, there exists a unique function f1 ∈ H1 such

that T (f1) = h and ‖f1‖ ≤ ‖g‖ for every g such that T(g) = h. Furthermore, the

function f1 ∈ N (T )⊥.

65

Proof. Let h ∈ R(T ). Then, there exists f ∈ H1 such that T (f) = h. Write

f = f1 + f2 where f1 ∈ N (T )⊥ and f2 ∈ N (T ). Then, T (f1) = h, since

T (f1) = T (f1) + T (f2) = T (f1 + f2) = h

Suppose T (g) = h. Then, the following argument shows that ‖f1‖ ≤ ‖g‖ with

equality if and only if g = f1. Since T (g − f1) = 0 and f1 ∈ N (T )⊥, we have

‖g‖2 = ‖f1 + g − f1‖2 = ‖f1‖2 + 2Re〈f1, g − f1〉+ ‖g − f1‖2

= ‖f1‖2 + ‖g − f1‖2

Lemma 3.3. Suppose H1, H2 are Hilbert spaces and T : H1 → H2 is a bounded

linear operator. Then, R(T )⊥ = N (T ∗). In particular, R(T ) is dense in N (T ∗)⊥.

Proof. Observe the following equivalences.

g ∈ R(T )⊥

⇔ 〈T (f), g〉 = 0 for every f ∈ H

⇔ 〈f, T ∗(g)〉 = 0 for every f ∈ H

⇔ g ∈ N (T ∗)

These lemmas inspire the definition of the reduction operator. Here, A⊥ repre-

sents the orthogonal complement of the subspace A. However, for general Banach

spaces, it represents the set of linear functions which annihilate A (see [21]).

66

Definition 3.4. The reduction of T , denoted T , is defined by the following.

T : N (T )⊥ → N (T ∗)⊥, T : f 7→ T (f)

Notice that from Lemmas 3.2 and 3.3, R(T ) is equal to R(T ) and dense in

N (T ∗)⊥. We will use T to denote the reduction of T , which comes from restricting

both the domain and range of T . We will also use the fact that the reduction of

the adjoint of T is the same as the adjoint of the reduction of T .

The following properties are often useful for describing a linear bounded op-

erator T : T is injective, R(T ) is closed and T ∗ is injective. They correspond

to the following properties concerning T : N (T )⊥ = H1, R(T ) = N (T ∗)⊥ and

N (T ∗)⊥ = H2.

Theorem 3.5. Suppose H1, H2 are Hilbert spaces and T : H1 → H2 is a bounded

linear operator. Then, the following statements are equivalent.

1. The reduction of T is a bijective map.

2. The reduction of T ∗ is a bijective map.

3. The reduction of T ∗T is a bijective map.

4. The reduction of TT ∗ is bijective map.

5. R(T ) is closed (R(T ) = N (T ∗)⊥).

6. R(T ∗) is closed (R(T ∗) = N (T )⊥).

67

7. R(T ∗T ) is closed (R(T ∗T ) = N (T )⊥).

8. R(TT ∗) is closed (R(TT ∗) = N (T ∗)⊥).

9. There exists A > 0 such that√A ‖f‖ ≤ ‖Tf‖ for every f ∈ N (T )⊥.

10. There exists A > 0 such that√A ‖f‖ ≤ ‖T ∗f‖ for every f ∈ N (T ∗)⊥.

11. There exists A > 0 such that AI ≤ T ∗T on N (T )⊥.

12. There exists A > 0 such that AI ≤ TT ∗ on N (T ∗)⊥.

If any of these statements hold, we may use the following value for A.

1A

=∥∥∥T−1

∥∥∥2

=∥∥∥(T ∗)−1

∥∥∥2

=∥∥∥(T ∗T )−1

∥∥∥ =∥∥∥(T T ∗)−1

∥∥∥Proof. Again, we will outline the proof. First, we make some opening comments.

Notice that R(T ) and R(T ) are the same and both are dense in N (T ∗)⊥ by

Lemmas 3.2 and 3.3. Thus, R(T ) is closed if and only if R(T ) contains or is equal

to N (T ∗)⊥. Similar statements hold for T ∗, T ∗T and TT ∗.

1 ⇔ 5. The definition of T implies that it is injective. Based on our opening

comments, R(T ) is closed if and only if T is bijective.

1⇔ 9. Suppose 1 holds. By the open mapping theorem T is open. Thus, T−1

is continuous, which is equivalent to being bounded for linear operators. Thus,

there exists an A > 0 such that ‖f‖ =∥∥∥T−1Tf

∥∥∥ ≤ 1√A‖Tf‖ for all f ∈ N (T ∗)⊥.

On the other hand, suppose 9 holds. Let g ∈ N (T ∗)⊥. Based on our opening

comments, there exists a sequence fn ∈ N (T )⊥ such that T (fn) is Cauchy and

68

converges to g. But the assumption implies that fn is Cauchy and thus converges

to some f ∈ N (T )⊥. By continuity, T (f) = g. Since g was arbitrary, R(T )

contains N (T ∗)⊥. Based on our opening comments, R(T ) is equal to N (T ∗)⊥.

5 ⇔ 6. Suppose 5 holds. Then, T−1 exists. Let f ∈ N (T )⊥. Then, for all

g ∈ N (T )⊥, we have 〈f, g〉 = 〈f, T−1T g〉 = 〈T ∗(T−1)∗f, g〉. Since N (T )⊥ is a

Hilbert space, f = T ∗(T−1)∗f . Based on our opening comments, f ∈ R(T ∗).

Since f was arbitrary, R(T ∗) contains N (T )⊥. Again, based on our opening

comments, R(T ∗) is closed. The reverse direction follows from duality.

5⇔ 7. Suppose 5 holds. Let f ∈ R(T ∗). Then, there exists g ∈ N (T ∗)⊥ such

that T ∗(g) = f . By the assumption, there exists h ∈ H1 such that T (h) = g.

Thus, T ∗T (h) = f . Since f was arbitrary, R(T ∗T ) contains R(T ∗). Since R(T ∗)

contains R(T ∗T ), R(T ∗T ) = R(T ∗) which is closed by the implication 5⇔ 6.

On the other hand, suppose 7 holds. Since R(T ∗) contains R(T ∗T ) = N (T )⊥,

based on our opening comments, R(T ∗) is closed. Thus, R(T ) is closed by the

implication 5⇔ 6.

We have shown 1 ⇔ 5 ⇔ 9 and 5 ⇔ 7. The implications 2 ⇔ 6 ⇔ 10 and

6⇔ 8 follow by replacing T and T ∗. The implications 3⇔ 7⇔ 11 and 4⇔ 8⇔

12 may be shown with arguments similar to those found in 1⇔ 5⇔ 9.

Definition 3.6. If any of the equivalences in Theorem 3.5 holds, then we refer to

the operator T−1 as the psuedo inverse of T .

69

Corollary 3.7. Suppose H1 and H2 are Hilbert spaces and T : H1 → H2 is a

bounded linear operator. Then, the following statements are equivalent.

1. T is a surjective map.

2. R(T ∗) is closed and T ∗ is an injective map.

3. TT ∗ is a bijective map.

4. There exists A > 0 such that√A ‖f‖ ≤ ‖T ∗f‖ for every f ∈ H2.

5. There exists A > 0 such that AI ≤ TT ∗ on H2.


1A

=∥∥∥(T )−1

∥∥∥2

=∥∥∥(T ∗)−1

∥∥∥2

=∥∥∥(T ∗T )−1

∥∥∥ = ‖(TT ∗)−1‖

Proof. Again, we outline the proof.

1 ⇔ 2. Suppose 1 holds. Since R(T ) is closed, Theorem 3.5 implies that

R(T ∗) is closed. Further, R(T ) = H2 is dense in N (T ∗)⊥ so T ∗ is injective. On

the other hand, suppose 2 holds. By Theorem 3.5, R(T ) is closed. Further, R(T )

is equal to its closure N (T ∗)⊥ = H2.

1 ⇔ 3. Suppose 1 holds. Since R(T ∗) ⊂ N (T ), TT ∗ is injective. Theorem

3.5 implies that R(T ∗) is equal to its closure N (T ∗)⊥. Thus, R(TT ∗) is equal to

R(T ) which is the same as R(T ). Thus, TT ∗ is surjective. On the other hand if,

3 holds then T must be surjective.

70

2 ⇔ 4. Suppose 2 holds. Since T ∗ is injective, N (T ∗)⊥ = H2. Since R(T ∗)

is closed, Theorem 3.5 implies 4. On the other hand, suppose 4 holds. Suppose

T ∗(f) = 0. Then,√A ‖f‖ ≤ 0 and f = 0. Since f was arbitrary, T ∗ is injective.

Theorem 3.5 implies that R(T ∗) is closed.

4 ⇔ 5. This implication follows from the fact that A ‖f‖2 = 〈AIf, f〉 and

‖T ∗f‖2 = 〈TT ∗f, f〉 for every A > 0 and f ∈ H2.

3.2 Bessel Sequences

In this section, we characterize Bessel sequences. There is a 1-1 corre-

spondence between bounded linear operators T : `2 → H and Bessel sequences

gn ⊂ H through the binding gn = T (δn). We also characterize Bessel sequences

as the sequences for which T , T ∗, G and S are well defined.

Theorem 3.8. Let `2 denote the space of square summable sequences and H

denote a Hilbert space. If gn ⊂ H, then the following are equivalent.

1. T : `2 → H, T : sn 7→∞∑n=1

sngn is well-defined.

2. T ∗ : H → `2, T ∗ : f 7→ 〈f, gn〉 is well-defined.

3. G : `2 → `2, G : sn 7→ ∞∑n=1

sn〈gn, gm〉 is well-defined (G = T ∗T ).

4. S : H → H, S : f 7→∞∑n=1

〈f, gn〉gn is well-defined (S = TT ∗).

Furthermore, if these operators are well-defined, they are bounded linear oper-

ators satisfying ‖T‖2 = ‖T ∗‖2 = ‖G‖ = ‖S‖.

71

Proof. Before proving the theorem, let us make a few comments. In 1 and 4,

well-defined means that the partial sums for the summations given converge with

respect to the norm for the Hilbert space H. In 2, well-defined means that the

sequence of inner products given is square summable. In 3, well-defined means

that the partial sums for the summations given converge with respect to the inner

product field (typically R or C) for each n ∈ Z and the sequence of terms, given

by evaluating these summations, is square summable.

In order to show partial sums converge, we will show that they are Cauchy

sequences. Once we show an operator is well-defined, we can also assume the

operator is linear and bounded by the Banach Steinhaus Theorem (see [23]). That

is, each of these operators may be written as the pointwise limit of linear and

bounded operators. Therefore, the Banach Steinhaus Theorem implies that they

are also linear and bounded operators.

1⇔ 2. Suppose 1 holds. We will show that T ∗ is well-defined and is the formal

adjoint of T . Let sn ∈ `2, f ∈ H. Then, we have

〈T (sn), f〉 = 〈∞∑n=1

sngn, f〉

=∞∑n=1

sn〈gn, f〉

= 〈sn, 〈f, gn〉〉

= 〈sn, T ∗(f)〉

72

On the other hand suppose 2 holds. Let sn ∈ `2. We will show that partial

sums in 1 form a Cauchy sequence and thus converge in H. Let N2, N1 ∈ N and

assume N2 ≥ N1. Then,∥∥∥∥ N2∑n=1

sngn −N1∑n=1

sngn

∥∥∥∥2

=

∥∥∥∥ N2∑n=N1

sngn

∥∥∥∥2

= sup‖f‖=1

∣∣∣⟨ N2∑n=N1

sngn, f⟩∣∣∣2

= sup‖f‖=1

∣∣∣ N2∑n=N1

sn〈gn, f〉∣∣∣2

≤ sup‖f‖=1

( N2∑n=N1

|sn|2)‖T ∗f‖2

≤ ‖T ∗‖2( N2∑n=N1

|sn|2)

≤ ‖T ∗‖2( N2∑n=1

|sn|2 −N1∑n=1

|sn|2)

1⇔ 3. Suppose 1 holds. Then, T ∗ is well-defined and G = T ∗T is well-defined.

On the other hand, suppose 3 holds. Let sn ∈ `2. Again, we will show that

partial sums in 1 form a Cauchy sequence and thus converge in H. Let N2, N1 ∈ N

and assume N2 ≥ N1. Then we have,∥∥∥∥ N2∑n=1

sngn −N1∑n=1

sngn

∥∥∥∥2

=

∥∥∥∥ N2∑n=N1

sngn

∥∥∥∥2

=∣∣∣⟨ N2∑

n=N1

sngn,N2∑

n=N1

sngn

⟩∣∣∣=

∣∣∣ N2∑m=N1

smN2∑

n=N1

sn〈gn, gm〉∣∣∣

≤( N2∑m=N1

|sm|2)1/2( N2∑

m=N1

∣∣∣ N2∑m=N1

sn〈gn, gm〉∣∣∣2)1/2

≤ ||G||( N2∑n=N1

|sn|2)

= ‖G‖( N2∑n=1

|sn|2 −N1∑n=1

|sn|2)

73

2⇔ 4. Suppose 2 holds. Then, T is well-defined and S = TT ∗ is well-defined.

On the other hand, suppose 4 holds. Let f ∈ H. Then,⟨ ∞∑n=1

〈f, gn〉gn, f⟩

=∞∑n=1

〈f, gn〉〈gn, f〉

=∞∑n=1

|〈f, gn〉|2

Definition 3.9. If any of the equivalences in Theorem 3.8 hold, then we refer to

the sequence gn ⊂ H as a Bessel sequence.

With respect to Bessel sequences, T is the synthesis or pre-frame operator,

T ∗ is the analysis operator, G is the Gram matrix and S is the frame operator.

The Gram matrix G has entries Gmn = 〈gn, gm〉 which are related to the inner

products involved in the Gram-Schmidt process. In addition, the Gram matrix

G : `2 → `2 is a synthesis operator for the Bessel sequence T ∗gn (recall the 1-1

correspondence between bounded linear operators and Bessel sequences).

3.3 Complete Bessel Sequences

In this section, we characterize Bessel sequences which are complete. There

is a 1-1 correspondence between bounded linear operators T : `2 → H with dense

range and complete Bessel sequences gn ⊂ H through the binding gn = T (δn).

Definition 3.10. Suppose gn is a sequence in a Hilbert space H. Then, gn

is said to be complete if its linear span is dense in H.

74

Lemma 3.11. The following are equivalent for a Bessel sequence gn.

1. gn is complete.

2. T ∗ is injective.


gn is complete.

⇔ (spangn)⊥ = 0.

⇔ f ∈ H and f ∈ (spangn)⊥ together imply f = 0.

⇔ f ∈ H and 〈f, gn〉 = 0 for every n ∈ N together imply f = 0.

⇔ f ∈ H and T ∗(f) = 0 together imply f = 0.

⇔ T ∗ is injective.

3.4 ω-Linearly Independent Bessel Sequences

In this section, we characterize Bessel sequences which are ω-linearly indepen-

dent. There is a 1-1 correspondence between bounded linear injective operators

T : `2 → H and ω-linearly independent Bessel sequences gn ⊂ H through the

binding gn = T (δn).

Definition 3.12. Suppose gn is a sequence in a Hilbert space H. Then, gn is

said to be ω-linearly independent if cn = 0 is the only sequence in `2 for which

the partial sumsN∑n=1

cngn converge to 0 in H.

75

Lemma 3.13. The following are equivalent for a Bessel sequence gn.

1. gn is ω-linearly independent.

2. T is injective.


gn is ω-linearly independent.

⇔ cn ∈ `2 andN∑n=1

cngn converges to 0 together imply cn = 0.

⇔ cn ∈ `2 and T (cn) = 0 together imply cn = 0.

⇔ T is injective.

3.5 Frame Sequences

In this section, we characterize frame sequences. There is a 1-1 correspon-

dence between bounded linear operators T : `2 → H with closed range and frame

sequences gn ⊂ H through the binding gn = T (δn). We also characterize frame

sequences using T , T ∗, G and S.

Corollary 3.14. Suppose gn is a Bessel sequence. Let T , T ∗, G and S be the

operators defined in Theorem 3.8. Then, the following statements are equivalent.

1. The reduction of T is a bijective map.

2. The reduction of T ∗ is a bijective map.

76

3. The reduction of G is a bijective map.

4. The reduction of S is bijective map.

5. R(T ) is closed (R(T ) = N (T ∗)⊥).

6. R(T ∗) is closed (R(T ∗) = N (T )⊥).

7. R(G) is closed (R(G) = N (T )⊥).

8. R(S) is closed (R(S) = N (T ∗)⊥).

9. There exists A > 0 such that√A ‖s‖ ≤ ‖T (s)‖ for every s ∈ N (T )⊥.

10. There exists A > 0 such that√A ‖f‖ ≤ ‖T ∗f‖ for every f ∈ N (T ∗)⊥.

11. There exists A > 0 such that AI ≤ G on N (T )⊥.

12. There exists A > 0 such that AI ≤ S on N (T ∗)⊥.


1A

=∥∥∥(T )−1

∥∥∥2

=∥∥∥(T ∗)−1

∥∥∥2

=∥∥∥(G)−1

∥∥∥ =∥∥∥(S)−1

∥∥∥Proof. This Corollary follows from Theorem 3.5. Notice that G, S are equal to

the compositions T ∗T and T T ∗ respectively.

Definition 3.15. If any of the equivalences in Corollary 3.14 hold, then we refer

to the sequence gn ⊂ H as a frame sequence.

77

3.6 Frames

In this section, we characterize frames. There is a 1-1 correspondence between

bounded linear surjective operators T : `2 → H and frames gn ⊂ H through

the binding gn = T (δn). We also characterize frames using T , T ∗, G and S.



1. gn is a complete frame sequence.

2. T is a surjective map.

3. R(T ∗) is closed and T ∗ is an injective map.

4. S is a bijective map.

5. There exists A > 0 such that√A ‖f‖ ≤ ‖T ∗f‖ for every f ∈ H.

6. There exists A > 0 such that AI ≤ S on H.


1A

=∥∥∥(T )−1

∥∥∥2

=∥∥∥(T ∗)−1

∥∥∥2

=∥∥∥(G)−1

∥∥∥ = ‖S−1‖

Proof. This Corollary follows from Corollary 3.7 and Lemma 3.11. Notice that G

and S are equal to the compositions T ∗T and T T ∗ respectively.


to the sequence gn ⊂ H as a frame.

78

Notice that if, gn is a frame, then we have the following reconstruction

formulas. For every f ∈ H,

f = SS−1f =∞∑

n=−∞〈f, S−1gn〉gn (35)

f = S−1Sf =∞∑

n=−∞〈f, gn〉S−1gn (36)

Further if gn is a frame, the sequence (S)−1gn is equal to the sequence

S−1gn and is referred to simply as the canonical dual frame. Any sequence hn

which may replace S−1gn in Equations 35 and 36 is known as a dual frame.

Also, notice that gn is a frame if there exists a lower frame bound A > 0

and an upper frame bound B > 0 such that for every f ∈ H,

A ‖f‖2 ≤∞∑n=1

|〈f, gn〉|2 ≤ B ‖f‖2(37)

3.7 Riesz Sequences

In this section, we characterize Riesz Sequences. There is a 1-1 correspondence

between bounded linear injective operators T : `2 → H with closed range and

Riesz sequences gn ⊂ H through the binding gn = T (δn). We also characterize

Riesz sequences using T , T ∗, G and S.

Definition 3.18. Suppose gn, hn are sequences in a Hilbert space H. Then,

gn, hn are said to be biorthogonal if 〈gm, hn〉 = 〈δm, δn〉 for every m,n ∈ Z.

Definition 3.19. We say that gn is minimal if there exists a sequence hn

such that gn, hn are biorthogonal.

79



1. gn is an ω-linearly independent frame sequence.

2. R(T ) is closed and T is an injective map.

3. T ∗ is a surjective map.

4. G is a bijective map.

5. There exists A > 0 such that√A ‖cn‖ ≤ ‖T (cn)‖ for every cn ∈ `2.

6. There exists A > 0 such that AI ≤ G on `2.

7. gn is a minimal frame sequence.

8. There exists a Bessel sequence hn such that gn, hn are biorthogonal.

9. S is a bijective map and gn, (S)−1gn are biorthogonal.


1A

=∥∥∥(T )−1

∥∥∥2

=∥∥∥(T ∗)−1

∥∥∥2

= ‖G−1‖ =∥∥∥(S)−1

∥∥∥Proof. First of all, equivalences 1-6 follow from Theorem 3.7 and Lemma 3.13.

Notice that G and S are equal to the compositions T ∗T and T T ∗ respectively.

2 ⇒ 9. Suppose 2 holds. By Corollary 3.14, S is a bijective map. Now, let

m,n ∈ Z. Since T is injective, δn is the only sequence in `2 such that T (δn) = gn.

Thus, 〈gm, (S)−1gn〉 = 〈(T )−1gm, (T )−1gn〉 = 〈δm, δn〉.

