chromatic derivatives

8/8/2019 Chromatic Derivatives

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Chromatic Derivatives, Chromatic Expansions and Associated Spaces(Extended Abstract)

Aleksandar IgnjatovicTo appear in: East Journal on Approximations

Let BL() be the space of continuous 2 functions with the Fourier transform supported within [, ] (i.e.,the space of band limited signals of finite energy), and let () be obtained by normalizing and scaling theLegendre polynomials, so that 1

2

()

()d = ( ).

We consider linear differential operators = (i)

i dd

; for such operators and every BL(),

[]() =

1

2

i ()

()eid.

We show that for BL() the values of[]() can be obtained in a numerically accurate and noiserobust way from samples of (), even for differential operators of high order.

Operators have the following remarkable properties, relevant for applications in digital signal processing.Proposition 0.1. Let : R R be a restriction of any entire function; then the following are equivalent:(a)

=0[](0)2 < ;(b) for all R the sum

=0[]()2 converges, and its values are independent of R;(c) BL().

Moreover, the following Proposition provides local representation of the usual norm, the scalar productand the convolution in BL().

Proposition 0.2. For all ,

BL() the following sums do not depend on R, and

=0

[]()2 =

()2;

=0

[]()[]() =

()();

=0

[]() [( )] =

()( ).

The following proposition provides a form of Taylors theorem, with the differential operators replacingthe derivatives and the spherical Bessel functions replacing the monomials.

Proposition 0.3. Let be the spherical Bessel functions of the first kind; then:

(1) for every entire function and for all C,() =

=0

(1)[](0)[0()] ==0

[](0) 2 + 1 ();

(2) if BL(), then the series converges uniformly onR and in 2.We give analogues of the above theorems for very general families of orthogonal polynomials. We also intro-

duce some nonseparable inner product spaces. In one of them, related to the Hermite polynomials, functions() = sin for all > 0 have finite positive norms and for every two distinct values 1 = 2 the corre-sponding functions 1() = sin 1 and 2() = sin 2 are mutually orthogonal. Related to the properties ofsuch spaces, we also make the following conjecture for families of orthonormal polynomials.

Conjecture 0.4. Let () be a family of symmetric positive definite orthonormal polynomials correspondingto a moment distribution function(),

() () d() = ( ),

and let > 0 be the recursion coefficients in the corresponding three term recurrence relation for suchorthonormal polynomials, i.e., such that

+1() =

() 1

1().

If satisfy 0 < lim

< for some 0 < 1, then 0 < lim

1

1

1=0

()2 < for all in the

support () of ().

Numerical tests with = for many [0, 1) indicate that the conjecture is true.

1


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CHROMATIC DERIVATIVES, CHROMATIC EXPANSIONS AND

ASSOCIATED SPACES

ALEKSANDAR IGNJATOVIC

Abstract. This paper presents the basic properties of chromatic derivativesand chromatic expansions and provides an appropriate motivation for intro-

ducing these notions. Chromatic derivatives are special, numerically robustlinear differential operators which correspond to certain families of orthogo-

nal polynomials. Chromatic expansions are series of the corresponding specialfunctions, which possess the best features of both the Taylor and the Shan-non expansions. This makes chromatic derivatives and chromatic expansions

applicable in fields involving empirically sampled data, such as digital signaland image processing.

1. MotivationSignal processing mostly deals with the signals which can be represented by con-

tinuous 2 functions whose Fourier transform is supported within [, ]; thesefunctions form the space BL() of band limited signals of finite energy. Foun-dations of classical digital signal processing rest on the WhittakerKotelnikovNyquistShannon Sampling Theorem (for brevity the Shannon Theorem): everysignal BL() can be represented using its samples at integers and the cardinalsine function sinc = sin /, as

(1) () =

=

() sinc ( ).

Such signal representation is of global nature, because it involves samples of the

signal at integers of arbitrarily large absolute value. In fact, since for a fixed thevalues of sinc() decrease slowly as grows, the truncations of the above seriesdo not provide satisfactory local signal approximations.

On the other hand, since every signal BL() is a restriction to R of an entirefunction, it can also be represented by the Taylor series,

(2) () =

=0

()(0)

!.