80

9 ⇒ 8. By Theorem 3.8, S−1(gn) is a Bessel sequence if the sequence

N∑n=1

cnS−1(gn) is a Cauchy sequence. However, N2 ≥ N1 implies

∥∥∥∥ N2∑n=1

cnS−1(gn)−

N1∑n=1

cnS−1(gn)

∥∥∥∥ =

∥∥∥∥S−1( N2∑n=N1

cngn

)∥∥∥∥≤

∥∥∥S−1∥∥∥ ‖T‖( N2∑

n=1

|cn|2 −N1∑n=1

|cn|2)1/2

8⇒ 7. Suppose 8 holds. Let Tgn and T ∗hn represent the synthesis and analysis

operators for the Bessel sequences gn and hn, respectively. By Definition 3.19,

gn is minimal. We show gn is a frame sequence by showing that R(Tgn) is

closed (see Definition 3.15 and Corollary 3.14).

Let f ∈ R(Tgn). For each l ∈ N, choose fl ∈ R(Tgn) such that liml→∞

fl = f . Find

a matrix C whose rows are in `2 such that fl =∞∑m=1

Cmlgm for each l ∈ N. The

following argument uses the fact that gn, hn are biorthogonal along with the

fact that Tgn , T ∗hn are continuous to show that f ∈ R(Tgn). Since f was arbitrary,

this implies that the range of Tgn is closed.

f = liml→∞

fl

= liml→∞

∞∑m=1

Cmlgm

= liml→∞

∞∑n=1

〈∞∑m=1

Cmlgm, hn〉gn

= liml→∞

Tgn(T ∗hn(fl))

= Tgn( liml→∞

T ∗hn(fl))

= Tgn(T ∗hn( liml→∞

fl))

= Tgn(T ∗hn(f))

81

7 ⇒ 1 Suppose 7 holds. Let hn be a sequence such that gn, hn are

biorthogonal. Suppose cn ∈ `2 and the partial sumsN∑n=1

cngn converge to zero.

If we can show that cn = 0, then we will know gn is ω-linearly independent

since cn was arbitrary. However, by continuity, for each m ∈ N,

cm = limN→∞

N∑n=1

cn〈gn, hm〉

=⟨

limN→∞

N∑n=1

cngn, hm

⟩= 0


to the sequence gn ⊂ H as a Riesz sequence.

Notice that if gn is a Riesz sequence, G is the synthesis operator for the

complete Riesz sequence T ∗gn. Also, notice that gn is a Riesz sequence if

there exists a lower Riesz bound A > 0 and an upper Riesz bound B > 0 such

that for every cn ∈ `2,

A∞∑n=1

|cn|2 ≤∥∥∥∥ ∞∑n=1

cngn

∥∥∥∥2

≤ B∞∑n=1

|cn|2 (38)

3.8 Riesz Basis

In this section, we characterize Riesz Basis. There is a 1-1 correspondence

between bounded linear bijective operators T : `2 → H and Riesz bases gn ⊂ H

through the binding gn = T (δn). We also characterize using T , T ∗, G and S.

82

Theorem 3.22. Suppose gn is a Bessel sequence. Let T , T ∗, G and S be the


1. gn is an ω-linearly independent frame.

2. gn is a complete Riesz sequence.

3. T is a bijective map.

4. T ∗ is a bijective map.

5. S is a bijective map and gn, S−1gn are biorthogonal.

Furthermore, ‖T−1‖2= ‖(T ∗)−1‖2

= ‖G−1‖ = ‖S−1‖.

Proof. This Theorem follows from Corollary 3.16 and Corollary 3.20.


to the sequence gn ⊂ H as a Riesz basis.

If gn is a Bessel sequence, the frame and Riesz upper bounds are equal and

may be given as the norm of either T 2, (T ∗)2, G or S. On the other hand, if

gn is a Riesz basis, the Riesz and frame lower bounds are also equal and may

be given as the norm of either T−2, (T ∗)−2, G−1 or S−1.

We conclude this section with Table 10 which summarizes the characterizations

given through this chapter. Notice that in the last chapter, we used the term

complete Riesz sequence to describe a Riesz basis.

83

Table 10: Characterizing Bessel Sequences

gn is complete gn is a frame sequence gn is ω-linearly independent

R(T ) is dense R(T ) is closed T is injective

T ∗ is injective R(T ∗) is closed R(T ∗) is dense in `2

gn is a frame

T is surjective

S is bijective

gn is a Riesz sequence

T ∗ is surjective

G is bijective

gn is a Riesz basis

This table presents a visual tool for describing the redundant teminology in the

literature surrounding Bessel sequences. The table assumes that gn is a Bessel

sequence and T , T ∗, G and S are the synthesis operator, analysis operator, Gram

matrix and frame operator associated to gn, respectively. Items occupying the

same set of columns are equivalent. As an example, ”gn is complete” is equiva-

lent to ”R(T ) is dense in H.” As another example, ”gn is a Riesz sequence” is

equivalent to the combination of the following properties: ”R(T ∗) is closed” and

”gn is ω-linearly independent.”

84

3.9 Comparison

In this section, we describe additional pieces presented in this chapter. We

also explain how our approach to pseudo inverses differs from that in the litera-

ture. We point out certain examples of characterizations which involve redundant

terminology. We also give another possible characterization for the four frame

operators. Finally, we give some additional insight to certain duality principles

discussed in the literature.

3.9.1 Additions

We have made many small contributions which are not currently found among the

literature in the context of Riesz Bases. Regarding pseudo inverses, we have added

the following equivalences of Theorem 3.5: 2, 3, 4, 7, 8, 11 and 12. Regarding

surjective bounded linear operators, we have added the following equivalences of

Corollary 3.7: 3 and 5. Regarding Bessel sequences, we have added the following

implications of Theorem 3.8: 2⇒ 1, 3⇒ 1 and 4⇒ 1. Regarding complete Bessel

sequences, we have added Lemma 3.10 of Section 3.3. Regarding frame sequences,

we have added the given implication and following equivalences of Corollary 3.14:

3 ⇒ 5, 2, 4, 10, 11 and 12. Regarding frames, we have added the implication

6 ⇒ 5 of Corollary 3.16. Finally, regarding Riesz sequences, we have added the

following equivalences of Theorem 3.20: 1, 5, 7, 8 and 9.

85

3.9.2 Psuedo Inverse Definition

Let us examine the steps taken to defining a pseudo inverse in [9].

1. Assume the bounded linear operator T : H1 → H2 has closed range.

2. Define the reduction T : N (T )⊥ → H2, T : f 7→ T (f).

3. Write T−1 : N (T ∗)⊥ → N (T )⊥.

4. Extend T−1 to the the pseudo inverse T † : H2 → H1.

In step 3 of his approach, the notation T−1 is abused. That is, the domain

and range of T should correspond to the range and domain of T−1, respectively.

Therefore, it makes more sense to rigorously define the reduction T by restricting

both the domain and range of T . Second, the definition of T † is unnecessary.

Rather than using T †, we found it easier to use T−1, which is the inverse of the

reduction operator. One should compare the implication 5 ⇔ 9 of Theorem 3.5

with the proof of Lemma 5.5.4 in [9] which uses the composition TT †T .

3.9.3 Banach Spaces and Pseudo Inverses

In the beginning of Section 3.1, we claimed that the proofs of the section could

be extended to Banach Spaces. Before continuing, see Table 11, which lists some

differences between Banach and Hilbert space notation.

86

Table 11: Banach and Hilbert Space Notation

Banach Space Notation Hilbert Space Notation

Dual X∗1 , X∗2 H∗1 = H1, H∗2 = H2

T T : X1 → X2 T : H1 → H2

T ∗ T ∗ : X∗1 → X∗2 T ∗ : H2 → H1

H1 quotient space(X1/N (T )

)∗= N (T )⊥ H1/N (T ) = N (T )⊥

H∗2 quotient space(⊥N (T ∗)

)∗= X∗2/N (T ∗) N (T ∗)⊥ = H∗2/N (T ∗)

T T : X1/N (T )→ ⊥N (T ∗) T : N (T )⊥ → N (T ∗)⊥

T ∗ T ∗ : X∗2/N (T ∗)→ N (T )⊥ T ∗ : N (T ∗)⊥ → N (T )⊥

This table presents differences in notation for Banach and Hilbert spaces. Hilbert

spaces, by definition, are reflexive which means that the dual of a Hilbert space

H is isometrically isomorphic to H. Other differences involve the quotient space,

pre-annihilator and annihilator. First, assume X is a Banach space. Then, the

quotient space of a subspace A ⊂ X is the collection of equivalence classes on X,

where two vectors are equivalent if their difference lies in A. The pre-annihilator

of a subspace A∗ ⊂ X∗ is the collection of elements in X which are mapped to 0

by the operators in A∗. The annihilator of a subspace A ⊂ X is the collection of

elements in X∗ which map A to 0. Next, assume H is a Hilbert space. Then, the

quotient space, pre-annihilator and annihilator of a subspace A = A∗ ⊂ H are all

isometrically isomorphic.

87

3.9.4 Frame Sequences and Frames

In [10], a frame sequence is defined as a sequence gn ⊂ H which is a frame for

its closed linear span spangn. Parts of Corollary 3.14 are proved in [10] only

after frames are characterized. Based on Table 10, this seems counterintuitive.

In [10], frames are characterized as sequences whose synthesis operator T sat-

isfies the following three properties: T is well-defined, R(T ) is closed and T ∗ is

injective. A separate characterization is given stating that T is well-defined and

surjective. From Section 3.1, we know that these two characterizations are equiv-

alent based on functional analysis. Another characterization is given stating that

T is bounded on N (T )⊥, T satisfies 9 of Corollary 3.14 and gn is complete. In a

similar fashion, we could pick one equivalence each from Theorem 3.8, Corollary

3.14 and Lemma 3.11 to give almost 100 separate characterizations.

Another interesting example can be found in [2] where a sequence is claimed

to be a frame if and only if T ∗ is well-defined, T is surjective and T satisfies 9 of

Corollary 3.14. It is more natural to state T is well-defined and the statement is

redundant because T satisfies 9 of Corollary 3.14 if and only if R(T ) is closed.

3.9.5 Riesz Bases

In [10], Theorem 3.6.6 gives several characterizations of Riesz Bases. The first

states that Riesz Bases are complete Riesz sequences. The second states that

gn is complete and G is a bijective map. Finally, the third states that gn is a

88

complete Bessel sequence and there exists a complete Bessel sequence hn which

is biorthogonal to gn. This is redundant because hn is automatically complete

given the other assumptions. Really, Theorem 3.6.6 should be used to characterize

Riesz sequences since the assumption that gn is complete is repeated. Notice

that we could give over 300 characterizations of Riesz bases by choosing different

equivalences presented in this chapter.

As another example, Theorem 6.1.1 of [10] gives eight equivalences for a frame

gn to be considered a Riesz basis. A frame gn is a Riesz basis if gn is

ω-linearly independent. However, many of the statements given in Theorem 6.1.1

include redundant assumptions which are automatically satisfied by frames.

3.9.6 Operators associated with Frames

We adopt the terminology from [10] for T , T ∗, G and S even though it may

seem awkward that the frame operator exists for Bessel sequences which are not

frames. We also state an additional characterization for each of these operators.

For instance, any bounded linear operator T : `2 → H is the synthesis operator

for some Bessel sequence. Any bounded linear operator T ∗ : H → `2 is the

analysis operator for some Bessel sequence. Any positive bounded linear operator

G : `2 → `2 is the Gram matrix for some complete Bessel sequence. Finally, any

positive bounded linear operator S : H → H over a separable Hilbert space H

with closed range is the frame operator for some Bessel sequence.

89

3.9.7 Duality Principles

In [10], we find the usual definition for Riesz bases, which states that a sequence

gn is a Riesz basis if there exists an orthonormal basis wn and a bounded lin-

ear bijective operator U such that gn = U(wn). However, this statement implies

that U = TgnT∗wn where Tgn and Twn are the synthesis operators for the sequences

gn and wn, respectively. From this form it is clear that well-defined, injec-

tive, closed range and dense range are all properties inherited from Tgn . Similar

definitions could be made for all of the sequences characterized in this chapter.

In [7], R-duals are discussed. Given a Bessel sequence gn and orthonormal

bases vn and wn one defines the R-dual of gn to be the sequence defined

by hn =∞∑m=1

〈vn, gm〉wm for every n ∈ Z. It is then proven explicitly that gn is

complete if and only if hn is ω-linearly independent and vice versa. However,

this definition implies that Thn = TwnT∗gnTvn . From this form it is clear that well-

defined, injective, closed range and dense range are all properties inherited from

the properties of Tgn . Unlike the case above, two properties are switched. That

is, if Tgn is injective, Thn will have dense range and vice versa.

3.10 Wavelet Based Noise Cancellation Algorithm

We now discuss an algorithm which was developed through a joint collabo-

ration between New Mexico State University (NMSU) and Los Alamos National

Laboratories (LANL). The algorithm uses a wavelet frame to denoise lightning

90

signals. We will introduce both wavelets and wavelet matrices (see [12] and [25])

describing their role in our denoising filter. Then, we will demonstrate the algo-

rithm using an actual lightning signal observation. This observation is coherent

taken at very high frequency (VHF) from the FORTE satellite using the Los

Alamos Portable Pulser (LAPP). While more rigorous approaches may be used to

quantify the reduction of the signal to noise ratio (SNR), they are not presented

here since they tend to involve modelling the lightning signal (see [15]).

Let hn and gn be any two sequences in `2 such that gn = (−1)nh1−n for

all n ∈ Z. We will refer to hn as the scaling filter and gn as the wavelet filter.

Definition 3.24. The approximation operator H : `2 → `2 is defined by the

following equation for all sn ∈ `2 and m ∈ Z.

(Hs)m =∑n∈Z

snhn−2m (39)

Definition 3.25. The detail operator G : `2 → `2 is defined by the following

equation for all sn ∈ `2 and m ∈ Z.

(Gs)m =∑n∈Z

sngn−2m (40)

The approximation operator H and detail operator G may be used to create

an analysis filter bank as shown in Figure 44. Since both operators are invertible,

there is also a synthesis filter bank (see [25] for more details). These filter banks

correspond to the analysis and synthesis operators discussed earlier.

91

Figure 44: 3 Stage Wavelet Transform Analysis Filter Bank

Let N ∈ N. Since H maps 2N -periodic sequences to 2N−1 periodic sequences,

H may be represented by a 2N−1 by 2N matrix HN . The entries of HN must be∑l∈Zhn+l2N−2m for m = 1, ..., 2N−1 and n = 1, ..., 2N . This is due to the fact that for

a 2N -periodic sequence sn and m = 1, ..., 2N−1, we have

(Hs)m =∑n∈Z

snhn−2m

=2N∑n=1

∑l∈Zsn+lNhn+l2N−2m

=2N∑n=1

sn∑l∈Zhn+l2N−2m

Similarly, G may be represented by a 2N−1 by 2N matrix GN where the en-

tries of GN are∑l∈Zgn+l2N−2m for m = 1, ..., 2N−1 and n = 1, ..., 2N . The vertical

concatenation of the matrices HN and GN creates a 2N by 2N matrix WN which

defines the Discrete Wavelet Transform. Just as the individual blocks of Figure

44 have matrix representations, the entire analysis filter bank of Figure 44 may

be represented by the following Wavelet matrix W .

92

W =

WN−2 0 0 0

0 I 0 00 0 I 00 0 0 I

[ WN−1 00 I

]WN

The Wavelet matrix W is invertible and its columns are referred to as wavelets.

Furthermore, W may be used to create an analysis operator defined for any se-

quence sn ∈ `2 using the following technique. First, express sn as a linear

combination of 2N new sequences. Then, break up each new sequence into con-

secutive blocks of length 2N , making sure blocks from separate sequences do not

start at the same index. For instance, the ith new sequence may contain blocks

which start at multiples of 2N but are offset by i (..., −2N + i, i, 2N + i, ...).

Finally, use W to transform each block into a set of wavelet coefficients.

The analysis operator described here is linear and bounded. Since this process

is invertible, the corresponding synthesis operator is surjective. Since sn may

be expressed using different linear combinations, the coefficients are not unique,

implying that the synthesis operator is not injective. From the results in this

section, we conclude that the analysis operator described here is based on a wavelet

frame. Each wavelet in the frame comes from zero padding and shifting the

columns of W .

An analysis filter bank makes use of the analysis operator to obtain a set of

coefficients. Similarly, a synthesis filter bank may reconstruct the signal from

these coefficients via the synthesis operator. Observe the following equalities.

93

sj = 12N

2N∑l=1

sj

= 12N

2N∑l=1

0∑n=1−l

sj−n2N∑m=1

W−1l,mWm,l+n + 1

2N

2N−1∑l=1

sj−n2N−l∑n=1

2N∑m=1

W−1l,mWm,l+n

= 12N

0∑n=1−2N

2N∑l=1−n

2N∑m=1

W−1l,mWm,l+nsj−n + 1

2N

2N−1∑n=1

2N−n∑l=1

2N∑m=1

W−1l,mWm,l+nsj−n

=2N−1∑

n=1−2Nrnsj−n

= (s ∗ r)j

where rn =

1

2N

2N∑m=1

2N∑l=1−n

W−1l,mWm,l+n if 1− 2N ≤ n ≤ 0

12N

2N∑m=1

2N−n∑l=1

W−1l,mWm,l+n if 1 ≤ n ≤ 2N − 1

0 otherwise

These equalities represent using the analysis operator to decompose s into

wavelet coefficients and using the synthesis operator to reconstruct s from wavelet

coefficients. These wavelet coefficients may be thought of as frequency coefficients

since the filters h and g may be thought of as lowpass and highpass filters, respec-

tively. However, instead of using the filter r given above, we will replace r with

the filter given below which accentuates wavelet bands where noise is not present

and suppresses wavelet bands where noise is present.

where rn =

12N

2N∑m=1

1

Meanj

2N∑l=1

Wm,lsl+j

2N∑l=1−n

W−1l,mWm,l+n if 1− 2N ≤ n ≤ 0

12N

2N∑m=1

1

Meanj

2N∑l=1

Wm,lsl+j

2N−n∑l=1

W−1l,mWm,l+n if 1 ≤ n ≤ 2N − 1

0 otherwise

94

In Figures 46-49, we demonstrate the effect of our denoising algorithm on a

lightning observation from the FORTE satellite. We use the Daubechies wavelet

filter with 10 vanishing moments (see [11]) and a 9 stage wavelet packet analysis

filter bank (see Figure 45). The numbers in Figure 45 order the wavelet coefficients

from lowest to highest frequency based on a concept called spectral flipping. The

wavelet matrix W associated with Figure 45 is given below.

Figure 45: 3 Stage Wavelet Packet Analysis Filter Bank

W =

WN−2 0 0 0

0 WN−2 0 00 0 WN−2 00 0 0 WN−2

[ WN−1 00 WN−1

]WN

The Matlab code to process our denoising algorithm is given in the Appendix.

One might try using different attenuation coefficients and make use of our algo-

rithm for calculating the spectral flipping indices is given in the appendix.

95

Figure 46: LAPP Lightning Pulse Before Denoising

Figure 47: LAPP Lightning Pulse After Denoising

The lightning occured between samples 5000 and 7000. This occurrence is more

apparent after using the denoising algorithm.

96

Figure 48: Wavelet Packet Matrix

This image matrix depicts the wavelet packet matrix of the denoising algorithm.

Figure 49: Wavelet Packet Coefficients

Each column of this image matrix depicts the wavelet coefficients for a block of

the denoised lightning signal.

97

4 SAMPLING

In this section, we generalize the Shannon Sampling equations for band-limited

functions. We first prove a Poisson Summation formula involving compactly sup-

ported functions in L2(R). We then develop the Shannon sampling formula for

functions in L2(R) based on the DTFT and give two generalizations involving

tiling sets and a general union of intervals. Finally, we develop two periodic

nonuniform sampling formulas based on frames. The first bridges the gap be-

tween the original Shannon formula and periodic nonuniform sampling formulas

found in recent literature. The second gives a slight generalization of the periodic

nonuniform sampling formula given in [24].

For each generalization presented in this section, we compare the sampling

rate to the Landau rate (see [18]), which represents the smallest possible average

sampling rate needed to perfectly reconstruct f from its samples. We also give an

example with respect to a specific set of spectral support.

4.1 Poisson Summation Formula for L2

In this section, we give another form of the Poisson Summation Formula

which will be used in the proofs throughout the chapter. Our approach involves

using the operator notation previously developed as opposed to writing out in-

tegral formulas. This methodology is just a matter of preference, but we prefer

98

this methodology since we may use the commutation and adjoint properties of the

operators involved. In addition, using operator notation gives a short hand nota-

tion for the ideas we are trying to express. We also use the fact that a function

f ∈ L2(R) whose Fourier Transform f is in L1(R) may be evaluated pointwise

using the following operator notation.

f(n) =

∫ ∞−∞

f(ω)e−2πiωtdω = 〈f ,Mn1R〉 (41)

Theorem 4.1. Suppose f ∈ L2(R) has compact support. Let Ω ∈ R. Then, for

any compact interval I, the following equality holds in L2(R).