Such a series converges uniformly on every finite interval, and its truncations providegood local signal approximations. Since the values of the derivatives ()(0) aredetermined by the values of the signal in an arbitrarily small neighborhood of zero,

2000 Mathematics Subject Classification. 41A58, 42C15, 94A12, 94A20.Key words and phrases. chromatic derivatives, chromatic expansions, orthogonal systems, orthog-onal polynomials, special functions, signal representation.Some of the results from this paper were presented at UNSW Research Workshop on FunctionSpaces and Applications, Sydney, December 2-6, 2008, and at SAMPTA 09, Marseille, May 18-22,

2009.

1


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2 ALEKSANDAR IGNJATOVIC

the Taylor expansion is of local nature. In this sense, the Shannon and the Taylorexpansions are complementary.

However, unlike the Shannon expansion, the Taylor expansion has found verylimited use in signal processing, due to several problems associated with its appli-cation to empirically sampled signals.

(I) Numerical evaluation of higher order derivatives of a function given by itssamples is very noise sensitive. In general, one is cautioned against numericaldifferentiation:

. . . numerical differentiation should be avoided whenever possible,particularly when the data are empirical and subject to appreciableerrors of observation [10].

(II) The Taylor expansion of a signal BL() converges non-uniformly on R;its truncations have rapid error accumulation when moving away from thecenter of expansion and are unbounded.

(III) Since the Shannon expansion of a signal BL() converges to in BL(),the action of a continuous linear shift invariant operator (in signal processingterminology, a filter) can be expressed using samples of and the impulseresponse [sinc] of :

(3) []() =

=

() [sinc]( ).In contrast, the polynomials obtained by truncating the Taylor series do notbelong to BL() and nothing similar to (3) is true of the Taylor expansion.

Chromatic derivatives were introduced in [11] to overcome problem (I) above; thechromatic approximations were introduced in [14] to obtain local approximationsof band-limited signals which do not suffer from problems (II) and (III).

1.1. Numerical differentiation of band limited signals. To understand theproblem of numerical differentiation of band-limited signals, we consider an arbi-

trary BL() and its Fourier transform (); then()() =

1

2

(i ) ()ei .

Figure 1 (left) shows, for = 15 to = 18, the plots of (/), which are, save afactor ofi, the symbols, or, in signal processing terminology, the transfer functionsof the normalized derivatives 1/ d/d. These plots reveal why there can be nopractical method for any reasonable approximation of derivatives of higher orders.Multiplication of the Fourier transform of a signal by the transfer function of anormalized derivative of higher order obliterates the Fourier transform of the signal,leaving only its edges, which in practice contain mostly noise. Moreover, the graphsof the transfer functions of the normalized derivatives of high orders and of thesame parity cluster so tightly together that they are essentially indistinguishable;see Figure 1 (left).1

However, contrary to a common belief, these facts do not preclude numericalevaluation of all differential operators of higher orders, but only indicate that, from

a numerical perspective, the set of the derivatives {, , , . . .} is a very poor base1If the derivatives are not normalized, their values can be very large and are again determined

essentially by the noise present at the edge of the bandwidth of the signal.


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CHROMATIC DERIVATIVES, CHROMATIC EXPANSIONS AND ASSOCIATED SPACES 3

2

2

0.5

0.5

2

2

2

1

1

2

Figure 1. Graphs of

(left) and of () (right) for = 15 18.

of the vector space of linear differential operators with real coefficients. We nowshow how to obtain a base for this space consisting of numerically robust lineardifferential operators.

1.2. Chromatic derivatives. Let polynomials () be obtained by normalizingand scaling the Legendre polynomials, so that

1

2 () ()d = ( ).We define operator polynomials2

= (i)

id

d

.

Since polynomials () contain only powers of the same parity as , operators have real coefficients, and it is easy to verify that

[ei ] = i () ei .Consequently, for BL(),

[]() = 12

i ()

() ei d.

In particular, one can show that

(4) [sinc ]() = (1) 2 + 1 j(),where j() is the spherical Bessel function of the first kind of order . Figure 1(right) shows the plots of (), for = 15 to = 18, which are the transferfunctions (again save a factor of i) of the corresponding operators . Unlike thetransfer functions of the (normalized) derivatives 1/ d/d, the transfer func-tions of the chromatic derivatives form a family of well separated, interleavedand increasingly refined comb filters. Instead of obliterating, such operators encodethe features of the Fourier transform of the signal (in signal processing terminol-ogy, the spectral features of the signal). For this reason, we call operators thechromatic derivatives associated with the Legendre polynomials.

Chromatic derivatives can be accurately and robustly evaluated from samples

of the signal taken at a small multiple of the usual Nyquist rate. Figure 2 (left)

2Thus, obtaining involves replacing in () with i

d/d for all . If is applied

to a function of a single variable, we drop index in .


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2

4

4

2

3

2

1

1

2

3

2

4

4

2

1

1

2

1

2

1

Figure 2. Transfer functions of 15 (left, black) and 15/15(right, black) and of their transversal filter approximations (gray).

shows the plots of the transfer function of a transversal filter given by 15[]() =64=64 (+/2) (gray), used to approximate the chromatic derivative 15[](),

and the transfer function of 15 (black). The coefficients of the filter were ob-tained using the Remez exchange method [16], and satisfy

< 0.2, (

64

64). The filter has 129 taps, spaced two taps per Nyquist rate interval, i.e., at adistance of 1/2. Thus, the transfer function of the corresponding ideal filter 15is 15(2) for /2, and zero outside this interval. The pass-band of the ac-tual transversal filter is 90% of the bandwidth [/2, /2]. Outside the transitionregions [11/20, 9/20] and [9/20, 11/20] the error of approximation is lessthan 1.3 104.

Implementations of filters for operators of orders 0 24 have been testedin practice and proved to be both accurate and noise robust, as expected from theabove considerations.

For comparison, Figure 2 (right) shows the transfer function of a transversal filterobtained by the same procedure and with the same bandwidth constraints, whichapproximates the (normalized) standard derivative (2/)15 d15/d15 (gray) andthe transfer function of the ideal filter (black). The figure clearly indicates thatsuch a transversal filter is of no practical use.

Note that (1) and (4) imply that(5)

[]() =

=

() [sinc]( ) =

=

() (1)

2 + 1 j(( )).

However, in practice, the values of[](), especially for larger values of, cannotbe obtained from the Nyquist rate samples using truncations of (5). This is due tothe fact that functions [sinc]( ) decay very slowly as grows; see Figure 3(left). Thus, to achieve any accuracy, such a truncation would need to containan extremely large number of terms. On the other hand, this also means thatsignal information present in the values of the chromatic derivatives of a signal

obtained by sampling an appropriate filterbank at an instant is not redundantwith information present in the Nyquist rate samples of the signal in any reasonablysized window around , which is a fact suggesting that chromatic derivatives couldenhance standard signal processing methods operating on Nyquist rate samples.


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15 10 5 5 10 15

0.4

0.2

0.2

0.4

0.6

0.8

1.0

10 5 5 10

1.5

1.0

0.5

0.5

1.0

1.5

Figure 3. left: Oscillatory behavior of sinc () (black), and15[sinc ]() (gray); right: A signal BL() (gray) and itschromatic and Taylor approximations (black, dashed)

1.3. Chromatic expansions. The above shows that numerical evaluation of thechromatic derivatives associated with the Legendre polynomials does not sufferproblems which precludes numerical evaluation of the standard derivatives ofhigher orders. On the other hand, the chromatic expansions, defined in Proposi-

tion 1.1 below, were conceived as a solution to problems associated with the use ofthe Taylor expansion.3

Proposition 1.1. Let be the chromatic derivatives associated with the Legendrepolynomials, let j be the spherical Bessel function of the first kind of order , andlet be an arbitrary entire function; then for all , C,

() =

=0

[]() [sinc ( )](6)

=

=0

(1) []() [sinc ]( )(7)

=

=0 []()

2 + 1 j((

))(8)

If BL(), then the series converges uniformly onR and in the space BL().The series in (6) is called the chromatic expansion of associated with the Le-

gendre polynomials; a truncation of this series is called a chromatic approximationof . As the Taylor approximation, a chromatic approximation is also a local ap-proximation; its coefficients are the values of differential operators []() at asingle instant , and for all ,

()() =d

d

=0

[]() [sinc ( )]

=

.

Figure 3 (right) compares the behavior of the chromatic approximation (black) ofa signal

BL() (gray) with the behavior of its Taylor approximation (dashed).

Both approximations are of order 16. The signal () is defined using the Shannon

3Propositions stated in this Introduction are special cases of general propositions proved in sub-

sequent sections.