QI

∞∑n=−∞

TnΩf = 1Ω

∞∑n=−∞

f( nΩ

)M nΩ1I (42)

Proof. Since both∞∑

n=−∞TnΩf and 1

Ω

∞∑n=−∞

f( nΩ

)M nΩ1R are periodic with period Ω, it

is enough to show the above equality holds for I = ΩΣ0. First of all, the function

QΩΣ0

∞∑n=−∞

TnΩf is in L2(ΩΣ0) since only finitely many terms in the summation

are non-zero. Further, the functions D 1ΩMn1Σ0 form an orthonormal basis for

L2(ΩΣ0). Thus, using the properties from Table 1, we have

QΩΣ0

∞∑n=−∞

TΩnf =∞∑

n=−∞〈QΩΣ0

∞∑k=−∞

TΩkf , D 1ΩMn1Σ0〉D 1

ΩMn1Σ0

=∞∑

n=−∞〈DΩf ,Mn1R〉D 1

ΩMn1Σ0

=∞∑

n=−∞D 1

Ωf(n)D 1

ΩMn1Σ0

= 1Ω

∞∑n=−∞

f( nΩ

)M nΩ1ΩΣ0

99

4.2 Basic Shannon Sampling

We now discuss a reconstruction formula related to that given in Section 2.7.

The main difference is that here the equality holds in L2(R) rather than S ′(R).

Another difference is that we are using the Ω-partition of unity 1ΩΣ0 .

Theorem 4.2. Let Σ0 = [−12, 1

2). Let Ω ∈ R. If f is band-limited to Σ = ΩΣ0,

then f satisfies the following equations in L2(R).

f = 1Ω

∞∑n=−∞

f( nΩ

)T nΩR1Σ (43)

f = 1Ω

∞∑n=−∞

f( nΩ

)M nΩ1Σ (44)

Proof. Using Theorem 4.1, we have the following.

f = QΩΣ0 f

= QΩΣ0

∞∑n=−∞

TΩnf

= 1Ω

∞∑n=−∞

f( nΩ

)M nΩ1ΩΣ0

Theorem 4.2 is based on the fact that the functions e−2πinω

Ω 1Σ form an orthog-

onal basis for L2(Σ). It is important to note that if a function is bandlimited

to an interval, Theorem 4.2 may be applied to some modulation of the function.

Equation 43 gives a formula for reconstructing a bandlimited signal after sampling

at the Landau rate Ω, which is equal to the Nyquist rate (supω∈Σ

ω − infω∈Σ

ω) for an

interval. The following theorem extends this result to tiling sets (see [3]).

100

Example 4.3. Suppose f is band-limited to the interval [−5, 5) as indicated by

Figure 50. Then, substituting Ω = 10 and Σ = [−5, 5) into Equation 43 yields

the following reconstruction formula.

f(t) =∞∑

n=−∞f( n

10)sinc(n− 10t)

Figure 50: Support for f from Example 4.3

4.3 Shannon Sampling with a Tiling Set

In many applications, such as Frequency Division Multiple Access (FDMA),

a function actually has frequency support on a union of M intervals. Since there

is an interval containing the M intervals, Theorem 4.2 may be applied to such

functions. However, the sampling rate typically will be much lower than the

Landau rate. For example, if the measure of the M intervals is only 2 percent of

the measure of the smallest interval containing these M intervals (convex hull),

then the Nyquist rate will be 50 times slower than the Landau rate. We now

generalize Theorem 4.2 to a special multiple interval case based on the comments

in Section 2.6 regarding tiling sets (see [3]).

101

Theorem 4.4. Suppose Σ is an Ω-tiling set which is the union of M disjoint

intervals of the form Im = Ω(Om + βm), where the union of the M intervals Om

is the unit interval Σ0 = [−12, 1

2) and βm ∈ Z for m = 0, ...,M − 1. If f is

band-limited to Σ, then f satisfies the following equations in L2(R).

f = 1Ω

∞∑n=−∞

f( nΩ

)M−1∑m=0

T nΩR1Im (45)

f = 1Ω

∞∑n=−∞

f( nΩ

)M−1∑m=0

M nΩ1Im (46)

Proof. Using the properties from Table 1 and Theorem 4.1, we have

f =M−1∑m=0

QΩ(Om+βm)f

=M−1∑m=0

TΩβmQΩOmT−Ωβm f

=M−1∑m=0

TΩβmQΩOm

∞∑n=−∞

TΩnf

= 1Ω

M−1∑m=0

TΩβm

∞∑n=−∞

f( nΩ

)M nΩ1ΩOm

= 1Ω

∞∑n=−∞

f( nΩ

)M−1∑m=0

M nΩ1Ω(Om+βm)

Theorem 4.4 is based on the fact that the functionsM−1∑m=0

e−2πinω

Ω 1Ω(Om+βm) form

an orthogonal basis for L2(Σ). Equation 45 describes reconstruction of a band-

limited function after sampling at the Landau rate Ω.

For an interval I, the formula for 1I will be useful when using Theorem 4.4.

Let |I| and I denote the length and midpoint of the interval I. Then,

1I(t) = |I|e−2πiItsinc(|I|t) (47)

102

Example 4.5. Suppose f is band-limited to the intervals shown in Figure 51.

Then, f has support in a 10-tiling set. Using Ω = 10, M = 2, O0 = [−5, 2.5),

β0 = −2, O1 = [2.5, 5) and β1 = 2, we may apply Theorem 4.4. Therefore, we

may reconstruct f after sampling at the Landau rate of 10 which is 1/5 the rate

required if we used Equation 43. One possible reconstruction formula is given by

the following equation.

f(t) = 110

∞∑n=−∞

f( n10

)M−1∑m=0

1Im( n10− t)


In Theorem 4.4, we may substitute f for T− γΩf given any γ ∈ R. The result

would be the effect of time limiting, modulating, translating and dilating an or-

thonormal basis over the unit interval. Making this substitution into Equation 46

yields the following equation.

f = 1Ω

∞∑n=−∞

f(γ+nΩ

)M−1∑m=0

M γ+nΩ1Im

= 1√Ω

∞∑n=−∞

f(γ+nΩ

)M−1∑m=0

e−2πiβmγD 1ΩTβmMγ+n1Om

103

4.4 General Shannon Sampling Formulas

In many situations, it may be difficult to find the appropriate parameters

necessary to use Theorem 4.4. In order to explore and discuss other possible

generalizations of Theorem 4.2, we will now present three different reconstruction

formulas for the Fourier Transform of a function f which is band-limited to M

disjoint intervals. These formulas are difficult to implement in practice, so we

leave the reconstruction formulas in terms of f . We will discuss advantages and

disadvantages after the theorem is proven.

Theorem 4.6. Suppose Σ has measure Ω and is the union of M disjoint intervals

of the form Im = Ω(Om + βm), where the union of the M intervals Om is the unit

interval Σ0 = [−12, 1

2) and βm ∈ R for m = 0, ...,M − 1. If f is band-limited to Σ,

then f satisfies the following equations in L2(R).

f = 1Ω

∞∑n=−∞

M−1∑m=0

(PImf)( nΩ

)M nΩ1Im (48)

f =∞∑

n=−∞

M−1∑m=0

1|Om|Ω(PImf)( n

|Om|Ω)M n|Om|Ω

1Im (49)

f = 1Ω

∞∑n=−∞

M−1∑m=0

e−2πiβmn(PImf)( nΩ

)M−1∑l=0

e2πiβlnM nΩ1Il

(50)

Proof. We outline the proof. For m = 0, ...,M − 1, the functions MΩβmPImf are

each band-limited to ΩΣ0. Thus, we may substitute f for MΩβmPImf in Equation

44 to obtain Equation 48.

104

Similarly, for m = 0, ...,M − 1, the functions MΩ(Om+βm)PImf are each band-

limited to |Om|ΩΣ0. Thus, we may substitute f for MΩ(Om+βm)PImf and Ω for

|Om|Ω in Equation 44 to obtain Equation 49.

Finally,M−1∑m=0

MΩβmPImf is band-limited to ΩΣ0. Thus, we may substitute f for

M−1∑m=0

MΩβmPImf in Equation 44 to obtain Equation 50.

Equation 48 represents reconstructing each piece PImf separately. The formula

requires M channels each with sampling rate Ω so the total sampling rate is M

times the Landau rate. One advantage of using this formula is that each channel

is sampled at the same points. On the other hand, using this formula requires the

complexity of calculating PImf .

Equation 49 also represents reconstructing each piece PImf separately. The

formula requires M channels each with different sampling rate and the total sam-

pling rate is actually equal to the Landau rate. However, this formula requires

sampling each channel at different points and the complexity of calculating PIm .

Equation 50 explores the concept of using modulation to manipulate the fre-

quency supports of f so that they form a tiling set. The formula requires M

channels each with sampling rate Ω. Unlike the cases above, the interpolation

functions are independent of the channel so the total sampling rate is equal to

the Landau rate. However, just like the cases above, this formula requires the

complexity of calculating PIm .

105

4.5 Periodic Nonuniform Sampling with Only Complete Aliasing

In this section, we introduce periodic nonuniform sampling, which involves

sampling over a periodic set. We view periodic nonuniform sampling as subsam-

pling after sampling at rate ΩK. Therefore, we must assume that no aliasing

occurs with respect to the sampling rate ΩK. On the other hand, aliasing may or

may not occur with respect to the sampling rate Ω.

In Section 4.3, we described how to reconstruct a function from its samples

when the M spectral slices Ω(Om + βm) formed an Ω-tiling set. In that case, no

aliasing occurs with respect to the sampling rate Ω. That is, given any two spectral

slices Im1 , Im2 , there does not exist an n ∈ N such that Im1 + Ωn intersects Im2 .

Our first periodic nonuniform sampling theorem describes the complementary case

in which complete aliasing occurs with respect to the sampling rate Ω. That is,

given any two spectral slices Ij1 , Ij2 there exists an n ∈ N such that Ij1 +Ωn = Ij2 .

See [24] for details on how to generate spectral slices from an arbitrary set.

For the first periodic nonuniform sampling theorem, we will assume that f

is band-limited to an ΩK-tiling set which is the union of J aliases with respect

to the sampling rate Ω. That is, when using the sampling rate Ω, the aliases are

indistinguishable, combining in accordance with the Poisson Summation Formula.

This is comparable to having one equation and J unknowns. In order to distin-

guish between aliases, J different sampling formulas are used. This is comparable

106

to having J equations and J unknowns. Likewise, only certain choices for the J

sampling formulas may be used to distinguish between aliases. We use J sam-

pling formulas based on the DFT. That is, we first choose a J by J matrix W by

extracting a submatrix from the K by K DFT matrix F where F is defined by

Fmn = e−2πimnK for m = 0, ..., K − 1 and n = 0, ..., K − 1. The J rows chosen will

correspond to the first J rows of F while the J columns chosen will depend on the

spacing of the spectral slices. Next, we create each sampling formula so that the

jth sampling formula involves multiplying the aliases by coefficients from the jth

row of W . We will be able to distinguish between aliases using the J sampling

formulas due to the fact that W is invertible (see [24]).

Theorem 4.7. Suppose Σ is an ΩK-tiling set which is the union of J intervals

of the form Ij = Ω(Σ0 + βj), where βj ∈ Z for j = 0, ..., J − 1. Let W represent

the J by J matrix whose entry in the jth row and kth column is [W ]jk = e−2πijβkK .

If f is band-limited to Σ, then f satisfies the following equations in L2(R).

f = 1Ω

∞∑n=−∞

J−1∑j=0

f( j+nKΩK

)J−1∑k=0

[W−1]kj[W ]jkT j+nKΩK

R1Ik (51)

f = 1Ω

∞∑n=−∞

J−1∑j=0

f( j+nKΩK

)J−1∑k=0

[W−1]kj[W ]jkM j+nKΩK

1Ik (52)

Proof. Notice the assumption that Σ is an ΩK-tiling set implies that the βj rep-

resent J of the K integers contained in some K-tiling set. This assumption is

similar to the tiling set assumptions used in Sections 2.8 - 2.10 and is sufficient

for W to be an invertible matrix (see [24]).

107

Before presenting the computations for the proof, let us review some basic

equalities using the operator notation. First of all, TΩβkT−Ωβl is the identity

operator if k = l. In addition, since W is an invertible matrix, we have

J−1∑j=0

[W−1]kj[W ]jl =

1 if k = l

0 otherwise

(53)

Now, assume n ∈ N and j, k ∈ 0, ..., J − 1. From the properties in Table 1,

we have the following equations.

[W ]jkM j+nKΩK

TΩβk = e2πijβkK M j+nK

ΩKTΩβk

= TΩβkM j+nKΩK

(54)

[W ]jkT−ΩβkF = M jΩKT−ΩβkM− j

ΩKF

= M jΩKT−ΩβkFT− j

ΩK

(55)

We now use the assumptions given in the theorem and assume f is band-

limited to Σ. Since Σ is an ΩK-tiling set, QΩΣ0+Ωnf = 0 if n 6∈ β0, ..., βJ−1.

Since T− jΩKf is also band-limited to Σ, we have the following equation.

QΩΣ0M jΩK

J−1∑k=0

T−Ωβk T− jΩKf = M j

ΩK

J−1∑k=0

T−ΩβkQΩΣ0+Ωβk T− jΩKf

= M jΩK

∞∑n=−∞

T−ΩβnQΩΣ0+ΩnT− jΩKf

= QΩΣ0M jΩK

∞∑n=−∞

T−ΩβnT− jΩKf

(56)

108

Note that Equations 53-56 will be applied in a different order than the order

in which they were presented. Applying these equations and using Theorem 4.1

yields the following equalities.

f =J−1∑k=0

QΩ(Σ0+βk)f

=J−1∑k=0

J−1∑l=0

(J−1∑j=0

[W−1]kj[W ]jl)TΩβkT−ΩβlQΩ(Σ0+βl)f

=J−1∑k=0

J−1∑j=0

[W−1]kjTΩβkQΩΣ0

J−1∑l=0

[W ]jlT−Ωβl f

=J−1∑k=0

J−1∑j=0

[W−1]kjTΩβkQΩΣ0M jΩK

J−1∑l=0

T−ΩβlT− jΩKf

=J−1∑k=0

J−1∑j=0


∞∑n=−∞

T−ΩnT− jΩKf

= 1Ω

J−1∑k=0

J−1∑j=0


∞∑n=−∞

T− jΩKf( n

Ω)M n

Ω1R

= 1Ω

∞∑n=−∞

J−1∑j=0

T− jΩKf( n

Ω)J−1∑k=0

[W−1]kjTΩβkM j+nKΩK

1ΩΣ0

= 1Ω

∞∑n=−∞

J−1∑j=0

T− jΩKf( n

Ω)J−1∑k=0

[W−1]kj[W ]jkM j+nKΩK

1Ω(Σ0+βk)

Theorem 4.7 is based on the fact that the functions e−2πi(j+nK)ω

ΩK 1Σ form a frame

for L2(Σ) where the functions are indexed by n ∈ Z and j = 0, ..., J − 1. The

functionsJ−1∑k=0

[W−1]kj[W ]jke− 2πi(j+nK)ω

ΩK 1Ik form a dual frame and are also indexed

by n ∈ Z and j = 0, ..., J − 1. Equation 51 gives a formula to reconstruct a band-

limited function after uniform sampling at the Landau rate ΩJ . This should be

compared to Equation 45 which gives a formula to reconstruct a band-limited

function after sampling at the rate ΩK.

109

Example 4.8. Suppose f is band-limited to the intervals shown in Figure 52.

Then, f has support in a 40-tiling set which contains 3 complete aliases with

respect to the sampling rate 10. Using Ω = 10, K = 4, J = 3, β0 = −4, β1 = −2

and β2 = 3, we may apply Theorem 4.7 to obtain following reconstruction formula.

f(t) = 110

∞∑n=−∞

2∑j=0

f( j+4n40

)2∑

k=0

[W−1]kj[W ]jk1Ik(j+4n

40− t)


4.6 Periodic Nonuniform Sampling

We now give a generalization of Theorems 4.4 and 4.7 which involves recon-

struction of a function f from its samples when f is band-limited to an ΩK-tiling

set Σ whose spectral slices Ijm form Ω-tiling sets in one sense and completely alias

in another sense. To be more precise, we assume the spectral slices Ijm satisfy the

following properties (see Table 12).

1. Given j,m1 6= m2, the intersection (Ijm1 +Ωn)∩ Ijm2 is empty for all n ∈ N.

2. Given j1 6= j2,m, there exists an n ∈ N such that (Ij1m + Ωn) = Ij2m.

110

Theorem 4.9. Suppose Σ is an ΩK-tiling set which is the union of J different Ω-

tiling sets Σj. Further suppose that each Ω-tiling set Σj is the union of M disjoint

intervals of the form Ijm = Ω(Om + βjm), where the union of the M intervals

Om is the unit interval Σ0 and βjm ∈ Z for j = 0, ..., J − 1, m = 0, ...,M − 1.

Let Wm represent the J by J matrix whose entry in the jth row and kth column

is [Wm]jk = e−2πijβkm

K . If f is band-limited to Σ, then f satisfies the following

equations in L2(R).

f = 1Ω

∞∑n=−∞

J−1∑j=0

f( j+nKΩK

)M−1∑m=0

J−1∑k=0

[W−1m ]kj[Wm]jkT j+nK

ΩKR1Ikm (57)

f = 1Ω

∞∑n=−∞

J−1∑j=0

f( j+nKΩK

)M−1∑m=0

J−1∑k=0

[W−1m ]kj[Wm]jkM j+nK

ΩK1Ikm (58)

Proof. Using the properties from Table 1 and Theorem 2.15, we have

f =M−1∑m=0

J−1∑k=0

QΩ(Om+βmk)f

=M−1∑m=0

J−1∑k=0

J−1∑l=0

(J−1∑j=0

[W−1m ]kj[Wm]jl)TΩβkmT−ΩβlmQΩ(Om+βlm)f

=M−1∑m=0

J−1∑k=0

J−1∑j=0

[W−1m ]kjTΩβkmQΩOm

J−1∑l=0

[Wm]jlT−Ωβlm f

=M−1∑m=0

J−1∑k=0

J−1∑j=0

[W−1m ]kjTΩβkmQΩOmM j

ΩK

J−1∑l=0

T−ΩβlmT− jΩKf

=M−1∑m=0

J−1∑k=0

J−1∑j=0


ΩK

∞∑n=−∞

T−ΩβnT− jΩKf

= 1Ω

M−1∑m=0

J−1∑k=0

J−1∑j=0


ΩK

∞∑n=−∞

T− jΩKf( n

Ω)M n

Ω1R

= 1Ω

∞∑n=−∞

J−1∑j=0

T− jΩKf( n

Ω)M−1∑m=0

J−1∑k=0

[W−1m ]kjTΩβkmM j+nK

ΩK1ΩOm

= 1Ω

∞∑n=−∞

J−1∑j=0

T− jΩKf( n

Ω)M−1∑m=0

J−1∑k=0

[W−1m ]kj[Wm]jkM j+nK

ΩK1Ω(Om+βkm)

111

Theorem 4.9 is based on the fact that the functions e−2πi(j+nK)ω

ΩK 1Σ form a frame

for L2(Σ). The functionsM∑m=1

J−1∑k=0

[W−1m ]kj[Wm]jke

− 2πi(j+nK)ωΩK 1Imk form a dual frame.

Equation 57 describes reconstruction of a band-limited function after uniform

sampling at the Landau rate ΩJ .

For comparison, we will rewrite Equations 17 and 18 from [24] using i for

the imaginary number, starting index sets at 0, choosing C = 0, ..., p − 1 and

making the following definitions: Ω = 1TL

, J = p and K = L. With these changes

Equations 17 and 18 of [24] may be written as follows.

f(t) =∞∑

n=−∞

J−1∑j=0

f( j+nKΩK

)φj(t− j+nKΩK

)

φj(ω) = 1Ω

M−1∑m=0

J−1∑l=0

[A−1m ]lj[Am]jl1Gm+Ωkm(l)

Our presentation has several differences from that of [24]. First of all, we give

contraints for spectral slices in terms of tiling sets so that the periodic nonuniform

sampling formula is actually a generalization of the Shannon sampling formula for

tiling sets. Second, the presentation is more rigorous in the sense that it is based

on the distribution theory presented in Chapter 2, giving results in L2(R). In

terms of notation, three other adjustments are made. We use Wm = Am to use

the DFT matrix instead of its inverse. The spectral slices Gm in [24] partition

[0,Ω] while the spectral slices ΩOm here partition [−Ω2, Ω

2) to facilitate comparisons

with the original Shannon formula. Finally, the shifts km(l) are written as βml.

112

Example 4.10. If f is band-limited to the intervals of Figure 53, then we may

use the parameters of Table 12 to show that f satisfies the reconstruction formula

of Table 12. The function f has support in a 40-tiling set which contains 3 10-

tiling sets, each consisting of two intervals (instead of one as in Figure 52). Each

interval from a 10-tiling set is a complete alias of exactly one interval from each

of the remaining 10-tiling sets. Here, Ω = 10, K = 4, J = 3 and M = 2.


Table 12: Uniform Sampling Example

Ω = 10, K = 4, O0 = [−.5, .25), O1 = [.25, .5)

f(t) = 110

∞∑n=−∞

J−1∑j=0

f( j+4n40

)M−1∑m=0

J−1∑k=0

[W−1m ]kj[Wm]jk1Imk(

j+4n40− t)

m = 0 m = 1

j = 0 I00 = Ω(O0 − 4) = [−45, 2− 7.5) I01 = Ω(O1 − 3)I01 = [−27.5,−25)

j = 1 I10 = Ω(O0 + 2) = [15, 22.5) I11 = Ω(O1 − 1) = [−7.5,−5)

j = 2 I20 = Ω(O0 + 3) = [25, 32.5) I21 = Ω(O1 + 2) = [22.5, 25)

113

4.7 More on ΩK-Tiling Sets

In this section, we discuss how to break up an ΩK-tiling set into intervals of

the form Ω(Om + βkm). This process is referred to as spectral slicing.