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expansion, with samples {() : () < 1, 32 32} which were randomlygenerated. The plot reveals that, when approximating a signal BL(), achromatic approximation has a much gentler error accumulation when moving awayfrom the point of expansion than the Taylor approximation of the same order.

Unlike the monomials which appear in the Taylor formula, functions [sinc]() =(1) 2 + 1 j() belong to BL() and satisfy [sinc]() 1 for all R.Consequently, the chromatic approximations also belong to BL() and are boundedon R.

Since by Proposition 1.1 the chromatic approximation of a signal BL()converges to in BL(), if is a filter, then commutes with the differentialoperators and thus for every BL(),

(9) []() =

=0

(1) []() [[ sinc]]( ).

A comparison of (9) with (3) provides further evidence that, while local just likethe Taylor expansion, the chromatic expansion associated with the Legendre poly-nomials possesses the features that make the Shannon expansion so useful in sig-nal processing. This, together with numerical robustness of chromatic derivatives,makes chromatic approximations applicable in fields involving empirically sampleddata, such as digital signal and image processing.

1.4. A local definition of the scalar product in BL(). Proposition 1.2 belowdemonstrates another remarkable property of the chromatic derivatives associatedwith the Legendre polynomials.

Proposition 1.2. Let : R R be a restriction of an arbitrary entire function;then the following are equivalent:

(a)

=0 [](0)2 < 0;(b) for all R the sum=0 []()2 converges, and its values are independent

of R;(c)

BL().

The next proposition is relevant for signal processing because it provides localrepresentations of the usual norm, the scalar product and the convolution in BL(),respectively, which are defined globally, as improper integrals.

Proposition 1.3. Let be the chromatic derivatives associated with the (rescaledand normalized) Legendre polynomials, and , BL(). Then the following sumsdo not depend on R and satisfy

=0

[]()2 =

()2;(10)

=0 []()[]() =

()();(11)

=0

[]() [( )] =

()( ).(12)


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1.5. We finish this introduction by pointing to a close relationship between theShannon expansion and the chromatic expansion associated with the Legendre poly-nomials. Firstly, by (7),

(13) () =

=0

[](0) (1)[sinc]().

Since

[sinc]() is an even function for even and odd for odd , (5) implies

(14) [](0) =

=

() (1)[sinc]().

Equations (13) and (14) show that the coefficients of the Shannon expansion of asignal the samples (), and the coefficients of the chromatic expansion of thesignal the simultaneous samples of the chromatic derivatives [](0), are relatedby an orthonormal operator defined by the infinite matrix

(1) [sinc]() : N, Z

=

2 + 1 j() : N, Z

.

Secondly, let [()] = ( + 1) be the unit shift operator in the variable ( might have other parameters). The Shannon expansion for the set of samplingpoints

{ + :

Z

}can be written in a form analogous to the chromatic

expansion, using operator polynomials = . . . , as() =

=0

( + ) sinc ( ( + ))(15)

=

=0

[]() [sinc ( )];(16)

compare now (16) with (6). Note that the family of operator polynomials { }Zis also an orthonormal system, in the sense that their corresponding transfer func-tions {ei }Z form an orthonormal system in 2[, ]. Moreover, the transferfunctions of the families of operators {}N and {}Z, where are thechromatic derivatives associated with the Legendre polynomials, are orthogonal on[

, ] with respect to the same, constant weight w() = 1/(2).

In this paper we consider chromatic derivatives and chromatic expansions whichcorrespond to some very general families of orthogonal polynomials, and prove gen-eralizations of the above propositions, extending our previous work [12].4 However,having in mind the form of expansions (6) and (16), one can ask a more general(and somewhat vague) question.

Question 1. What are the operators for which there exists a family of operatorpolynomials {()}, orthogonal under a suitably defined notion of orthogonality,such that for an associated function m(),

() =

()[]() ()[m( )]

for all functions from a corresponding (and significant) class ?4Chromatic expansions corresponding to general families of orthogonal p olynomials were firstconsidered in [5]. However, Proposition 1 there is false; its attempted proof relies on an incorrectuse of the Paley-Wiener Theorem. In fact, function defined there need not be extendable to

an entire function, as it can be shown using Example 4 in Section 4 of the present paper.