Theorem 4.11. Suppose Σ is the union of M disjoint intervals, each having finite

measure. Then, there exists Ω, K such that Ω is contained in an ΩK-tiling set Σ.

Furthermore, there exists a partition of [−Ω2, Ω

2), denoted by O0, ..., OM−1 where

M ≤ 2M + 1 such that the following two properties are satisfied.

1. Σ =K−1∪k=0

M−1∪m=0

Ω(Om + βkm) for some βkm ∈ N

2. Σ =J−1∪j=0

Mj−1

∪m=0

Ω(Om +βjm) for some J ≤ K and 0 ≤MJ−1 ≤ ... ≤M0 ≤ M .

We omit the proof of this theorem, since this process is described in detail in

[24]. However, let us summarize the process. First of all, one possible choice for

Σ is the convex hull of Σ. Once the ΩK-tiling set Σ is chosen, Σ may be broken

up into intervals of the form Ikm = Ω(Om + βkm) where the union over a subset

of these intervals forms Σ.

If we place the intervals Ikm into a K by M matrix, as we did in Table 12, we

will obtain a matrix satisfying several properties. First of all, the union over all

intervals is the ΩK-tiling set Σ. Second, the union over all intervals in each row

is an Ω-tiling set. Finally, the entries within a given column are all aliases of each

other with respect to the sampling rate Ω. In fact, we may permute the columns

114

of this matrix or the entries within a column without changing these properties.

Therefore, we may make assumptions about the locations of the intervals whose

union is Σ (ie 0 ≤MJ−1 ≤ ... ≤M0 ≤ M).

Consider the spectral slicing representation shown in Table 13. Notice, for

example, each interval in the third column is an alias of [.1, .25) under the sampling

rate Ω = .5. Since Σ is contained in an ΩK-tiling set, Theorem 4.2 states that any

function which is band-limited to Σ may be reconstructed after sampling at the

rate ΩK which is the sum of the measures of all the intervals in Table 13. Theorem

4.9 reduces this rate to ΩJ which is the sum of the measures of all intervals in the

first J rows (each of these rows has an interval which is contained in Σ). Finally,

the Landau rate is |Σ|, which is the sum of the measures of all intervals which are

contained in Σ.

In Table 13, the double horizontal line separates the first J rows from the

remaining and the ∗ marks the intervals contained in Σ. We have ΩK = 2.5,

ΩJ = 2 and the Landau rate is 1.2. Notice that the efficiency gain between

Shannon sampling and Uniform sampling is ΩK − ΩJ , represented by the rows

which do not have an interval contained in Σ. There still may be an efficiency

gain between Uniform sampling and some other type of sampling. The maximum

efficiency gain possible is ΩJ − |Σ| which is represented by the intervals in the

first J rows which are not contained in Σ.

115

Table 13: Spectral Splicing

Σ = [−2.75,−2.4) ∪ [−2.25,−2) ∪ [−1.75,−1.5) ∪ [1.5, 1.6) ∪ [1.75, 2)

Σ = [−2.75,−1.25) ∪ [1.25, 2.25)

Ω = .5, K = 5, J = 4, O0 = [−.5, 0), O1 = [0, .2), O2 = [.2, .5)

m = 0 m = 1 m = 2

k = 0 I∗00 = Ω(O0 − 5) I∗01 = Ω(O1 − 5) I02 = Ω(O2 − 5)

I∗00 = [−2.75,−2.5) I∗01 = [−2.5,−2.4) I02 = [−2.4,−2.25)

k = 1 I∗10 = Ω(O0 − 4) I∗11 = Ω(O1 + 3) I12 = Ω(O2 − 4)

I∗10 = [−2.25,−2) I∗11 = [1.5, 1.6) I12 = [−1.9,−1.75)

k = 2 I∗20 = Ω(O0 − 3) I21 = Ω(O1 − 4) I22 = Ω(O2 − 3)

I∗20 = [−1.75,−1.5) I21 = [−2,−1.9) I22 = [−1.4,−1.25)

k = 3 I∗30 = Ω(O0 + 4) I31 = Ω(O1 − 3) I32 = Ω(O2 + 3)

I∗30 = [1.75, 2) I31 = [−1.5,−1.4) I32 = [1.6, 1.75)

k = 4 I40 = Ω(O0 + 3) I41 = Ω(O1 + 4) I42 = Ω(O2 + 4)

I40 = [1.25, 1.5) I41 = [2, 2.1) I42 = [2.1, 2.25)

116

5 TIME-FREQUENCY LOCALIZATION OPERATORS

Many sampling formulas use infinitely many samples. However, applications

typically use samples which are finite in number. In order, to reduce the associated

error, one might explore options presented by time-frequency localizations. In this

section, we discuss the time-frequency localization operator PΣQS.

We start with basic properties of these operators and their eigenvalues. We

move on to discuss related sampling formulas and discuss why understanding

the eigenvalues is critical to using these formulas. We then discuss results of

Landau which describe the behavior of the eigenvalues and provide some additional

improvements and discussion. Then, we build on the work of [16], presenting a

theorem which may be used to approximate eigenvalues numerically. Finally, we

discuss the error associated with these numerical approximations.

5.1 Prolate Spheriodal Wave Functions

Here, we will analyze the operator PΣQS where S is a union of M intervals

of total measure T and Σ is a single interval of measure Ω. We refer the reader to

[27] for a similar discussion regarding the case of one time interval. First of all,

using Theorem 2.13, we have PΣQSf = F−1(F (f1S)1Σ) = (f1S) ∗ (F−11Σ) for

any f ∈ L2(R). This expression may be expanded into the following equation.

PΣQSf(t) =∫S1Σ(x− t)f(x)dx (59)

117

Since we will be discussing the eigenvalues and eigenfunctions of PΣQS, let us

quickly define what it means to be an eigenvalue or eigenfunction.

Definition 5.1. Let T be an operator. A non-zero function φ is an eigenfunction

with eigenvalue λ if Tφ = λφ.

We know that neither PΣ or QS, by themselves, are compact operators, since

the unit ball in an infinite dimensional space is not compact. However, the compo-

sition PΣQS satisfies Equation 59 which is an integral equation over a compact set

with a continuous positive definite symmetric kernel. By Mercer’s theorem, PΣQS

is compact, its eigenvalues (non-negative) sum to the integral of the kernel and its

eigenfunctions form an orthogonal basis for L2(S). If we denote the eigenvalues

as λ0, λ1, ... and place them in decreasing order, we have the following equation.

∞∑n=0

λn =∫S1Σ(x− t)dx

The prolate spheroidal wave functions (PSWF) are the eigenfunctions of the

operator PΣQS. We will use φn to represent the normalized nth eigenfunction

corresponding to the eigenvalue λn. Using the fact that the PSWFs are band-

limited to Σ and are eigenfunctions of Equation 59, we have the following two

equations.

φn(t) =∫∞−∞ 1Σ(x− t)φn(x)dx (60)

λnφn(t) =∫S1Σ(x− t)φn(x)dx (61)

118

We may use properties of PΣ andQS to compute the inner product between two

eigenfunctions over the interval S. That is, using the fact that PΣ is idempotent

we have the following equalities.∫Sφm(x)φn(x)dx = 1

λn

∫Sφm(x)PΣQSφn(x)dx

= 1λn〈QSφm, PΣQSφn〉

= 1λn〈PΣQSφm, PΣQSφn〉

= λm〈φm, φn〉= λmδmn

We already know that the eigenfunctions are orthogonal over the whole real

line so combining the previous result yields the following dual orthogonality.

∫∞−∞ φm(x)φn(x)dx = δmn (62)

∫Sφm(x)φn(x)dx = λmδmn (63)

In a sense, the eigenfunctions with the largest eigenvalues are the functions

which are band-limited to Σ and have the most localization in S. That is, as a

special case of Equation 63, we have the following equation.

∫S|φn(x)|2dx = λn (64)

Finally, the PSWFs form an orthonormal basis for the space R(PΣ) and are

band-limited to Σ. Therefore, 1Σ(k− t) has the following orthonormal expansion.

1Σ(k − t) =∞∑n=0

∫∞−∞ 1Σ(k − x)φn(x)dx φn(t)

=∞∑n=0

φn(k)φn(t)(65)

119

5.2 Related Operators

In this section, we will discuss operators whose eigenfunctions and eigenvalues

are related to those of PΣQS. Following the theorems, we present the results of

the section in Table 14.

Theorem 5.2. Let n ≥ 1. Then, the following are equivalent.

1. φ is an eigenfunction of (PΣQS)n with eigenvalue λ.

2. Fφ is an eigenfunction of (QΣP−S)n with eigenvalue λ.

3. Dαφ is an eigenfunction of (PαΣQ 1αS)n with eigenvalue λ for any real α 6= 0.

4. Tβφ is an eigenfunction of (PΣQS+β)n with eigenvalue λ for any real β.

5. Mγφ is an eigenfunction of (PΣ−γQS)n with eigenvalue λ for any real γ.

6. Rφ is an eigenfunction of (P−ΣQ−S)n with eigenvalue λ.

Proof. The fact that F , Dα, Tβ, Mγ and R are all unitary operators shows that

if one of the given functions is a non-zero function, then all of the given functions

are non-zero functions. Let α 6= 0, β and γ be any real values. Then, the following

equations may be used to obtain the properties given above.

1. (QΣP−S)nF = F (PΣQS)n.

2. (PαΣQ 1αS)nDα = Dα(PΣQS)n.

120

3. (PΣQS+β)nTβ = Tβ(PΣQS)n.

4. (PΣ−γQS)nMγ = Mγ(PΣQS)n.

5. (P−ΣQ−S)nR = R(PΣQS)n.

Theorem 5.3. Let n ≥ 1. Suppose φ is an eigenfunction of (PΣQS)n with non-

zero eigenvalue λ. Then,

1. QSφ is an eigenfunction of (QSPΣ)n with eigenvalue λ.

2. PΣφ is an eigenfunction of (PΣQS)n with eigenvalue λ.

Proof. 1. Since λ is non-zero real and φ is a non-zero function, λφ is a non-zero

function. Thus, (PΣQS)nφ is a non-zero function which implies that QSφ is a

non-zero function. Further, (QSPΣ)nQS = QS(PΣQS)n

2. First of all, (PΣQS)nφ = λφ. Since λ is non-zero, φ is in the range of PΣ.

Since PΣ is idempotent, φ = PΣφ.

Theorems 5.2 and 5.3 may be generalized to make similar statments. For

example, suppose φ is an eigenfunction of (PΣQS)n with non-zero eigenvalue λ.

Then, FQSφ is an eigenfunction of (P−SQΣ)n with eigenvalue λ. Also, for any

real value γ, FMγφ is an eigenfunction of (QΣ−γP−S)n with eigenvalue λ. Finally,

for any m ∈ N, (PΣQS)mPΣφ is an eigenfunction of (PΣQS)n with eigenvalue λ.

121

Theorem 5.4. Let n ≥ 1. The following are equivalent.

1. φ is an eigenfunction of (PΣQS)n with non-zero eigenvalue λ.

2. φ is an eigenfunction of (PΣQSPΣ)n with non-zero eigenvalue λ.

Proof. 1⇔ 2. Suppose φ is an eigenfunction of (PΣQS)n with non-zero eigenvalue

λ. Then, (PΣQS)nφ = λφ. Since λ is non-zero, φ is in the image of PΣ. Since PΣ

is idempotent, this implies that φ = PΣφ. Thus,

(PΣQSPΣ)nφ = (PΣQS)nPΣφ

= (PΣQS)nφ

= λφ

Now, suppose φ is an eigenfunction of (PΣQSPΣ)n with non-zero eigenvalue λ.

Then, (PΣQSPΣ)nφ = λφ. Again, since λ is non-zero, φ = PΣφ. Therefore,

(PΣQS)nφ = (PΣQS)nPΣφ

= (PΣQSPΣ)nφ

= λφ

Theorem 5.5. Let m ≥ 1. Suppose φ is an eigenfunction of PΣQS with non-zero

eigenvalue λ. Then,

1. φ is an eigenfunction for PΣQR/S with eigenvalue 1− λ.

2. φ is an eigenfunction for (PΣQS)m with eigenvalue λm.

122

Proof. 1. Again, since λ is non-zero, φ = PΣφ. We have,

PΣQR/Sφ = (PΣ − PΣQS)φ

= (1− λ)φ

2. (PΣQS)mφ = (PΣQS)m−1λφ = ... = λmφ.

Theorem 5.3 shows how PΣQS and QSPΣ are related. Using arguments similar

to that of Theorem 5.4, we know that the non-zero eigenvalues and corresponding

eigenfunctions for both (QSPΣ)n and (QSPΣQS)n are the same. In a similar

fashion, notice that PΣQS, QΣP−S and P−SQΣ all have the same eigenvalues.

In particular, PΣQS and P−SQΣ have the same eigenvalues. See Table 14 for a

summary of related operators.

Table 14: Related Operators

Fφ, (QΣP−S)n, λ QSφ, (QSPΣ)n, λ

Dαφ, (PαΣQ 1αS)n, λ PΣφ, (PΣQS)n, λ

Tβφ, (PΣQS+β)n, λ φ, (PΣQSPΣ)n, λ

Mγφ, (PΣ−γQS)n, λ φ, PΣQR/S, 1− λ

Rφ, (P−ΣQ−S)n, λ φ, (PΣQS)m, λm

This table shows eigenfunctions and eigenvalues for operators related to PΣQS.

We assume that φ is an eigenfunction for (PΣQS)n with eigenvalue λ where n ∈ N.

We also assume α 6= 0, β and γ are real whereas m ∈ N.

123

Many of the proofs in this chapter involve multiple time intervals and a single

frequency interval but the theorems also hold for a single time interval and multiple

frequency intervals. In fact, when S is a single interval and Σ is the union of M

intervals, the corresponding PSWFs form an orthonormal basis for R(PΣQS).

5.3 PSWF Sampling Formula

In this section, we will derive a sampling formula based on the PSWFs.

This sampling formula differs from the Walter-Shen formula (see [27]) because we

assume Σ to be a union of M intervals rather than a single interval. One may

consider a multiband signal to be the sum of several single band signals. However,

one would encounter the same issues discussed in Section 4.4. We aim at providing

a formula which does not require the complexity and error gained when band-

limiting the signal before sampling. In this section, we will use φΣ,n to denote an

eigenfunction of PΣQS to distinguish between different sets of eigenfunctions.

Theorem 5.6. Suppose S is an interval of measure T and Σ is a union of M

intervals of total measure Ω. Then, if f is band-limited to Σ, f satisfies the

following reconstruction formula in L2(R).

〈f, φIm,n〉 =∞∑

ν=−∞

1|Im|(PImf)( ν

|Im|)φIm,n( ν|Im|)

(66)

f(t) =M−1∑m=0

∞∑n=0

〈f, φIm,n〉φIm,n(t) (67)

124

Proof. Assume S is an interval of measure T , Σ is a union of M intervals of total

measure Ω and f is band-limited to Σ. Further, suppose m ∈ 0, ...,M − 1 and

n ∈ N. Then, from Equations 4.4 and 65, we have the following calculation.

〈f, φIm,n〉 =∫∞−∞ f(x)φIm,n(x)dx

=∫∞−∞

M−1∑µ=0

1|Iµ|

∞∑ν=−∞

(PIµf)( ν|Iµ|)1Iµ( ν

|Iµ| − x)φIm,n(x)dx

= 1|Im|

∞∑ν=−∞

(PImf)( ν|Im|)

∫∞−∞ 1Im( ν

|Im| − x)φIm,n(x)dx

= 1|Im|

∞∑ν=−∞

(PImf)( ν|Im|)φIm,n( ν

|Im|)

This calculation shows Equation 66. Notice that the eigenfunctions from the

set n ∈ N : φIm,n form an orthonormal basis for R(PImQS), the space of func-

tions bandlimited to Im. Furthermore, if m 6= µ, for any n, ν ∈ N, we have

〈φIm,n, φIµ,ν〉 = 0. This follows from Parseval’s equality and the fact that φIm,n

has frequency support in Im. Since Σ =M−1∪m=0

Im, the eigenfunctions from the set

φIm,n : m = 0, ...,M − 1, n ∈ N form an orthonormal basis for R(PΣQS). Using

basic Hilbert Space theory (see [23]), we obtain Equation 67.

Theorem 5.6 takes the approach of band-limiting f to each interval indepen-

dently and then combining several reconstruction formulas. However, as men-

tioned in Section 4.4, this is not very useful in practice due to the complexity

of band-limiting f . Instead, we may build upon Theorem 4.9 and use periodic

nonuniform sampling to gain the following result.

125

Theorem 5.7. Suppose Σ is an ΩK-tiling set which is the union of J different Ω-

tiling sets Σj. Further, suppose that each Ω-tiling set Σj is the union of M disjoint

intervals of the form Ijm = Ω(Om + βjm), where the union of the M intervals Om

is the unit interval Σ0 and βjm ∈ Z for j = 0, ..., J − 1, m = 0, ...,M − 1. Let

Wm represent the J by J matrix whose entry in the jth row and kth column is

[Wm]jk = e−2πijβkm

K . For simplicity, define Cjkm = [W−1m ]kj[Wm]jk/Ω. If f is

band-limited to Σ, then f satisfies the following reconstruction formulas in L2(R).

〈f, φIkm,n〉 =J−1∑j=0

∞∑ν=−∞

Cjkmf( j+νKΩK

)φIkm,n( j+νKΩK

) (68)

f(t) =M−1∑m=0

J−1∑k=0

∞∑n=0

〈f, φIkm,n〉φIkm,n(t) (69)

f(t) =M−1∑m=0

J−1∑k=0

∞∑n=0

∞∑l=0

〈f, φIkm,n〉〈φΣ,l, φIkm,n〉φΣ,l (70)

Proof. Assume f satisfies the assumptions stated in Theorem 5.7. Further, sup-

pose m ∈ 0, ...,M − 1, k ∈ 0, ..., J − 1 and n ∈ N. Then, from Equations 4.9

and 65 we have the following calculation.

〈f, φIkm,n〉 =∫∞−∞ f(x)φIkm,n(x)dx

=J−1∑j=0

∞∑ν=−∞

Cjkmf( j+νKΩK

)∫∞−∞ 1Ikm( j+νK

ΩK− x)φIkm,n(x)dx

=J−1∑j=0

∞∑ν=−∞

Cjkmf( j+νKΩK

)φIkm,n( j+νKΩK

)

Notice that the eigenfunctions from the set φIkm,n : n ∈ N form an orthonor-

mal basis for R(PIkmQS). For reasoning stated in the proof of Theorem 5.6, the

eigenfunctions from the set φIkm,n : k = 0, ..., J − 1,m = 0, ...,M − 1, n ∈ N

form an orthonormal basis for R(PΣQS) and Equation 69 is satisfied.

126

Now, both f and φΣ,l satisfy the assumptions of Theorem 5.7. Therefore,

they satisfy Equation 69. Using Parseval’s Equality, we have the following inner

product computation.

〈f, φIΣ,l〉 =M−1∑m=0

J−1∑k=0

∞∑n=0

〈f, φIkm,n〉〈φΣ,l, φIkm,n〉

Recall that the eigenfunctions from the set φΣ,l : l ∈ N form an orthonormal

basis for R(PΣQS). Therefore, from basic Hilbert Space theory and using the fact

that both f and φΣ,l satisfy Equation 69, we obtain Equation 70.

5.4 Eigenvalues Greater Than 1/2

In Section 5.1, we learned that the PSWFs are the band limited functions

with greatest localization in S where the amount of localization is directly tied to

the eigenvalue associated with the PSWF. In Section 5.3, we demonstrated how a

band-limited signal could be reconstructed from the PSWFs. Therefore, in order

to reconstruct a signal which is well localized in S, we may use only the PSWFs

with high eigenvalues.

In order to explore the behavior of eigenvalues, we follow the work of Landau,

which is distributed over a series of papers (see [17]-[19]). We will quantify which

eigenvalues are greater than and less than 1/2. Before presenting these estimates,

let us recall some properties from Hilbert Space theory. For the proof of the next

theorem, we refer the reader to [23] and [17].

127

Theorem 5.8. Suppose T is a compact self-adjoint operator on L2(R) so that its

positive eigenvalues λ0, λ1, ... may be listed in non-increasing order. If we let H

denote the set of subspaces of L2(R) with dimension d, then we have

1. λd−1 = maxH∈H

minf∈H

〈Tf,f〉‖f‖2 .

2. λd−1 ≥ minf∈H

〈Tf,f〉‖f‖2 for every H ∈H .

3. λd = minH∈H

maxf∈H⊥

〈Tf,f〉‖f‖2 .

4. λd ≤ maxf∈H⊥

〈Tf,f〉‖f‖2 for every H ∈H .

We now apply this theorem to the operator PΣQS where Σ is a single interval

of measure Ω and S is a union of M intervals of total measure T . Since PΣQS is

compact, PΣQSPΣ is also compact by Theorem 5.4. Using the fact that PΣ and

QS are both self adjoint, we may replace 〈PΣQSPΣf, f〉 with ‖QSf‖2 to obtain

the following corollary.

Corollary 5.9. Let λ0, λ1,... denote the eigenvalues of PΣQS in non-increasing

order. Let H denote the set of subspaces of L2(R) with dimension d. Then,

1. λd−1 = maxH∈H ,H∈R(PΣ)

minf∈H

‖QSf‖2

‖f‖2 .

2. λd−1 ≥ minf∈H

‖QSf‖2

‖f‖2 for every H ∈H such that H ⊂ R(PΣ).

3. λd = minH∈H

maxf∈H⊥∩R(PΣ)

‖QSf‖2

‖f‖2 .

4. λd ≤ maxf∈H⊥∩R(PΣ)

‖QSf‖2

‖f‖2 for every H ∈H .

128

In order to estimate the minimum and maximum number of eigenvalues greater

than 1/2, we will apply Property 2 of Corollary 5.9 to a specific subspace H1 and

apply Property 4 of Corollary 5.9 to a separate subspaceH2. Our next two lemmas

concern the dimensions of these subspaces and a function which will be used to

define H1 and H2.

Lemma 5.10. Suppose Σ is an interval of measure Ω and S is a union of M

disjoint intervals of total measure T . Then, N1 ≥ bΩT c − 2M + 2 and N2 ≤

dΩT e+ 2M − 2 where N1 and N2 are defined by the following equations.

N1 = supβ∈R

∣∣∣N1(β)∣∣∣, N1(β) = n ∈ Z : ΩS + β contains Σ0 + n (71)

N2 = infβ∈R

∣∣∣N2(β)∣∣∣, N2(β) = n ∈ Z : ΩS + β intersects Σ0 + n (72)

Proof. Express ΩS + β in the formM∪m=1

[am, am + qm) using β = −12− Ω min

t∈St.

Notice thatM∑m=1

qm = ΩT and a1 = −12

by the choice of β. Analogous to the

definition of N1 and N2, let N1m and N2m represent the number of intervals of the

form Σ0 + n for which [am, am + qm) contains and intersects Σ0 + n, respectively.

Let brc represent the greatest integer less than or equal to r and dre = −b−rc.

Also, recall that for r ∈ R and n ∈ N, we have the following properties.

1. r ≤ n⇔ dre ≤ n and n ≤ r ⇔ n ≤ brc

2. n < r ⇔ n ≤ dr − 1e and r < n⇔ br + 1c ≤ n

3. dre+ n = dr + ne and brc+ n = br + nc

129

Now, [am, am+qm) will contain Σ0+n if both am ≤ −12+n and 1

2+n ≤ am+qm.

From Property 1, this is equivalent to dam + 12e ≤ n and n ≤ bam + qm − 1

2c.

Similarly, [am, am + qm) will intersect Σ0 + n if am < 12

+ n or −12

+ n < am + qm.

From Property 2, this is equivalent to bam + 12c ≤ n or n ≤ dam + qm − 1

2e.

Therefore, by Property 3, we have the following two equations:

N1m = bam + qm −1

2c − dam +

1

2e+ 1 = bam + qm +

1

2c − dam +

1

2e

N2m = dam + qm −1

2e − bam +

1

2c+ 1 = dam + qm +

1

2e − bam +

1

2c

In addition to the properties above, if r ∈ R, we have brc > r − 1 and

dre < r + 1. Therefore, using the fact that a1 = −12

yields N11 > q1 − 1,

N1m > qm − 2, N21 < q1 + 1 and N2m < qm + 2 for m = 2, ...,M . We have

N1 ≥∣∣∣N1(−1

2− Ω min

t∈St)∣∣∣ =

M∑m=1

N1m > ΩT − 2M + 1

N2 ≤∣∣∣N2(−1

2− Ω min

t∈St)∣∣∣ =

M∑m=1

N2m < ΩT + 2M − 1

From Properties 2 and 3, we obtain N1 ≥ bΩT − 2M + 2c = bΩT c − 2M + 2

and N2 ≥ dΩT + 2M − 2e = dΩT e+ 2M − 2.

Lemma 5.11. Let h(t) = 2 cos(πt)1Σ0(t) where Σ0 = [−12, 1

2). Then, ‖h‖2 = 2

and if f is band-limited to Σ0, then we have the following inequality.

‖f‖2 ≤∞∑

n=−∞|∫

Σ0+nf(t)h(n− t)dt|2 (73)

130

Proof. First of all, from the trigonometric identity cos2(t) = 12

+ 12

cos(2t) we know

h2(t) =(

2 + 2 cos(2πt))1Σ0(t). Therefore, by direct integration, ‖h‖2 = 2.

Now, h = (M− 12

+ M 12)1Σ0 so h = (T− 1

2+ T 1

2)sinc. If we use V to denote the

function |ω| − 1 due to its shape, then sinc ≥ −V on [−1, 1] which implies that

h ≥ −(T− 12

+ T 12)V = 1 on Σ0. Using the fact that h is supported on Σ0 along

with Theorems 2.12, 4.2 and Bessel’s Inequality, we have

‖f‖2 ≤ ‖f ∗ h‖2

=∞∑

n=−∞|f ∗ h(n)|2

=∞∑

n=−∞|∫

Σ0+nf(t)h(n− t)dt|2

(74)

We are now ready to present Landau’s eigenvalue estimate (see [17]). However,

we give a slight improvement, since if ΩT = 7 and M = 3, our estimate implies

λ2 ≥ .5 and λ12 ≤ .5 while Landau’s estimate implies λ1 ≥ .5 and λ14 ≤ .5.

Theorem 5.12. Suppose Σ is a single interval of measure Ω and S is a union of

M intervals of total measure T . Let λ0, λ1, ... denote the eigenvalues of PΣQS in

non-increasing order. Then λN1−1 ≥ 12

and λN2 ≤ 12

where N1 and N2 are defined

as in Lemma 5.10. In particular, λbΩT c−2M+1 ≥ 12

and λdΩT e+2M−2 ≤ 12.

Proof. Let α = 1Ω

, β = −12− Ω min

t∈St and γ = 1

2+ 1

Ωminω∈Σ

ω. By Theorem 5.2,

the eigenvalues of PΣQS and PαΣ−γQ 1αS+β = PΣ0QΩS+β are the same. We will use

Corollary 5.9 to estimate the eigenvalues of PΣ0QΩS+β.

131

Define N1(β) and N1 as in Lemma 5.10 and h as in Lemma 5.11. For n ∈

N1(β), let g1n satisfy g1n ∗ h = Tn(sinc). Let H = spang1n : n ∈ N1(β) and

f ∈ H. We will show that‖QΩS+βf‖2

‖f‖2 ≥ 12

so that by Corollary 5.9, λbΩT c−2M+1 ≥

λN1−1 ≥ 12. Notice that g1n ∗ h(n) = 0 if n 6∈ N1(β). Therefore,

‖f‖2 ≤∞∑

n=−∞|∫


=∑

n∈N1(β)

|∫


≤∑

n=N1(β)

∫Σ0+n

|f(t)|2dt∫

Σ0+n|h(n− t)|2dt

≤ ‖h‖2 ∫t∈ΩS+β

|f(t)|2dt

= 2 ‖QΩS+β‖2

Define N2(β) and N2 as in Lemma 5.10 and h as in Lemma 5.11. For n ∈

N2(β), let g2n(t) = h(n − t). Let H = spang2n : n ∈ N2(β) and f ∈ R(PΣ0) ∩

H⊥. We will show that‖QΩS+βf‖2

‖f‖2 ≤ 12

so that by Corollary 5.9, λdΩT e+2M−2 ≤

λN2 ≤ 12. Notice that 〈f, g2n〉 = 0, if n ∈ N2(β). Therefore,

‖f‖2 ≤∞∑

n=−∞|∫


=∑

n6∈N2(β)

|∫


≤∑

n 6∈N2(β)

∫Σ0+n

|f(t)|2dt∫

Σ0+n|h(n− t)|2dt

≤ ‖h‖2 ∫t6∈ΩS+β

|f(t)|2dt

= 2(‖f‖2 − ‖QΩS+β‖2)

132

It turns out that if the M intervals have endpoints which lie on the integer

lattice, then we may greatly improve the estimate given above. This is due to the

fact that both N1 and N2 will be equal to ΩT .

Corollary 5.13. Suppose Σ is a single interval of measure Ω and S is a union of

M intervals whose total measure is T . Further suppose, each of the M intervals

has the form 1Ω

(Σ0 +n+β0) for some β0 ∈ R. Let λ0, λ1, ... denote the eigenvalues

of PΣQS in non-increasing order. Then, λΩT−1 ≥ 12

and λΩT ≤ 12.

5.5 When No Significant Eigenvalues Exist

Theorem 5.12 suggests that if Σ is a pairwise disjoint union of a large number

of short intervals, PΣQS might fail to have an eigenvalue larger than 1/2, even

though the product area ΩT could be much larger than one. This is due to the

fact that if Σ is a union of M intervals which are highly separated, then the time

localization operators for these intervals are almost orthogonal.

In this section, we will give a bound on the largest eigenvalue λ0 for PΣQS0

where S0 = [−12, 1

2) and Σ is a union of M intervals. However, this bound will

only be useful for the case in which the M intervals all have measure less than 1

and are well separated. At the end of the section, we will give a Corollary that

describes how to contstruct a set Σ based on an arbitrarily large product area ΩT

such that PΣQS0 has no eigenvalues greater than 1/2. In order to prove the bound

on λ0, we will require several lemmas.

133

Lemma 5.14. Suppose S0 = [−12, 1

2) and I is an interval of measure Ω. Further,

suppose ψ ∈ R(PIQS0) where ‖ψ‖ = 1. Then, we have the following estimate.

∫S0|ψ|2dt ≤ |I| (75)

Proof. Suppose ψ ∈ R(PIQS0) where ‖ψ‖ = 1. Then, ψ is the inverse Fourier

Transform of ψ. Using the Cauchy-Schwartz inequality yields the following.

∫S0|ψ|2dt =

∫S0|∫Iψ(ω)e2πiωtdω|2dt

≤∫S0

∫I|ψ(ω)|2dω

∫I|e2πiωt|2dωdt

=∫S0|I|dt

= |I|

Lemma 5.15. Suppose that f1, f2 ∈ L2(R) are band-limited to [−Ω/2,Ω/2]. Then

f1f2 = f1 ∗ f2 is supported in [−Ω,Ω] and |f1f2| ≤ ‖f1‖ ‖f2‖.

Proof. First of all, f1f2(ω) = (f1∗f2)(ω) =∫∞−∞ f1(ξ)f2(ω−ξ)dξ. Suppose |ω| > Ω

and ξ ∈ R. In the case where |ξ| > Ω/2, we have f1(ξ) = 0. In the other case

where |ξ| ≤ Ω/2, we have |ω − ξ| > Ω/2 and f2(ω − ξ) = 0. Therefore, f1 ∗ f2

is supported in [−Ω,Ω]. In addition, from the Cauchy-Schwartz inequality and

Parseval’s equality, we have the following.

|∫∞−∞ f1(ξ)f2(ω − ξ)dξ| ≤

( ∫∞−∞ |f1(ξ)|2dν

)1/2( ∫∞−∞ |f2(ω − ξ)|2dν

)1/2

= ‖f1‖ ‖f2‖

134

Lemma 5.16. Suppose S0 = [−12, 1

2) and I1, I2 are two intervals of measure Ω.

Further, suppose ψ1 ∈ R(PS0QI1) and ψ2 ∈ R(PS0QI2) where ‖ψ1‖ = ‖ψ2‖ = 1.

Finally, assume the distance ω0 between the midpoints I1 and I2 is greater than

Ω. Then, we have the following estimate.

∫S0ψ1(t)ψ2(t)dt ≤ 2Ω

ω0−Ω(76)

Proof. Define f1 = MI1ψ1 and f2 = MI2

ψ2. Then, from Table 1, f1 and f2 are

bandlimited to [−Ω/2,Ω/2]. According to Lemma 5.15 and Theorem 2.12, we

have the following inequalities.

|∫S0ψ1(t)ψ2(t)dt| = |

∫∞−∞ f1(t)f2(t)1S0(t)e−2πiω0t dt|

=∣∣∣ ∫∞−∞ f1f2(ξ)1S0(ω0 − ξ)dξ

∣∣∣≤

∫ Ω

−Ω|sinc(ω0 − ξ)|dξ

≤ 2Ωω0−Ω

Theorem 5.17. Assume that S0 = [−12, 1

2) and Σ is the union of K intervals

I0,...,IK−1 each having measure Ω. Further suppose that the minimum distance

ω0 between any two midpoints I0, ..., IK−1 is greater than Ω. Then, we have the

following estimate.

λ0 ≤ Ω + 2Ωω0−Ω

(K2 −K) (77)

135

Proof. Let φ0 denote the eigenfunction of PΣQS0 which has the largest eigenvalue

λ0 and assume ‖φ0‖ = 1. Write φ0 =K−1∑k=0

akψk where ψk ∈ R(PS0QIk) and

‖ψk‖ = 1. Notice thatK−1∑k=0

|ak|2 = 1 and λ0 =∫S0|φ0(t)|2dt. Using Lemma 5.14

and Lemma 5.16, we have the following equalities.∫S0|φ0(t)|2dt =

∣∣∣K−1∑j=0

K−1∑k=0

ajak∫S0ψj(t)ψk(t)dt

∣∣∣=

K−1∑k=0

|ak|2∫S0|ψk(t)|2dt+

K−1∑j=0

K−1∑k=0,k 6=j

ajak∫S0ψj(t)ψk(t)dt

≤ ΩK−1∑k=0

|ak|2 +K−1∑j=0

K−1∑k=0,k 6=j

2Ωω0−Ω

≤ Ω + 2Ωω0−Ω

(K2 −K)

Corollary 5.18. For every N ∈ N, there exists a K and a set Σ such that Σ has

measure N , Σ is the union of K intervals of discrete measure Ω and PΣQS0 has

no eigenvalues larger than 1/2.

Proof. Choose Ω = 14

and K = 4N . Next, choose ω0 such that 2Ωω0−Ω

(K2−K) ≤ 14.

Finally, choose K intervals I0, ..., IK−1 each of measure Ω such that Ik = 14

+ kω0

for k = 0, ..., K − 1. For Σ =K−1∪k=0

Ik, Theorem 5.17 shows that λ0 ≤ 12

where λ0 is

the largest eigenvalue of PΣQS0 .

One interpretation of Theorem 5.17 is that a signal cannot be well localized in

time and be bandlimited to highly separated intervals of small measure. However,

even for such a case, the PSWFs have the best time localization and their use will

require the fewest samples for reconstruction.

136

5.6 Eigenvalue Calculations

In this section, we give a method for numerically estimating the eigenvalues

of the time-frequency localization operator PΣQS. Following the theorem, we

present an example demonstrating this method. One should take note of how this

algorithm was coded by referencing the Matlab code given in the Appendix.

Theorem 5.19. Suppose Σ and S are subsets of R. Suppose we have the sampling

formula f(t) =∑n∈N

f( nΩ

)hn(t) for all f band-limited to Σ. Define the matrix A

(possibly infinite) such that Amn =∫S1Σ(x− m

Ω)hn(x)dx for m ∈ Z and n ∈ N .

Then, if φ is an eigenfunction of the operator PΣQS with eigenvalue λ, then the

vector φ( nΩ

) : n ∈ N is an eigenvector of the matrix A with eigenvalue λ.

Conversely, if the vector φ( nΩ

) : n ∈ N is an eigenvector of the matrix A

with eigenvalue λ and the function φ(t) =∑n∈N

φ( nΩ

)hn(t) converges, then φ is an

eigenfunction of the operator PΣQS with eigenvalue λ.

Proof. Now, suppose φ is an eigenfunction of the operator PΣQS with eigenvalue

λ. Then, φ is band-limited to Σ. Therefore, φ(t) =∑n∈N

φ( nΩ

)hn(t). Plugging

this into the expression for PΣQSφ, yields λφ(mΩ

) = PΣQSφ(mΩ

) =∑n∈N

Amnφ( nΩ

).

Therefore, φ( nΩ

) : n ∈ N is an eigenvector of the matrix A with eigenvalue λ.

On the other hand suppose φ( nΩ

) : n ∈ N is an eigenvector of the matrix A

with eigenvalue λ and the function φ(t) =∑n∈N

φ( nΩ

)hn(t) converges. Then, φ and

PΣQS are both band-limited to Σ. Therefore,

137

PΣQSφ(t) =∑m∈N

PΣQSφ(mΩ

)hm(t)

=∑m∈N

∑n∈N

Amnφ( nΩ

)hm(t)

= λ∑m∈N

φ(mΩ

)hm(t)

= λφ(t)

Notice that it is possible to obtain more accurate estimates by using a sampling

formula with better efficiency. However, using small spectral slices may cause the

DFT matrix to be ill conditioned.

Example 5.20. Let S = [−.5, .5) and Σ = [0, 1.3)∪ [4.5, 5). We will calculate the

eigenvalues of PΣQS. For these sets, the parameters may be chosen as in Table

5.20 or as in Table 5.20. The eigenvalues calculated for each case are shown in

Tables 5.20. In both cases, we truncated the matrix A to a 21 by 21 matrix. One

should take particular note of how the algorithm was coded up to depend on the

parameters chosen (see Appendix).

Table 15: Eigenvalues for Example 5.20

n = 0 n = 1 n = 2 n = 3 n = 4

λn, Set 1 0.88342 0.47761 0.32722 0.03670 0.02495

λn, Set 2 0.86616 0.50594 0.32567 0.04507 0.01162

138

Table 16: First Set of Parameters for Example 5.20

Ω = 5, K = 1, O0 = [0, .26), O1 = [.26, .5), O2 = [−.5,−.1), O3 = [−.1, 0)

hn(x) = 1Ω

(1[0,1.3) + 1[4.5,5)(t)

)|t= n

Ω−x

m = 0 m = 1 m = 2 m = 3

j = 0 I0 = Ω(O0 + 0) I1 = Ω(O1 + 0) I2 = Ω(O2 + 1) I3 = Ω(O3 + 1)

I0 = [0, 1.3) I1 = [1.3, 2.5) I2 = [2.5, 4.5) I3 = [4.5, 5)

Table 17: Second Set of Parameters for Example 5.20

Ω = .5, K = 10, O0 = [0, .5), O1 = [−.5,−.4), O2 = [−.4, 0)

hj+nK(x) = 1Ω

2∑m=0

3∑k=0

[W−1m ]kj[Wm]jk1Ikm(t)|t= j+nK

ΩK−x

m = 0 m = 1 m = 2

j = 0 I00 = Ω(O0 + 0) I01 = Ω(O1 + 1) I02 = Ω(O2 + 1)

I00 = [0, .25) I01 = [.25, .3) I02 = [.3, .5)

j = 1 I10 = Ω(O0 + 1) I11 = Ω(O1 + 2) I12 = Ω(O2 + 2)

I10 = [.5, .75) I11 = [.75, .8) I12 = [.8, 1)

j = 2 I20 = Ω(O0 + 2) I21 = Ω(O1 + 3) I22 = Ω(O2 + 3)

I20 = [1, 1.25) I21 = [1.25, 1.3) I22 = [1.3, 1.5)

j = 3 I30 = Ω(O0 + 9) I31 = Ω(O1 + 10) I32 = Ω(O2 + 10)

I30 = [4.5, 4.75) I31 = [4.75, 4.8) I32 = [4.8, 5)

139

5.7 Matrix Error Bounds

In section 5.6 we established a very general result relating projections onto

Paley-Wiener spaces with samples of corresponding interpolating functions. In

order to give the result practical value, one would like to know how accurately

one can approximate the samples φ( nΩ

) : n ∈ N of the eigenvectors from

finite dimensional approximations of the matrix Amn =∫SR1Σ(m

Ω− x)hn(x)dx.

Since the Amn are defined in terms of integrals, there is a secondary question

of how accurately one can estimate these entries. We will ignore this issue. In

practical applications, the singular value decomposition of A would be a one time

calculation. On the other hand, slow decay of interpolating functions suggests

that a large truncation is needed to obtain accurate sample estimates. In this

section we will not address truncation errors for the general case. Instead we

will focus specifically on the case of the matrix corresponding to the traditional

truncations to time and frequency interval pairs with the goal of obtaining sharp

estimates on the norm of the truncation remainder.

Our method will not address the accuracy of truncations of A when restricted

to specific eigenspaces. In the case of the prolate spheroidal wave functions this

question was already addressed, in effect, in the work of Walter and Shen (see [27]

and [26]). Their Fourier transform is normalized as f(ξ) =∫f(t)eitξ dt. With ϕn

140

the λn-eigenfunction of P[−π,π]Q[−T,T ], Walter and Shen proved that

∑|k|>T

ϕn(k)2 ≤ (1− λn) + 4πT(1− λn

3λn

)1/2

≡ (1− λn)1/2C(n, T ) (78)

As a consequence, they showed that if f ∈ span ϕ1, . . . , ϕN, N ≤ 2T , then∫ T

−T

∣∣∣f(t)−∑|k|≤T

f(k) [2T ]∑n=0

ϕn(k)ϕn(t)1[−T,T ]

∣∣∣2 ≤ ‖f‖2

N∑n=0

λn(1− λn)1/2C(n, T )

Thus, one can approximate f very effectively in L2 using a finite number of samples

if f is in the subspace of time-frequency localized signals.