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2. Basic Notions

2.1. Families of orthogonal polynomials. Let : R be a linearfunctional on the vector space of real polynomials in the variable and let = (). Such is a moment functional, is the moment of of order and the Hankel determinant of order is given by

=

0 . . . 1 . . . +1. . . . . . . . . . . . 2

.The moment functionals which we consider are assumed to be:

(i) positive definite, i.e., > 0 for all ; such functionals also satisfy 2 > 0;(ii) symmetric, i.e., 2+1 = 0 for all ;

(iii) normalized, so that (1) = 0 = 1.For functionals which satisfy the above three conditions there exists a family

{ ()}N of polynomials with real coefficients, such that(a) { ()}N is an orthonormal system with respect to , i.e., for all , ,

(

()

()) = (

);

(b) each polynomial () contains only powers of of the same parity as ;(c) 0 () = 1.

A family of polynomials is the family of orthonormal polynomials correspondingto a symmetric positive definite moment functional just in case there exists asequence of reals > 0 such that for all > 0,

(17) +1() =

()

1

1().

If we set 1 = 1 and 1() = 0, then (17) holds for = 0 as well.We will make use of the Christoffel-Darboux equality for orthogonal polynomials,

(18) (

)

=0 () () = (+1() () +1() ()),and of its consequences obtained by setting = in (18) to get

(19)

=0

2+1()2

=0

2 ()2

= 2+1

2+2()

2+1(),

and by by letting in (18) to get

(20)

=0

()2 = (

+1() () +1() ()).

For every positive definite moment functional there exists a non-decreasingbounded function (), called a moment distribution function, such that for the

associated Stieltjes integral we have

(21)

d() =


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and such that for the corresponding family of polynomials { ()}N(22)

()

() d() = ( ).

We denote by 2() the Hilbert space of functions : R C for which theLebesgue-Stieltjes integral

()2 d() is finite, with the scalar product de-

fined by , () = () () d(), and with the corresponding norm de-noted by ().We define a function m : R R as

(23) m() =

eid().

Since

(i ) ei d()

2d()

d()

1/2=

2 < ,

we can differentiate (23) under the integral sign any number of times, and obtainthat for all s,

m(2)(0) = (

1)2;(24)

m(2+1)(0) = 0.(25)

2.2. The chromatic derivatives. Given a moment functional satisfying con-ditions (i) (iii) above, we associate with a family of linear differential operators{}N defined by the operator polynomial5

=1

i (i ) ,

and call them the chromatic derivatives associated with . Since is symmetric,such operators have real coefficients and satisfy the recurrence

(26) +1 = 1

( ) + 1

1,

with the same coefficients > 0 as in (17). Thus,

(27) [i ] = i () i and

(28) [m]() =

i () ei d().

The basic properties of orthogonal polynomials imply that for all ,,

(29) ( )[m](0) = (1)( ),and, if < or if is odd, then(30) ( )[m](0) = 0.

5Thus, to obtain , one replaces in

() by i

, where

[] = d

d (). We use the

square brackets to indicate the arguments of operators acting on various function spaces. If is a linear differential operator, and if a function (, ) has parameters , we write [] todistinguish the variable of differentiation; if () contains only variable , we write [()] for

[()] and [()] for [()].


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The following Lemma corresponds to the Christoffel-Darboux equality for or-thogonal polynomials and has a similar proof which uses (26) to represent the lefthand side of (31) as a telescoping sum.

Lemma 2.1 ([12]). Let{}N be the family of chromatic derivatives associatedwith a moment functional, and let , ; then

(31)

=0

[]

[] = (+1[] [] + [] +1[]).2.3. Chromatic expansions. Let be infinitely differentiable at a real or com-plex ; the formal series

CE[, ]() =

=0

[]() [m( )](32)

=

=0

(1)[]() [m]( )

is called the chromatic expansion of associated with , centered at , and

CA[, , ]() =

=0(1)[]()[m]( )

is the chromatic approximation of of order .From (29) it follows that the chromatic approximation CA[, , ]() of order

of () for all satisfies

[CA[, , ]()]

==

=0

(1)[]() ( )[m](0) = []().

Since is a linear combination of derivatives for , also ()() = [CA

[, , ]()]

=for all . In this sense, just like the Taylor approxi-

mation, a chromatic approximation is a local approximation. Thus, for all ,

(33) ()() = [CA[, , ]()]

==

=0(1) []() ( )[m](0).