Now we are ready to consider approximations via truncations of the matrix

A. Specifically, let A denote the “time-frequency localization matrix” with entries

Ak` =∫ T/2−T/2 sinc(x−k) sinc(x−`) dx. We have seen that the eigenvectors of A are

the samples of the eigenfunctions of PΣQT . Thus one can compute the projection

of any f ∈ R(P[−π,π]Q[−T,T ]) onto the eigenfunctions by means of sample inner

products in `2(Z). This requires all samples of f . One would like, instead, to

estimate the projections from finitely many samples as Shen and Walter did for

eigenspaces. Since the PSWFs with eigenvalues close to one are also the most

localized on [−T/2, T/2] one expects these also to be the ones whose samples

decay fastest off the time support. However, this leaves open the problem of

estimating the projections onto the PSWFs in the transition region. Ideally one

would like direct estimates on the decay of the eigenvectors of A and estimates

on some truncation of A that respects these decay estimates. What we propose

in this section is really more basic: we want to prove `2-bounds on the remainder

141

A = AN obtained by truncating the entry Ak` to zero if max|k|, |`| > N .

Obtaining bounds actually is not difficult. It takes a little more work to show that

the bounds are essentially sharp and to provide a formula from which numerical

approximations of the eigenfunctions can be computed efficiently.

In the following we denote by A = A(T ) the matrix with entries

Ak` =

∫ T

−Tsinc(x− k) sinc(x− `) dx, k, ` ∈ Z (79)

Unless specifically stated otherwise we will assume that T ∈ N though, again,

this is really only for notational convenience. Now if |x| ≤ T and |k| > T , then

x−k ∈ [−T −k, T −k] so that |sinc(x−k)| ≤ 1π

max 1|T+k| ,

1|T−k| = 1

π1

|k|−T . Since

the same holds for `, if |k|, |`| > T , we have

|Ak`| ≤ 2T B(k)B(`) where B(k) =1

π

1

|k| − T(80)

If one of k or ` is no bigger than T then we can get a corresponding dependence

logarithmic in T . For example, if |k| < T then

|Ak`| =∣∣∣∫ T

−Tsinc(x− k) sinc(x− `) dx

∣∣∣≤ 2

πB(`)

∫ T

−T

1

1 + |x− k|dx

=2

πB(`)

∫ T−k

−T−k

1

1 + |u|dx

≤ 2

πB(`) maxln |T − k|, ln |T + k| ≤ CB(`) ln(T ) (81)

There will always be essentially 2T values of k and ` that are less than T in

absolute value.

142

Proposition 5.21. Let A = A(T ) be as in (79) and let Atr = AtrN be such that

Atrk` = Ak` if max|k|, |`| ≤ NT and Atr

k` = 0 otherwise. Then A = A− Atr is `2

bounded with norm at most C ln(T )N−1/2 where C is independent of N and T .

Proof. We use (80) and (81) to obtain elementwise bounds. Fix s = s` ∈ `2(Z).

Case 1: |k| ≤ T .

The only nonzero terms of Ak` then correspond to |`| > NT . If |k| ≤ T then

|Ak`| ≤ C ln(T )|`|−T with |`| ≥ NT . In this case, we have

∣∣∑`

Ak`s`∣∣ ≤ C ln(T )

(−NT−1∑−∞

+∞∑

NT+1

1

|`| − T|s`|)

We can consider the cases ` > 0 and ` < 0 separately. For the positive terms,

∞∑NT+1

1

|`| − T|s`| ≤

( ∞∑NT+1

1

(|`| − T )2

)1/2( ∞∑NT+1

|s`|2)1/2

Together with the terms for ` < 0, when |k| ≤ T , one obtains

∣∣∑`

Ak`s`∣∣ ≤ C ln(T )

( ∞∑NT+1

1

(|`| − T )2

)1/2

‖s‖`2 ≤ Cln(T )√NT‖s‖`2

Case 2: T < |k| ≤ NT .

That Ak` = 0 unless |`| > NT still pertains, but now the Ak` satisfy the

inequality |Ak`| ≤ 2T B(k)B(`) ≤ 2T (|k|−T )−1(|`|−T |)−1 and otherwise arguing

just as before, when T < |k| ≤ NT , we obtain

∣∣∑`

Ak`s`∣∣ ≤ C

T

(|k| − T )√NT‖s‖`2

143

Case 3: |k| > NT . In this case Ak` can be nonzero for any ` but we have∑|`|<T +

∑|`|≥T min

ln(T ), T

|`|−T

|s`|

≤ ln(T )(∑

|`|<T 1)1/2(∑

|`|<T |s`|2)1/2

+(∑

|`|≥TT

(1+|`|−T )2

)1/2(∑|`|≥T |s`|2

)1/2

≤ C√T ln(T )‖s‖`2

so that for |k| > NT , we have

∣∣∑` Ak`s`

∣∣ ≤ C ln(T )√T

(|k|−T )‖s‖`2

To estimate the norm of A we then have

‖As‖2 =∑|k|≤T

+∑

T<|k|≤NT

+∑|k|>NT

∣∣∑`

Ak`s`∣∣2

≤ C ln(T )2

NT‖s‖2

`2 +C

N‖s‖2

`2

( ∑T<|k|≤NT

1

1 + (|k| − T )2

)+ CT ln(T )2‖s‖2

`2

∑|k|>NT

1

(|k| − T )2

≤ C ′ ln(T )2

N‖s‖2

`2

If the bounds A ≈ T B(k)B(`) are sharp then one can improve the `2 bound

at most by removal of the factor ln(T ). This is because in the case of separable

bounds one cannot get any faster decay than B(k). However, showing that the

bounds are effectively separable requires more precise estimates of the entries of A

that will be supplied below. Removal of the logarithmic factor ln(T ) in the norm

bound for A would require additional cancellation in the integral defining Ak`. We

will leave the question of whether this logarithmic improvement is possible open.

144

The error should be smaller when Atr is applied to estimate sample vectors of

PSWFs that are localized on [−T, T ]. Part of the reason comes from the case in

which k ≤ NT . There, only contributions from ` ≥ NT apply and the PSWF

samples decay fast away from [−T, T ]. On the other hand, when k is large one is

effectively getting the k-th entry of the eigenvector which is known to decay.

We now wish to provide an alternative formula for the entries of A from which

it can be inferred that the only potential improvement on the error bounds given

by Proposition 5.21 would be removal of the factor ln(T ) in the norm estimate for

A. As such, we will focus primarily on the entries Ak` in which K ≥ 0 and ` ≥ 0.

We will see that the matrix has a great deal of symmetry as well. But, in order

to show that the estimate of Proposition 5.21 cannot be substantially improved,

it is enough to show that it cannot be substantially improved when operating on

s supported in N = Z+.

Diagonal terms. Akk =∫ T−T

sin2 π(x−k)π2(x−k)2 dx. Setting u = π(x− k) we can write

Akk =1

π

∫ π(T−k)

−π(T+k)

sin2 u

u2du

Using the half angle formula one replaces this with

Akk =1

2π

∫ π(T−k)

−π(T+k)

1− cos 2x

x2dx

First we assume that k 6= ±T .

145

Case 1. 0 ≤ k < T . One can integrate by parts, setting u = 1 − cos 2x and

v = −1/x, to get

12π

[cos 2x−1

x

∣∣∣π(T−k)

−π(T+k)+ 1

π

∫ π(T−k)

−π(T+k)sin 2xx

dx

= 1π

[cosx−1

x

∣∣∣2π(T−k)

−2π(T+k)+ 1

π

∫ 2π(T−k)

−2π(T+k)sinxxdx

= T (cos 2πT−1)π2(T 2−k2)

+ 1π

(Si(2π(T + k)) + Si(2π(T − k))

)by replacing x for 2x. Here Si(x) =

∫ x0

sin ttdt is the sine integral function.

Case 1′. −T < k ≤ 0. Here we have k = −|k| and the same integration by

parts, keeping track of signs, gives

Akk =T (cos 2πT − 1)

π2(T 2 − k2)+

1

π

(Si(2π(T − |k|)) + Si(2π(T + |k|))

)(82)

Case 2. |k| > T . We use integration by parts just as before, but rearrange

terms to insure the argument of the sine integral is positive. This yields

Akk =T cos(2πT − 1)

π2(T 2 − k2)+

1

π

(Si(2π(T + k))− Si(2π(k − T ))

), (k > T )

Akk =T (cos 2πT − 1)

π2(T 2 − k2)+

1

π

(Si(2π(|k|+ T ))− Si(2π(|k| − T ))

), (k < −T )

Note that, just as in case 1, A−k,−k = Ak,k which can also be seen directly from

the symmetry of the integral defining A.

Case 3. |k| = T . In this case T ∈ Z and one can easily check that the evaluation

of cosx−1 at the given endpoints yields zero. One is left simply with the estimate

Akk = ATT = 1πSi(4πT ).

146

Off diagonal terms.

Lemma 5.22. If k 6= ` then

sinc(x− k) sinc(x− `) = 2(−1)k−`

k − `

(1− cos 2π(x− k)

2π(x− k)− 1− cos 2π(x− `)

2π(x− `)

)(83)

Proof. First we recall that

sin π(x− k) sin π(x− `) =1

2

[cosπ(`− k)− cos π((x− k) + (x− `))

]=

1

2

[(−1)k−` − cos π((2x− (k + `))

]=

(−1)k−`

2

[1− cos 2π(x− k)

]=

(−1)k−`

2

[1− cos 2π(x− `)

]

where we used that fact that cosπ(2x− n) = (−1)n cos 2πx. We also have

1

(x− k)(x− `)=

1

(k − `)

[ 1

(x− k)− 1

(x− `)

]Putting these together we get

sinc(x− k) sinc(x− `)

= (−1)k−`

2

[1− cos 2π(x− k)

](1

π2(x−k)(x−`)

)= (−1)k−`

2π2(k−`)

[1− cos 2π(x− k)

][1

(x−k)− 1

(x−`)

]= 2 (−1)k−`

(k−`)

(1−cos 2π(x−k)

2π(x−k)

)−(

1−cos 2π(x−`)2π(x−`)

)which is what was to be proved.

147

The cosine integral Ci(x) is ordinarily defined as

Ci(x) = −∫ ∞x

cos t

tdt = γ + lnx+

∫ x

0

cos t− 1

tdt

where γ is Euler’s number.

In view of (83) it is possible to express the matrix elements in terms of cosine

integral values. Assume, as above, that we are truncating in time over [−T, T ].

Then, we have the following equalities.

Ak` =

∫ T

−Tsinc(t− k) sinc(t− `) dt

= 2(−1)k−`

(k − `)

∫ T

−T

(1− cos 2π(t− k)

2π(t− k)

)−∫ T

−T

(1− cos 2π(t− `)2π(t− `)

)Setting u = 2π(t− k) in the first integral and u = 2π(t− `) in the second,

Ak` =

∫ T

−Tsinc(t− k) sinc(t− `) dt

=(−1)k−`

π(k − `)

∫ 2π(T−k)

−2π(T+k)

(1− cosu

u

)du−

∫ 2π(T−`)

−2π(T+`)

(1− cosu

u

)du

In the following we will assume k < ` since Ak` is symmetric. There are different

cases that we need to consider depending on whether k or ` is between −T and

T . We will first phrase the cases in terms of k. Then, we will consider both k and

` together.

148

Case 1. T + k > 0, T − k < 0 (k > T ). Since the integrand is odd, we may

switch the limits of integration using∫ BA

= −∫ AB

to obtain∫ 2π(T−k)

−2π(T+k)

(1−cosu

u

)du

=∫ 0

−2π(T+k)

(1−cosu

u

)du+

∫ 2π(T−k)

0

(1−cosu

u

)du

= −∫ 2π(T+k)

0

(1−cosu

u

)du−

∫ 0

2π(T−k)

(1−cosu

u

)du

= −∫ 2π(T+k)

0

(1−cosu

u

)du+

∫ 2π(k−T )

0

(1−cosu

u

)du

= −Ci(2π(T + k)) + γ + ln(2π(T + k)) + (Ci(2π(k − T ))− γ − ln(2π(k − T )))

= Ci(2π(k − T ))− Ci(2π(k + T )) + ln(k+Tk−T

).

Case 1′. T + k < 0, (k < −T ). We may use a calculation similar to the one

shown previously in Case 1 to show that if M(k;T ) =∫ 2π(T−k)

−2π(T+k)

(1−cosu

u

)du then

M(−|k|;T ) = −M(|k|;T ) when |k| > T .

Case 2. k ≥ 0, T − k > 0 (k ∈ [0, T )). Since the integrand is odd,∫ 2π(T−k)

−2π(T+k)

(1−cosu

u

)du

=∫ 0

−2π(T+k)

(1−cosu

u

)du+

∫ 2π(T−k)

0

(1−cosu

u

)du

= −∫ 2π(T+k)

0

(1−cosu

u

)du+

∫ 2π(T−k)

0

(1−cosu

u

)du

= Ci(2π(T − k))− γ − ln(2π(T − k))− (Ci(2π(T + k))− γ − ln(2π(T + k)))

= Ci(2π(T − k))− Ci(2π(T + k)) + ln(T+kT−k

)Case 2′. k ≤ 0, T + k > 0 (k ∈ (−T, 0]). Using the oddness of the integrand

again, yields the following equalities.∫ 2π(T−k)

−2π(T+k)

(1−cosu

u

)du =

∫ 0

−2π(T+k)

(1−cosu

u

)du+

∫ 2π(T−k)

0

(1−cosu

u

)du

=∫ 0

−2π(T−|k|)

(1−cosu

u

)du+

∫ 2π(T+|k|)0

(1−cosu

u

)du

= −∫ 2π(T−|k|)

0

(1−cosu

u

)du+

∫ 2π(T+|k|)0

(1−cosu

u

)du

so, once again, we have M(−|k|;T ) = −M(|k|;T ).

149

Case 3. T = k. In this case the definition of the cosine integral gives

∫ 2π(T−k)

−2π(T+k)

(1− cosu

u

)du = −

∫ 4πT

0

(1− cosu

u

)du

= −Ci(4πT ) + γ + ln(4πT )

Case 3′. T = −k. Just as in Case 2, the integral is odd in the argument k.

Putting all of these cases together yields the following proposition.

Proposition 5.23. Define M(k;T ) =∫ 2π(T−k)

−2π(T+k)

(1−cosu

u

)du. In addition, define

C(k;T ) = Ci(2π|k − T |) − Ci(2π|k + T |) and L(k;T ) = ln∣∣∣T+kT−k

∣∣∣. When k 6= T ,

we can write M(k;T ) = C(k;T ) + L(k;T ). On the other hand, when k = T , we

can write M(k;T ) = −Ci(4πT ) + γ + ln(4πT ) for k > 0.

The terms M(k;T ) provide a recipe for writing the terms of Ak` as follows.

Corollary 5.24. Let M(k;T ) =∫ 2π(T−k)

−2π(T+k)

(1−cosu

u

)du. Then, for k 6= `, we can

write Ak` = (−1)k−`

π(k−`) (M(k)−M(`)).

Size estimates on Ak`. We are concerned with the decay of Ak` so we are

concerned with the cases in which both k, ` are large and positive, or both large and

negative – symmetric to the large positive case – or when one is large and positive

and the other is large and negative. We can express the differences M(k)−M(`)

in terms of integrals involving (cosu)/u (cosine terms) and involving 1/u (log

terms). We are justified in separating these terms and treating them as principal

150

value integrals because limu→0

(1− cosu)/u = 0. Our goal is to show that the cosine

terms in M(k)−M(`) are relatively insignificant when k or ` and k − ` is large.

Then we want to show that the log terms look like the products B(k)B(`) with

B(k) ∼ (|k| − T ), or a close variant of them, as discussed above. This will allow

us to show that the norm bound for the remainder term A as discussed above is

essentially sharp.

We have (for k, ` both larger than T and positive, and with ` > k, say)

M(k)−M(`) = M(k;T )

=

∫ 2π(T−k)

−2π(T+k)

−∫ 2π(T−`)

−2π(T+`)

(1− cosu

u

)du

=

∫ 2π(T+`)

0

−∫ 2π(T+k)

0

+

∫ 2π(T−k)

0

−∫ 2π(T−`)

0

(1− cosu

u

)du

=

∫ 2π(T+`)

2π(T+k)

−∫ 0

2π(T−k)

+

∫ 0

2π(T−`)

(1− cosu

u

)du

=

∫ 2π(T+`)

2π(T+k)

+

∫ 2π(k−T )

0

−∫ 2π(`−T )

0

(1− cosu

u

)du

=

∫ 2π(`+T )

2π(k+T )

−∫ 2π(`−T )

2π(k−T )

(1− cosu

u

)du

We can analyze the logarithmic term and the term∫

cosuudu separately. In

particular, we have the following equalities.

151

∫ 2π(n+1)

2πncosuudu

=∫ π

0cosu

(1

(2πn+u)− 1

(2πn+π+u)

)du

=∫ π

0cosu

(π

(2πn+u)(2πn+π+u)

)du

=∫ π/2

0cosu

(π

(2πn+u)(2πn+π+u)− π

(2πn+π−u)(2πn+2π−u)

)du

=∫ π/2

0cosu

(π[(2πn+π−u)(2πn+2π−u)−(2πn+u)(2πn+π+u)]

(2πn+u)(2πn+π+u)(2πn+π−u)(2πn+2π−u)

)du

=∫ π/2

0cosu

(4π2(n+1)(π−u)

(2πn+u)(2πn+π+u)(2πn+π−u)(2πn+2π−u)

)du

Since each term in the denominator of the large factor is at least 2πn it follows

that the integrand is bounded by (n + 1) cosu/(4πn4) so the integral on [0, π/2]

is bounded by (n + 1)/(4πn4). This indicates that the contribution of the cosine

part decays like 1/n3 and the sum over those n larger than some fixed N decays

like 1/N2. We summarize this as follows.

Proposition 5.25. For ` ≥ k > T the contribution of the cosine term found in

M(k)−M(`) is bounded in modulus by a constant (essentially 1/(4π)) times 1/k2

independent of `. Consequently, by Corollary 5.24 the cosine contribution to Ak`

is essentially bounded by 1/(4π2k2(`− k)) when ` > k ≥ T .

The logarithmic terms. We can treat the logarithmic terms in M(k)−M(`) as

principal value integrals, since the integrals defining M(k)−M(`) are absolutely

convergent and the parts involving an integrand 1/u with u small cancel upon

subtraction. As such, bounding M(k)−M(`) reduces to bounding the logarithm

parts of the integrals above. Using these ideas, along with the assumption that

152

` > k > T yields the following approximation.

∫ 2π(`+T )

2π(k+T )

−∫ 2π(`−T )

2π(k−T )

(1

u

)du = log

((`+ T )(k − T )

(`− T )(k + T )

)= log

(k − Tk + T

)− log

(`− T`+ T

)= log

(1− 2T

k + T

)− log

(1− 2T

`+ T

)≈ 2T

`+ T− 2T

k + T

= 2Tk − `

(k + T )(`+ T )

where we have applied the Taylor estimate log(1 + t) = t + o(t2) as t → 0. That

is, the approximation of the log term log((k − T )/(k + T )) is accurate up to a

constant times (2T/(k+T ))2 when k > T and similarly for `. In view of Corollary

5.24 the total contribution of the logarithmic terms to Ak` will be essentially

Lk` =(−1)k−`

π(k − `)(L(k)− L(`))

≈ 2T(−1)k−`

π(k − `)k − `

(k + T )(`+ T )

=(−1)k−`

π

2T

(k + T )(`+ T ).

Putting all these estimates together yields the following.

Proposition 5.26. When ` > k ≥ T one has

Ak` =(−1)k−`

π

2T

(k + T )(`+ T )+O

( 1

k2(`− k)

), as k, `→∞.

153

A closer inspection of the cosine term shows that the second term above is

actually more like 1k2`

. In particular, this error term will always be at most a

fraction of the term 2T(k+T )(`+T )

. This shows that `2-bound for the error matrix A

given by Proposition 5.21 is sharp except possibly for the factor ln(T ).

154

6 CONCLUSION

In our discussion we have learned several methods for dealing with analog

signals in the digital setting. In Chapter 2, we learned that band-limited signals

could be represented by countably infinite samples due to the Fourier Series and

that a good approximation for the Fourier Transform of a Schwartz function can

be found using the Discrete Time Fourier Transform. In addition, we learned

that for f in specific classes of functions, f may be represented by N samples

whose Discrete Fourier Transform consists of N samples of f and also represents

f . For these specific classes of functions, sample trunctation does not lead to error

in the representations of f or f implying that only quantization is necessary to

completely tie the analog signal to its digital counterpart.

In addition to relating four different settings of the Fourier Transform, we

included the distribution theory which is often missing in the literature. This

included Poisson summation formulas for the class of Schwartz functions S (R),

the square integrable functions L2(R) and the tempered distributions S ′(R).

In Chapters 3 and 4, we explored the idea that generalized bases lead to

generalized sampling formulas. Several generalized bases were brought into one

framework giving a new type of characterization. In addition, the gaps between

Shannon sampling and periodic nonuniform sampling were bridged. Familiar con-

cepts in these areas were extended to include tiling sets. An additional level of

155

rigor, often missing in the literature, was used to apply concepts from distribution

theory to show equality in L2(R) for the sampling formulas presented. Finally,

new methods for applying this theory were also introduced.

In Chapter 5, we discussed time-frequency localization operators and the ad-

vantages of using time-frequency localized eigenfunctions. We developed a new

sampling formula which may be applied to a function which is band-limited to

several intervals. Since the key to using this sampling formula is an understand-

ing of eigenvalues, we included an additional discussion regarding eigenvalues.

We reviewed several known concepts, improved certain results and provided new

estimates.

156

7 APPENDIX

This appendix contains Matlab code (Version 7.3) used to generate the figures

presented here. For each section, two procedure based script files were created to

generate the plots. The code to generate the signals and matrices is shown in the

first section and is separated from the code to generate the plots which is shown

in the last section. One reason for separating this code is so that the plots may

be configured without repeating calculations.

Each script file accesses three main types of methods. The first type of method

returns a digital signal representation. The second type of method performs an

algorithm based on theory. These methods where used to create a small library

for the sake of modularization and code reuse. The final type of method is a

configuration method used to help configure the plots. Examples of how to use

latex along with how to save the workspace or figures are included.

In writing code, we attempted to make clear the theory being used. For

example, there is a simple algorithm to compute the FT of a periodic function

such as f(t) = sin(t). This algorithm is demonstrated by Line 50 in the first

Computation Script. First, we time limit the function to one period. Then, we

use the DTFT to compute a Riemann sum approximation to the FT. In many

cases, the DTFT converges uniformly to the FT. Finally, we impulse sample.

One last item to point out is that the parameters for the script files are placed

157

at the beginning of the script. Therefore, one can easily change these parameters

to explore other examples.

7.1 Computation Scripts

1 % This script calculates functions satifying various FT relations

2 % Author S. Izu, NMSU, 2009

3

4 close all; clear;

5 t = −5:.01:5; omega = −60:.01:60;

6 T = .5; Omega = 6;

7 vector = [4 1 2]; %Dimensions are 2 by T*Omega

8

9 %Basic Schwartz Functions

10 rhoA = rho(16*t+6); %Uses rho

11

12 rho = rho(t); %Method creates rho

13 rhohat = DTFT(rho, 100, omega);

14

15 psi = unity(1,0,t); %Partition of Unity

16 psihat = DTFT(psi, 100, omega);

17

18 psi1 = psi1(t); %Method creates psi1

19 psi1hat = DTFT(psi1, 100, omega);

158

20

21 psi2hat = psi2hat(omega); %Method creates psi2hat

22 psi2 = reverse(DTFT(psi2hat, 100, t));

23

24 %psi1 and f satisfy the FSII Equations

25 %f = f1+f2, f1 is a Schwartz function, f2 is an Impulse

26 f1 = rhoA; f1(1,:) = t; f2 = [3/8; .02];

27 fhat = add(DTFT(f1, 100, omega), impulseFT(f2, omega));

28

29 %a and b satisfy FS1 Equations

30 %b, c and d satisfy DFTI Equations

31 a = a(t,Omega*T*ifft(vector),Omega,T); %Method creates a (vector)

32 A = DTFT(a,100,omega);

33

34 b = periodize(a,.5,t);

35 B = Isample(A,T,1);

36

37 c = Isample(b,Omega,1);

38 C = periodize(B, Omega, −8/T:1/T:8/T);

39

40 D = D(omega,T*vector,T,Omega); %Method creates D (vector)

41 d = reverse(DTFT(D,100,t));

42

43 %g0g1 satisfies FSII Equations

44 %g0, c and d satisfy DFTI Equations

159

45 %g0 and g0g1 satisfy DFTII Equations

46 %g0 and g0g2 satsify DFTII Equations

47 %g0g2 satisfies DFTIII Equations

48 g0 = g0(t, vector, Omega, T); %Method creates g0 (vector)

49 g0hat = Isample(DTFT(time limit(g0, 0, T), 100, omega), T, 1);

50

51 g0g1 = multiply(g0, unity(T,2,t));

52 g0g1hat = DTFT(g0g1, 100, omega);

53

54 g0g2 = multiply(g0, g2(t, Omega, T)); %Method creates g2

55 g0g2hat = DTFT(g0g2, 100, omega);

56

57 %Shifts of psi

58 psiRshift = shift(psi, 1);

59 psiLshift = shift(psi, −1);

60

61 %Approximations for f and fhat

62 fApp = reverse(multiply(DTFT(fhat, T, t), psi1));

63 fhatApp = interpolate(fhat, T, psi1hat,1);

64

65 %Alternate function for a

66 Alta = time limit(b, 0, T);

67 AltAhat = DTFT(Alta, 100, omega);

68

69 %Approximations for c and C

160

70 cApp1 = reverse(DTFT(D, T, t));

71 CApp1 = DTFT(b, Omega, omega);

72

73 cApp2 = periodize(d, T, t);

74 CApp2 = periodize(B, Omega, −8/T:1/T:8/T);

75

76 %Approximations for g0 and g0hat

77 g0App1 = interpolate(g0, Omega, psi2, 1);

78 g0hatApp = multiply(DTFT(g0, Omega, omega), psi2hat);

79

80 g0App2 = reverse(DTFT(g0g1hat, T, t));

81 g0App3 = periodize(g0g1, T, t);

82 save('FourierTransform.mat');

1 % This script applies a filtering algorithm to a signal s


3

4 clear; close all;

5 N = 7; skip = 1;

6 [x, y] = textread('lappFile001.dat','%f %f');

7 s = x(1:10000+2ˆN); %s comes from the input file above

8 h = [ 0.03771716; 0.26612218; 0.74557507; ...

9 0.97362811; 0.39763774; −0.35333620; ...

10 −0.27710988; 0.18012745; 0.13160299; ...

161

11 −0.10096657; −0.04165925; 0.04696981; ...

12 5.10043697e−3; −0.01517900; 1.97332536e−3; ...

13 2.81768659e−3; −9.69947840e−4; −1.64709006e−4; ...

14 1.32354367e−4; −1.875841e−5]; %Daubechies 20 filter

15

16 %Compute Filter and denoise signal

17 W = wavelet packet matrix(h, N, N);

18 beforeS = scalogram(s, W, skip);

19 r = denoising filter(beforeS, W);

20 filtered = conv(s,r);

21 afterS = scalogram(filtered, W, skip);

22 save('Wavelet.mat');

1 % This script calculates eigenvalues


3

4 clear; close all; tic;

5 Sigma = [ 0.0 1.3 ;

6 4.5 5.0 ];

7 Omega = 5; K = 1; %Omega = .5; K = 10;

8 beta = [ 0 0 1 1 ] ; %beta = [ 0 1 1 ;

9 % 1 2 2 ;

10 % 2 3 3 ;

11 % 9 10 10 ];

162

12 O = [ 0.0 0.26 ; %O = [ 0.0 0.5 ;

13 0.26 0.5 ; % −0.5 −0.4 ;

14 −0.5 −0.1 ; % −0.4 0.0 ];

15 −0.1 0.0 ];

16 save('g.mat', 'Sigma'); %Used by g.m

17

18 %Step 1: Calculate necessary Intervals and Matrices

19 %I jm = Omega(O m+ beta jm)

20 %[W m] jk = exp(−(2 pi i j beta jm)/K)

21 %(W m)ˆ(−1)

22 J = size(beta,1); M = size(beta,2); %M = size(O,1);

23 const = −2*pi*i/K;

24 for m=0:(M−1)

25 for j=0:(J−1)

26 I(j+1,m+1,1:2) = Omega*(O(m+1,1:2)+beta(j+1,m+1));

27 W(j+1,1:J,m+1) = exp(const*j*beta(1:J,m+1));

28 end

29 Winv(1:J,1:J,m+1) = W(1:J,1:J,m+1)ˆ(−1);

30 end

31 save('h.mat', 'J', 'M', 'I', 'W', 'Winv'); %Used by h.m

32

33 %Step 2: Calculate A

34 %A mn = integral S(FT 1 Sigma(x−m/Omega) * h n(x) dx)

35 M = 2*ceil(Omega*K); N = 2*ceil(Omega*K);

36 A = zeros(2*M+1, 2*N+1);

163

37 Kst = ['(' num2str(Omega) ',' num2str(K) ','];

38 for m=−M:M

39 for n=−M:M

40 %st = h(Omega, K, n, x).*g(Omega, K, m, x)

41 st = ['h' Kst num2str(n) ',x).*g' Kst num2str(m) ',x)'];

42 A(m+M+1,n+N+1) = quadl(st, −.5, .5, 1e−10);

43 end

44 end

45 [V, D] = eig(A); d=real(sort(diag(D)));

46 save('Eigenvalue.mat');

7.2 Fourier Transform Digital Signal Representations

1 function rho = rho(t)

2 % Shwartz function

3 % rho(t) = eˆ(tˆ2/(tˆ2−1)) if −1<t<1


5

6 ind = logical((−1<t).*(t<1));

7 tsquared = t(ind).ˆ2;

8 rho(1,:) = t;

9 rho(2,ind) = exp(tsquared./(tsquared − 1));

164

1 function psi1 = psi1(t)

2 % psi1(t) = psi0(8t+3) + psi0(8t−3)


4

5 psi1(1,:) = t;

6 y1 = unity(.5,2,8*t+3);

7 y2 = unity(.5,2,8*t−3);

8 psi1(2,:) = 3*y1(2,:) + 3*y2(2,:);

1 function psi2hat = psi2hat(omega)

2 % psi2hat(omega)

3 % = rho(omega/2+2)+rho(omega/2−3)+rho(omega/2−5)


5

6 psi2hat(1,:) = omega;

7 y1 = rho(omega/2+2);

8 y2 = rho(omega/2−3);

9 y3 = rho(omega/2−5);

10 psi2hat(2,:) = y1(2,:) + y2(2,:) + y3(2,:);

1 function a = a(t, v, Omega, T)

2 % M = Omega*T

3 % a(t) = sum(m=0:M−1) v(m)*rho(Omega*t−m+mod(m,2)*M)

165

4 % Assumes size(v,2) = M and M is an integer

5 % a(t, [24 6 12], 6, .5) = 24rho(6t) + 6rho(6t+2) + 12rho(6t−2)


7

8 M = Omega*T;

9 a = zeros(2,size(t,2));

10 a(1,:) = t;

11 for m=0:M−1

12 temp = rho(Omega*t−m+mod(m,2)*M);

13 a(2,:) = a(2,:)+ v(m+1)*temp(2,:);

14 end

1 function D = D(omega,v,T,Omega)

2 % M = Omega*T

3 % D(t)

4 % = sum(n=−infty:infty) sum(m=1:M)

5 % v(m)*rho(2ˆ|n |(T*omega−m−N*n)

6 % Assumes size(v,2) = N and N is an integer

7 % D(omega,[24 6 12],6,.5)

8 % = sum(n=−infty:infty) 24 rho(2ˆ|n|(6omega−0−3n))

9 % + 6 rho(2ˆ|n|(6omega−1−3n))

10 % + 12 rho(2ˆ|n|(6omega−2−3n))


12

166

13 M = Omega*T;

14 D = zeros(2,size(omega,2));

15 D(1,:) = omega;

16 flr = floor(omega/Omega);

17 for m=0:M−1

18 temp = rho(2.ˆabs(flr).*(T*omega−m−M*flr));

19 D(2,:) = D(2,:)+v(m+1)*temp(2,:);

20 end

21 temp = rho(2.ˆabs(flr+1).*(T*omega−M−M*flr));

22 D(2,:) = D(2,:)+v(1)*temp(2,:);

1 function g0 = g0(t, v, Omega, T)

2 % M = Omega*T

3 % g0(t) = sum(m=0:M−1) v(m)eˆ(−2*pi*i(m+(−1)ˆm*M)t/T)


5 % g0(t, [24 6 12], 6, .5)

6 % = 6eˆ(−8*pi*i*t)+24eˆ(12*pi*i*t)+12eˆ(20*pi*i*t)


8

9 M = Omega*T;

10 C1 = 2*pi*i/T;

11 g0 = zeros(2,size(t,2));

12 g0(1,:) = t;

13 for m=0:M−1

167

14 g0(2,:) = g0(2,:) + v(m+1)*exp(C1*(m+(−1)ˆm*M)*t);

15 end

1 function g2 = g2(t, Omega, T)

2 % M = Omega*T

3 % g2(t) = sum(m=0:M−1) psi(Omega*t−m+mod(m,2)*M)


5 % g2(t, 6, .5) = psi(6t) + psi(6t+2) + psi(6t−2)


7

8 M = Omega*T;

9 g2 = zeros(2,size(t,2));

10 g2(1,:) = t;

11 for m=0:M−1

12 temp = unity(1,0,Omega*t−m+mod(m,2)*M);

13 g2(2,:) = g2(2,:) + temp(2,:);

14 end

7.3 Fourier Transform Library

1 function added = add(f1,f2)

2 % Adds functions

3 % Assumes domains are equal, ie f1(1,:) = f2(1,:);

168


5

6 added(1,:) = f1(1,:);

7 added(2,:) = f1(2,:) + f2(2,:);

1 function F = DTFT(f,Omega,omega)

2 % Calculates DTFT and may be used to approximate FT

3 % F(omega)

4 % = (1/Omega) sum(n=−N:N) f(n/Omega) eˆ(−2 pi i omega n/Omega)


6

7 F(1,:) = omega;

8 F(2,:) = zeros(1,size(omega,2));

9 deltat = f(1,2) − f(1,1); %Assume evenly spaced sample locations

10 N = floor(max(f(1,:))*Omega);

11 for n=−N:N

12 samp = sample(f,n/Omega,deltat);

13 F(2,:) = F(2,:) + samp*exp(−2*pi*i*omega*n/Omega);

14 end

15 F(2,:) = (1/Omega)*F(2,:);

1 function F = impulseFT(f,omega)

2 % Returns FT for an Impulse Set

169


4

5 F(1,:) = omega;

6 F(2,:) = zeros(1,size(omega,2));

7 for k=1:size(f,2)

8 F(2,:) = F(2,:) + f(2,k).*exp(−2*pi*i*f(1,k).*omega);

9 end

1 function answer = interpolate(f,Omega,psi,multiply)

2 % Interpolates samples of f using translates of psi

3 % Returns sum(n=−N:N) f(n/Omega)*psi(t − n/Omega)

4 % multipy == 1 multiplies summation by 1/Omega


6

7 answer = zeros(size(f));

8 answer(1,:) = psi(1,:);


10 deltat = f(1,2) − f(1,1);

11 for n=−N:N

12 samp = sample(f, n/Omega, deltat);

13 psi shift = shift(psi, n/Omega);

14 answer(2,:) = answer(2,:) + samp.*psi shift(2,:);

15 end

16 answer(2,:) = (1/Omega)ˆ(multiply)*answer(2,:);

170

1 function answer = intervalFT(a,b,t)

2 % Calculates FT of the characteristic function 1 [a,b)


4

5 len = b−a;

6 mid = (a+b)/2;

7 answer = len*exp(−2*pi*i*mid*t).*sinc(len*t);

1 function impulses = Isample(f,Omega,multiply)

2 % Impulse samples f, Returns f(n/Omega)

3 % multiply == 1 multiplies set values by 1/Omega


5


7 deltat = f(1,2) − f(1,1);

8 impulses(1,:) = (1/Omega)*(−N:N);

9 C1 = (1/Omega)ˆ(multiply);

10 for n=−N:N

11 impulses(2, n + N + 1) = C1*sample(f,n/Omega,deltat);

12 end

1 function multiplied = multiply(f1,f2)

2 % Multiplies functions

171

3 % Assumes domains are equal, ie f1(1,:) = f2(1,:);


5

6 multiplied(1,:) = f1(1,:);

7 multiplied(2,:) = f1(2,:).*f2(2,:);

1 function answer = periodize(f, periodL, t)

2 % Periodizes f, Returns sum(−N:N) f(t−n*periodL)

3 % t holds the domain for answer


5

6 %Find location of zero

7 tol = 10ˆ−10;

8 deltat = f(1,2) − f(1,1);

9 samps per period = ceil(periodL/deltat + tol);

10 zeroind = find( abs(f(1,:)) <= deltat/2 + tol);

11

12 %Calculate one period

13 temp(1,:) = f(1,zeroind(1):zeroind(1)+samps per period−1);

14 temp(2,:) = zeros(1,samps per period);

15 for l=1:size(f,2)

16 ind = mod(l−zeroind(1),samps per period)+1;

17 temp(2,ind) = temp(2,ind) + f(2,l);

18 end

172

19

20 %Extend the period to all of t

21 answer(1,:) = t;

22 for l=1:size(answer,2)

23 ind = mod(answer(1,l)+deltat/2,periodL)−deltat/2;

24 answer(2,l) = sample(temp,ind,deltat);

25 end

1 function reversed = reverse(f)

2 % Reverses f, f(−t)


4

5 reversed(1,:) = −1*f(1,size(f,2):−1:1);

6 reversed(2,:) = f(2,size(f,2):−1:1);

1 function samp = sample(f,t,deltat)

2 % Samples f

3 % Returns f(t0) where |t−t0|<deltat

4 % t cannot be a vector


6

7 tol = 10ˆ−10; %Account for machine tolerance

8 samp = f(2,logical( abs(f(1,:)−t) <= deltat/2+tol ));

173

9 samp = samp(1);

1 function shifted = shift(f,shift)

2 % Shifts f, f(t−shift)


4

5 L = size(f,2);

6 shifted(1,:) = f(1,:);

7 shifted(2,:) = zeros(1,L);

8 shiftind = floor(−shift/(f(1,2)−f(1,1)));

9 if shiftind >= 0 %shift left

10 shifted(2, 1:L−shiftind) = f(2, 1+shiftind:L);

11 else

12 shifted(2, 1−shiftind:L) = f(2, 1:L+shiftind);

13 end

1 function y=sinc(x)

2 % y=sin(pi*x)/(pi*x)

3 % Author D. Menemenlis, MIT, 7 feb 94 (289B)

4 % Modified S. Izu, NMSU, 2009

5

6 ix=find(x==0);

7 x(ix)=ones(size(ix));

174

8 y=sin(pi*x)./(pi*x);

9 y(ix)=ones(size(ix));

1 function time limited = time limit(f,a,b)

2 % Time limits f to [a,b]


4

5 time limited(1,:) = f(1,:);

6 time limited(2,:) = (a<=f(1,:)).*(f(1,:)<b).*f(2,:);

1 function unity = unity(T,N,t)

2 % Creates a T−partition of unity which is flat over N periods

3 % The trailing edges have a specific form

4 % y1 = rho(t/T+N/2), y2 = rho(t/T+N/2+1)

5 % y3 = rho(t/T−N/2), y4 = rho(t/T−N/2−1)

6 % unity(t) = y1/((N+1)*(y1+y2)) if −T*N/2−T < t < −T*N/2

7 % unity(t) = 1/(N+1) if −T*N/2 <= t <= T*N/2

8 % unity(t) = y3/((N+1)*(y3+y4)) if T*N/2 < t < T*N/2+T


10

11 unity(1,:) = t;

12 x2 = T*N/2; x3 = x2+T; x1 = −x2; x0 = −x3;

13 y1 = rho(t/T+N/2); y2 = rho(t/T+N/2+1);

175

14 y3 = rho(t/T−N/2); y4 = rho(t/T−N/2−1);

15

16 ind = logical((x0<t).*(t<x1));

17 unity(2,ind) = y1(2,ind)./(y1(2,ind)+y2(2,ind));

18

19 ind = logical((x1<=t).*(t<=x2));

20 unity(2,ind) = 1;

21

22 ind = logical((x2<t).*(t<x3));

23 unity(2,ind) = y3(2,ind)./(y3(2,ind)+y4(2,ind));

24

25 unity(2,:) = unity(2,:)/(N+1);

7.4 Wavelet Library

1 function W = wavelet packet matrix(h, N, N 0)

2 % Generates the 2ˆN by 2ˆN Wavelet Packet matrix W

3 % h is the scaling filter

4 % N 0 <= N is the number of levels in the wavelet tree


6

7 W = eye(2ˆN);

8 for n=0:N 0−1

9 %Place wavelet matrices along the diagonal of W level n

176

10 C1 = 2ˆ(N−n);

11 indeces = 1:C1;

12 W level n = zeros(2ˆN);

13 W level n(1:C1,1:C1) = wavelet matrix(h, N−n);

14 for i=2:2ˆn

15 indeces = indeces + C1;

16 W level n(indeces, indeces) = W level n(1:C1,1:C1);

17 end

18 W = W level n*W;

19 end

20

21 %Account for spectral flipping

22 W = W(spectral flip(N),:);

1 function W N = wavelet matrix(h, N)

2 % Generates the 2ˆN by 2ˆN Wavelet Transform matrix W N

3 % h is the scaling filter whose indeces are 0 through L−1

4 % W N combines Approximation H N and Detail G N


6

7 %Zeropad h so its length is a multiple of 2ˆN

8 L = size(h,1); C1 = 2ˆN; cols = ceil(L/C1);

9 h = [h; zeros(cols*C1−L,1)];

10

177

11 %Calculate first row of Approximation H N and Detail G N

12 H N = zeros(C1/2,C1); G N = H N;

13 H N(1,:) = sum(reshape(h, C1, cols), 2);

14 G N(1,:) = (−1).ˆ(0:C1−1).* conj(H N(1,[2 1 C1:−1:3]));

15

16 %Shift each row two indeces to right to obtain next row

17 indeces = [C1−1 C1 1:C1−2];