Similarly, since

=0 ()()( )/!

== ()() for , we also

have

(34) []() =

=0

()()( )/!

=

=

=0

()() /! (0).Equations (33) and (34) for = relate the standard and the chromatic bases

of the vector space space of linear differential operators,

=

=0

(1) ( )[m](0) ;(35)

=

=0

/! (0)

.(36)

Note that, since for > all powers of in /! are positive, we have(37) > /! (0) = 0.


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3. Chromatic Moment Functionals

3.1. We now introduce the broadest class of moment functionals which we study.

Definition 3.1. Chromatic moment functionals are symmetric positive definite

moment functionals for which the sequence {1/ /}N is bounded.If

is chromatic, we set

(38) = lim sup

!

1/= e limsup

1/

< .

Lemma 3.2. Let be a chromatic moment functional and such that (38) holds.Then for every such that 0 < 1/ the corresponding moment distribution() satisfies

(39)

ed() < .

Proof. For all > 0,

ed() =

=0 /!

d(). For even wehave

d() . For odd we have < 1 + +1 for all , and thus

d() <

d() +

+1d()

1 + +1. Let = if is even,

and = 1 + +1 if is odd. Then also ed() =0 /!. Sincelimsup(/!)

1/ = and 0 < 1/, the last sum converges to a finitelimit.

On the other hand, the proof of Theorem 5.2 in II.5 of [7] shows that if (39)holds for some > 0, then (38) also holds for some 1/. Thus, we get thefollowing Corollary.

Corollary 3.3. A symmetric positive definite moment functional is chromatic justin case for some > 0 the corresponding moment distribution function() satis-

fies (39).

Note: For the remaining part of this section we assume that is a chromaticmoment functional.

For every > 0, we let () = { C : Im() < }. The following Corollarydirectly follows from Lemma 3.2.

Corollary 3.4. If ( 12 ), then ei 2().We now extend function m() given by (23) from R to the complex strip ( 1 ).

Proposition 3.5. Let for ( 1 ),

(40) m() =

ei d().

Then m() is analytic on the strip ( 1 ).

Proof. Fix and let = + i with < 1/; then for every > 0,

(i )ei d()

ed() =

=0

!

+d().


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As in the proof of Lemma 3.2, we let = + for even values of + , and = 1 + ++1 for odd values of + ; then the above inequality implies

(i )ei d()

=0

!

,

and it is easy to see that for every fixed , limsup(/!)1/ = .

Proposition 3.6. Let () 2(); we can define a corresponding function :( 12 ) C by

(41) () =

()i d().

Such () is analytic on (1

2 ) and for all and ( 12 ),

(42) []() =

i () () e

i d().

Proof. Let = + i , with < 1/(2). For every and > 0 we have

(i )()ei d()

()ed()

=0

!

+()d()

=0

!

2+2d()

()2d()1/2

.

Thus, also

(i )()ei d() ()()

=0

!

2+2.

The claim now follows from the fact that lim sup 22+2/

! = 2 for everyfixed .

Lemma 3.7. If is chromatic, then{ ()}N is a complete system in2().Proof. Follows from a theorem of Riesz (see, for example, Theorem 5.1 in II.5of [7]) which asserts that if lim inf (/!)

1/ < , then { ()}N is acomplete system in 2().

6

Proposition 3.8. Let () 2(); if for some fixed ( 12 ) the function()ei also belongs to 2(), then in

2() we have

(43) ()i =

=0(i)[]() (),6Note that we need the stronger condition lim sup (/!)

1/ < to insure that function

m() defined by (40) is analytic on a strip (Proposition 3.5).


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and for () given by (41) we have

(44)

=0

[]()2 =()ei2

()< .

Proof. By Proposition 3.6, if ( 12 ), then equation (42) holds for the corre-sponding given by (41). If also ()ei

2(), then (42) implies that

(45) ()ei, ()() = (i)[]().Since { ()}N is a complete orthonormal system in 2(), (45) implies (43),and Parsevals Theorem implies (44).

Corollary 3.9. For every () 2() and every R, equality (43) holds and

(46)

=0

[]()2 = ()2() .

Thus, the sum

=0 []()2 is independent of R.Proof. If R, then ()ei 2() and ()ei2() = ()2().