18 for m = 2:C1/2

19 H N(m,:) = H N(m−1,indeces);

20 G N(m,:) = G N(m−1,indeces);

21 end

22 W N = [H N; G N];

1 function indeces = spectral flip(N)

2 % Calculates indeces according to spectral flipping

3 % N is the number of levels in the tree


5

6 indeces = 1;

7 for i=1:N

8 J = size(indeces,2);

9 indeces(2*J:−1:J+1) = indeces+J;

10 end

11 indeces = indeces(size(indeces,2):−1:1);

178

1 function r = denoising filter(S, W)

2 % Calculates the filter for the denoising algorithm

3 % S is a Scalogram matrix

4 % W is a Wavelet matrix


6

7 N = log2(size(W,1));

8 MeanS = mean(S,2);

9 Winv = pinv(W);

10 r = zeros(2ˆ(N+1)−1,1);

11 for n=1−2ˆN:2ˆN−1

12 if( n <= 0)

13 lvect = 1−n:2ˆN;

14 else

15 lvect = 1:2ˆN−n;

16 end

17 outersum = 0;

18 for m=1:2ˆN

19 innersum = 0;

20 for l=lvect

21 innersum = innersum + Winv(l,m)*W(m,l+n);

22 end

23 outersum = outersum + (1/MeanS(m))*innersum;

24 end

179

25 r(n+2ˆN) = 1/(2ˆN)*outersum;

26 end

1 function S = scalogram(s, W, skip)

2 % Calculates the scalogram matrix

3 % s is the input signal

4 % W is the Wavelet matrix

5 % skip >= 1 gives the number of blocks to skip


7

8 C1 = size(W,1);

9 C2 = size(s,1)−C1+1;

10 S = zeros(C1, C2);

11 for i = 1:skip:C2

12 indeces = i:(i+C1−1);

13 S(:,i) = abs(W*s(indeces));

14 end

7.5 Eigenvalue Calculations

1 function h = h(Omega, K, N, x)

2 % Assumes h.mat contains J,M,I,W and Winv


180

4

5 load('h.mat');

6 h = 0;

7 j = mod(N,K); %N = j + nK = mod(N,K) + floor(N/K)*K

8 if(0 <= j && j < J)

9 t = N/(Omega*K)−x;

10 for m=0:(M−1)

11 for k=0:(J−1)

12 C = conj(Winv(k+1,j+1,m+1)*W(j+1,k+1,m+1));

13 h = h + C*intervalFT(I(k+1,m+1,1),I(k+1,m+1,2),t);

14 end

15 end

16 h = (1/Omega)*h;

17 end

1 function g = g(Omega, K, m, x)

2 % Assumes g.mat contains Sigma


4

5 load('g.mat');

6 g = 0;

7 t = x−m/(Omega*K);

8 for l=1:size(Sigma)

9 g = g + intervalFT(Sigma(l,1),Sigma(l,2), t);

181

10 end

7.6 Plots and Configuration

1 % This script plots functions satifying various FT relations


3

4 close all; clear;

5 flag = 1; %0−use sub plots, 1−separate figures

6

7 load('FourierTransform.mat');

8 taxis = [−.6 .6 −1.5 1.5]; Omegaaxis = [−12 12 −5 5];

9 taxisI = taxis.*[1 1 Omega Omega]; %Time Impulse Sample

10 OmegaaxisI = Omegaaxis.*[1 1 T T]; %Frequency Impulse Sample

11

12 %Basic Schwartz functions

13 figure(1); if(˜flag); subplot(2,1,1); end; hold on;

14 plotf(rho,['−b';'−r'],0); %rho

15 label([−1.1 1.1 −.1 1.1], '$t$', '$\rho(t)$');

16 if(flag); saveas(gcf, 'images/rho.png'); end

17

18 if(flag); figure(2); else subplot(2,1,2); end; hold on;

19 plotf(rhohat,['−b';'−r'],0); %rhohat

20 label([−6 6 −.2 1.4], '$\omega$', '$\widehat\rho(\omega)$');

182

21 if(flag); saveas(gcf, 'images/rhohat.png'); end

22


24 plotf(psi,['−b';'−r'],0); %psi

25 label([−1.1 1.1 −.1 1.1], '$t$', '$\psi(t)$');

26 if(flag); saveas(gcf, 'images/psi.png'); end

27


29 plotf(psihat,['−b';'−r'],0); %psihat

30 label([−6 6 −.2 1.4], '$\omega$', '$\widehat\psi(\omega)$');

31 if(flag); saveas(gcf, 'images/psihat.png'); end

32


34 plot([−2.1 2.1], [1 1], 'k');

35 plotf(psi,['−b';'−r'],0); %psi(t)

36 plotf(psiRshift,['−g';'−r'],0); %psi(t−1)

37 plotf(psiLshift,['−m';'−r'],0); %psi(t+1)

38 label([−2.1 2.1 −.1 1.1], '$t$', '$\psi(t+1),\psi(t),\psi(t−1)$');

39 if(flag); saveas(gcf, 'images/psi shifts.png'); end

40

41 %FSII Equations


43 plotf(psi1,['−b';'−r'],0); %psi

44 label([−1 1 −.1 1.1], '$t$', '$\psi(t)$');

45 if(flag); saveas(gcf, 'images/fs psi.png'); end

183

46


48 plotf(psi1hat,['−b';'−r'],0); %psihat

49 label([−30 30 −.4 .4], '$\omega$', '$\widehat\psi(\omega)$');

50 if(flag); saveas(gcf, 'images/fs psihat.png'); end

51


53 plotf(f1,['−b';'−r'],0); %Schwartz component of f

54 plotf(f2,['−b';'−r'],1); %Impulse component of f

55 label([−1 1 −.1 1.1], '$t$', '$f(t)$');

56 if(flag); saveas(gcf, 'images/fs f.png'); end

57


59 plotf(fhat,['−b';'−r'],0); %fhat = f1hat+f2hat

60 label([−30 30 −.1 .1], '$\omega$', '$\widehatf(\omega)$');

61 if(flag); saveas(gcf, 'images/fs fhat.png'); end

62


64 plotf(fApp,['−b';'−r'],0); %f approx

65 label([−1 1 −.4 1.4], '$t$', '$\tildef(t)$');

66 if(flag); saveas(gcf, 'images/fs fapp.png'); end

67


69 plotf(fhatApp,['−b';'−r'],0); %fhat approx

70 label([−30 30 −.1 .1],'$\omega$','$\tilde\widehatf(\omega)$');

184

71 if(flag); saveas(gcf, 'images/fs fhatapp.png'); end

72

73 %First Plot, FS Equations


75 plotf(a,['−b';'−r'],0); %a

76 label(taxisI, '$t$', '$a(t)$');

77 if(flag); saveas(gcf, 'images/a.png'); end

78


80 plotf(A,['−b';'−r'],0); %A

81 plotf(Isample(A,T,0),['−−b';'−−r'],1); %A sampled

82 label(OmegaaxisI, '$\omega$', '$A(\omega)$');

83 if(flag); saveas(gcf, 'images/ahat.png'); end

84


86 plotf(b,['−b';'−r'],0); %b

87 plotf(Isample(b,Omega,0),['−−b';'−−r'],1); %b sampled

88 label(taxisI, '$t$', '$b(t)$');

89 if(flag); saveas(gcf, 'images/b.png'); end

90


92 plotf(B,['−b';'−r'],1); %B

93 label(Omegaaxis, '$\omega$', '$B(\omega)$');

94 if(flag); saveas(gcf, 'images/bhat.png'); end

95

185


97 plotf(c,['−b';'−r'], 1); %c

98 label(taxis, '$t$', '$c(t)$');

99 if(flag); saveas(gcf, 'images/c.png'); end

100


102 plotf(C,['−b';'−r'],1); %C

103 label(Omegaaxis, '$\omega$', '$C(\omega)$');

104 if(flag); saveas(gcf, 'images/chat.png'); end

105


107 plotf(d,['−b';'−r'],0); %d

108 label([−.75 .75 −30 30], '$t$', '$d(t)$');

109 if(flag); saveas(gcf, 'images/d.png'); end

110


112 plotf(D,['−b';'−r'],0); %D

113 plotf(Isample(D,T,0),['−−b';'−−r'],1); %D sampled

114 label(OmegaaxisI, '$\omega$', '$D(\omega)$');

115 if(flag); saveas(gcf, 'images/dhat.png'); end

116

117 %Secont Plot, FS Equations


119 plotf(g0g1,['−b';'−r'],0); %g0g1

120 label(taxisI, '$t$', '$g 0 g 1(t)$');

186

121 if(flag); saveas(gcf, 'images/g0g1.png'); end

122


124 plotf(g0g1hat,['−b';'−r'],0); %g0g1hat

125 plotf(Isample(g0g1hat,T,0),['−−b';'−−r'],1); %g0g1hat sampled

126 label(OmegaaxisI, '$\omega$', '$\widehatg 0 g 1(\omega)$');

127 if(flag); saveas(gcf, 'images/g0g1hat.png'); end

128


130 plotf(g0,['−b';'−r'],0); %g0

131 plotf(Isample(g0,Omega,0) ,['−−b';'−−r'],1); %g0 sampled

132 label(taxisI, '$t$', '$g 0(t)$');

133 if(flag); saveas(gcf, 'images/g0.png'); end

134


136 plotf(g0hat,['−b';'−r'],1); %g0

137 label(Omegaaxis, '$\omega$', '$\widehatg 0(\omega)$');

138 if(flag); saveas(gcf, 'images/g0hat.png'); end

139


141 plotf(g0g2,['−b';'−r'],0); %g0g2

142 plotf(Isample(g0g2,Omega,0),['−−b';'−−r'],1); %g0g2 sampled

143 label(taxisI, '$t$', '$g 0 g 2(t)$');

144 if(flag); saveas(gcf, 'images/g0g2.png'); end

145

187


147 plotf(g0g2hat,['−b';'−r'],0); %g0g2hat

148 plotf(Isample(g0g2hat,T,0),['−−b';'−−r'],1); %g0g2hat sampled

149 label(OmegaaxisI, '$\omega$', '$\widehatg 0 g 2(\omega)$');

150 if(flag); saveas(gcf, 'images/g0g2hat.png'); end

151


153 plotf(Isample(g0g2,Omega,1),['−b';'−r'], 1); %g0g2 Discrete

154 label(taxis, '$t$', '$(1/\Omega)g 0 g 2(n/\Omega)$');

155 if(flag); saveas(gcf, 'images/g0g2D.png'); end

156


158 plotf(Isample(g0g2hat,T,1),['−b';'−r'],1); %g0g2hat Discrete

159 label(Omegaaxis, '$\omega$', '$(1/T)\widehatg 0 g 2(m/T)$');

160 if(flag); saveas(gcf, 'images/g0g2hatD.png'); end

161

162 %Third Plot, FS Equations, Alternate A


164 plotf(Alta,['−b';'−r'],0); %Alta

165 label(taxisI, '$t$', '$A(t)$');

166 if(flag); saveas(gcf, 'images/alta.png'); end

167


169 plotf(AltAhat,['−b';'−r'],0); %Ahat

170 plotf(Isample(AltAhat,T,0),['−−b';'−−r'],1); %Ahat sampled

188

171 label(OmegaaxisI, '$\omega$', '$A(\omega)$');

172 if(flag); saveas(gcf, 'images/altahat.png'); end

173

174 %Fourth Plot, FS Equations, Approximate c and C


176 plotf(cApp1,['−b';'−r'],0); %D samples interpolated

177 label([−.75 .75 −150 150], '$t$', '$c(t)$');

178 if(flag); saveas(gcf, 'images/capp.png'); end

179


181 plotf(CApp1,['−b';'−r'],0); %b samples interpolated

182 label([−12 12 −45 45],'$\omega$','$C(\omega)$');

183 if(flag); saveas(gcf, 'images/chatapp.png'); end

184


186 plotf(cApp2,['−b';'−r'],0); %d periodized

187 label([−.75 .75 −90 90], '$t$', '$c(t)$');

188 if(flag); saveas(gcf, 'images/capp2.png'); end

189


191 plotf(CApp2,['−b';'−r'],1); %B periodized

192 label(Omegaaxis , '$\omega$', '$C(\omega)$');

193 if(flag); saveas(gcf, 'images/chatapp2.png'); end

194

195 %Fifth Plot, FS Equations, Approximate g0 and g0hat

189


197 plotf(g0App1,['−b';'−r'],0); %g0 samples interpolated

198 label(taxisI, '$t$', '$g 0(t)$');

199 if(flag); saveas(gcf, 'images/g0app.png'); end

200


202 plotf(g0hatApp,['−b';'−r'],0); %g0 samples interpolated

203 label([−12 12 −50 50],'$\omega$','$\widehatg 0(\omega)$');

204 if(flag); saveas(gcf, 'images/g0hatapp.png'); end

205


207 plotf(g0App2,['−b';'−r'],0); %g0g1hat samples interpolated

208 label(taxisI, '$t$', '$g 0(t)$');

209 if(flag); saveas(gcf, 'images/g0app2.png'); end

210


212 plotf(g0App3,['−b';'−r'],0); %g0*g 1 periodized

213 label(taxisI, '$t$', '$g 0(t)$');

214 if(flag); saveas(gcf, 'images/g0app3.png'); end

1 function plotf(f,color,impulses)

2 % Plots distribution f

3 % color(1) is the color to plot real(f)

4 % color(2) is the color to plot imag(f)

190

5 % impulses == 1 will treat f like a set of impulses

6

7 if(impulses) %Distribution is set of impulses

8 for i=1:size(f,2)

9 loc = f(1,i); weight = f(2,i);

10 wt = [real(weight) imag(weight)];

11 %Plot longer impulse first

12 imagFirst = wt(1)*wt(2)*(abs(wt(2))−abs(wt(1)))>0;

13 ind = mod(imagFirst,2)+1;

14 plot([loc loc], [0, wt(ind)], color(ind,:));

15 ind = mod(imagFirst+1,2)+1;

16 plot([loc loc], [0, wt(ind)], color(ind,:));

17 end

18 else %Distribution is regular function

19 plot(f(1,:),real(f(2,:)),color(1,:));

20 plot(f(1,:),imag(f(2,:)),color(2,:));

21 end

1 function label(Axis,xstring,ystring)

2 % Use latex to place x−axis/y−axis labels


4

5 axis(Axis);

6

191

7 xloc = (Axis(1)+Axis(2))/2;

8 yloc = (Axis(3)−(Axis(4)−Axis(3))/10);

9 H = text(xloc,yloc,xstring);

10 set(H, 'Interpreter', 'latex');

11 set(H, 'FontSize', 14);

12 set(H, 'HorizontalAlignment','Center');

13

14 % xloc = (Axis(1)−(Axis(2)−Axis(1))/5); %Alt option

15 xloc = (Axis(1)−(Axis(2)−Axis(1))/10);

16 yloc = (Axis(3)+Axis(4))/2;

17 H = text(xloc,yloc,ystring);

18 set(H, 'Rotation', 90);

19 set(H, 'Interpreter', 'latex');

20 set(H, 'FontSize', 14);

21 set(H, 'HorizontalAlignment','Center');

1 % This script plots a signal s before and after denoising


3

4 load('Wavelet.mat');

5 figure(1); %Plot the scaling filter h

6 plot(h); xlabel('n'); ylabel('h n');

7 saveas(gcf, 'images/h.png');

8

192

9 figure(2); %Calculate the Wavelet packet matrix W

10 imagesc(W);

11 saveas(gcf, 'images/W.png');

12

13 figure(3); %Plot the original signal

14 plot(s); xlabel('n'); ylabel('s n');

15 saveas(gcf, 'images/s.png');

16

17 figure(4); %Plot the scalogram of the original signal

18 imagesc(beforeS); colorbar;

19 saveas(gcf, 'images/sScalogram.png');

20

21 figure(5); %Calculate the filter r

22 plot(r); xlabel('n'); ylabel('r n');

23 saveas(gcf, 'images/filter.png');

24

25 figure(6); %Plot the filtered signal

26 plot(filtered); xlabel('n'); ylabel('r*s n');

27 saveas(gcf, 'images/filtereds.png');

28

29 figure(7); %Plot the scalogram of the filtered signal

30 imagesc(afterS); colorbar;

31 saveas(gcf, 'images/filteredsScalogram.png');

193

1 % This script plots eigenvalues


3

4 load('Eigenvalue.mat');

5 figure(1); imagesc(abs(A));

6 figure(2); plot(1:size(D,2),d,'x');

7 display(['sum = ' num2str(sum(d)) ' time = ' num2str(toc)]);

194

REFERENCES

[1] M. Laura Arias, Gustavo Corach, and Miriam Pacheco. Characterization ofBessel sequences. Extracta Math., 22(1):55–66, 2007.

[2] John J. Benedetto and Shidong Li. The theory of multiresolution analy-sis frames and applications to filter banks. Appl. Comput. Harmon. Anal.,5(4):389–427, 1998.

[3] J. L. Jr. Brown. Sampling expansions for multiband signals. IEEE. Trans.Acoustics., 33(1):312–315, 1985.

[4] Peter G. Casazza and Ole Christensen. Frames and Schauder bases. InApproximation theory, volume 212 of Monogr. Textbooks Pure Appl. Math.,pages 133–139. Dekker, New York, 1998.

[5] Peter G. Casazza, Ole Christensen, and Nigel J. Kalton. Frames of translates.Collect. Math., 52(1):35–54, 2001.

[6] Peter G. Casazza and Gitta Kutyniok. Frames of subspaces. In Wavelets,frames and operator theory, volume 345 of Contemp. Math., pages 87–113.Amer. Math. Soc., Providence, RI, 2004.

[7] Peter G. Casazza, Gitta Kutyniok, and Mark C. Lammers. Duality principlesin frame theory. J. Fourier Anal. Appl., 10(4):383–408, 2004.

[8] Ole Christensen. Frames and pseudo-inverses. J. Math. Anal. Appl.,195(2):401–414, 1995.

[9] Ole Christensen. Frames, Riesz bases, and discrete Gabor/wavelet expan-sions. Bull. Amer. Math. Soc. (N.S.), 38(3):273–291 (electronic), 2001.

[10] Ole Christensen. An introduction to frames and Riesz bases. Applied andNumerical Harmonic Analysis. Birkhauser Boston Inc., Boston, MA, 2003.

[11] A. Cohen and Ingrid Daubechies. Orthonormal bases of compactly supportedwavelets. III. Better frequency resolution. SIAM J. Math. Anal., 24(2):520–527, 1993.

[12] Ingrid Daubechies. Ten lectures on wavelets, volume 61 of CBMS-NSF Re-gional Conference Series in Applied Mathematics. Society for Industrial andApplied Mathematics (SIAM), Philadelphia, PA, 1992.

[13] S. Haykin and B. Veen. Signals and systems. John Wiley & Sons Inc., NewYork, 1997.

195

[14] R. F. Hoskins and J. Sousa Pinto. Theories of generalised functions. HorwoodPublishing Limited, Chichester, 2005. Distributions, ultradistributions andother generalised functions, Revised reprint of the 1994 original.

[15] Abram R. Jacobson and Xuan-Min Shao. Polarization observations of broad-band vhf signals by the forte satellite. Radio Science, 36(6):1573–1589, 2001.

[16] Kedar Khare and Nicholas George. Sampling theory approach to prolatespheroidal wavefunctions. J. Phys. A, 36(39):10011–10021, 2003.

[17] H. J. Landau. The eigenvalue behavior of certain convolution equations.Trans. Amer. Math. Soc., 115:242–256, 1965.

[18] H. J. Landau. On the density of phase-space expansions. IEEE Trans. Inform.Theory, 39(4):1152–1155, 1993.

[19] H. J. Landau and H. Widom. Eigenvalue distribution of time and frequencylimiting. J. Math. Anal. Appl., 77(2):469–481, 1980.

[20] Michael Reed and Barry Simon. Methods of modern mathematical physics.II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jo-vanovich Publishers], New York, 1975.

[21] Walter Rudin. Functional analysis. McGraw-Hill Book Co., New York, 1973.McGraw-Hill Series in Higher Mathematics.

[22] Walter Rudin. Fourier analysis on groups. Wiley Classics Library. JohnWiley & Sons Inc., New York, 1990. Reprint of the 1962 original, A Wiley-Interscience Publication.

[23] Charles Swartz. Measure, integration and function spaces. World ScientificPublishing Co. Inc., River Edge, NJ, 1994.

[24] Raman Venkataramani and Yoram Bresler. Perfect reconstruction formulasand bounds on aliasing error in sub-Nyquist nonuniform sampling of multi-band signals. IEEE Trans. Inform. Theory, 46(6):2173–2183, 2000.

[25] David F. Walnut. An introduction to wavelet analysis. Applied and NumericalHarmonic Analysis. Birkhauser Boston Inc., Boston, MA, 2002.

[26] G. Walter and T. Soleski. A new friendly method of computing prolatespheroidal wave functions and wavelets. Appl. Comput. Harmon. Anal.,19(3):432–443, 2005.

[27] Gilbert G. Walter and Xiaoping A. Shen. Sampling with prolate spheroidalwave functions. Sampl. Theory Signal Image Process., 2(1):25–52, 2003.

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