Corollary 3.10. Let > 0; then for all ( 12 )

(47)

=0

[m]()2 0, if ( 12 ) then

=

[m]()

=

2

=

[m]()21/2

=

21/2 e( 12)

(),

which implies that the series converges absolutely and uniformly on ( 12 ).Consequently, () = =0 [m]() is analytic on ( 12 ), and

[](0) =

=0

( )[m](0) = (1).

Thus,

=0 [](0)2 =

=0 2 < and so () L2. Proposition 3.16provides the opposite direction.

Note that Proposition 3.17 implies that for every > 0 functions () L2

arebounded on the strip ( 12 ) because

()

=0

[](0)21/2

e(

1

2)

().

3.3. A function space with a locally defined scalar product.

Definition 3.18. 2

is the space of functions () : R C obtained from functionsin L2

by restricting their domain to R.

Assume that , 2

; then (45) implies that for all R,

=0

[]()[]() = []()ei, []()ei()

= [](), []()().Note that for all , R,

=0 []() [( )]=

=0

[]() ( i) () []() ei()d().


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By (43), the sum

=0( i)[]() () converges in 2() to []() ei.Since []() ei() 2(), and since

[]() []() eid() < []()() []()() ,

we have that for all , R,

=0

[]() [( )] =

[]() []() eid() < .

Proposition 3.19 ([12]). We can introduce locally defined scalar product, an as-sociated norm and a convolution of functions in 2

by the following sums which

are independent of R :

2

=

=0

[]()2 = []()2() ;(52)

,

=

=0

[]()[]()(53)

=

[](),

[]()

();

(

)() =

=0

[]() [( )](54)

=

[]() []()eid().

Letting () m() in (54), we get (

m)() = CE[, ]() = () for all() 2

, while by setting = 0, = and = /2 in (54), we get the following

lemma.

Lemma 3.20 ( [12]). For every , 2

and for every R,

=0(1) []() [](0) =

=0(1) [](0) []()

=

=0

(1) [](/2) [](/2).

Since m() is analytic on ( 1 ), so are [m]() for all ; thus, since by (29)=0( )[m](0)2 = 1, we have [m]() 2 for all . Let be a fixed real

parameter; consider functions () = [m( )] = (1)[m]( ) 2.Since

(), ()(55)

=

=0

( )[m( )]( )[m( )]

=

=0

(1)+( )[m]( )( )[m]( )=

= ( ),


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the family { ()}N is orthonormal in 2 and for all N and all R,

(56)

=0

( )[m]()2 = 1.

By (29), for 2

,

, [m( )] =

=0

[]()( )[m( )]=

=

=0

(1)[]()( )[m](0)

= []().(57)Proposition 3.21 ([12]). The chromatic expansionCE[, ]() of() 2

is the

Fourier series of () with respect to the orthonormal system{[m( )]}N.The chromatic expansion converges to () in 2

; thus, {[m( )]}N is a

complete orthonormal base of 2

.

Proof. Since

[()

CE[, , ]()]

= equals 0 for

and equals

[]()

for > , CE[, , ]

==+1 []()2 0. Note that using (56) with = 0 we get

() CE[, , ]()

=+1

[]()[m]( )

=+1

[]()2

=+1

[m]( )21/2

=

=+1[]()2

1/21

=0[m]( )2

1/2.(58)

Let

() =

1

=0

[m]()21/2

;

then, using Lemma 2.1, we have

() = +1[m]() [m]()

1

=0

[m]()21/2

.

Since ( )[m](0) = 0 for all 0 1, we get that () () = 0 for all 2 + 1. Thus, (0) = 0 and () is very flat around = 0, as the followinggraph of15() shows, for the particular case of the chromatic derivatives associatedwith the Legendre polynomials. This explains why chromatic expansions provideexcellent local approximations of signals BL().


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30 20 10 10 20 30

0.2

0.4

0.6

0.8

Figure 4. Error bound 15() for the chromatic approximationof order 15 associated with the Legendre polynomials.

3.4. Chromatic expansions and linear operators. Let be a linear operatoron 2

which is continuous with respect to the norm

. If is shift invariant,

i.e., if for every fixed , [( + )] = []( + ) for all 2

, then commuteswith differentiation on 2

and

[]() =

=0

(1)[]() [[m]]( ) = (

[m])().

Consequently, the action of such on any function in 2

is uniquely determinedby [m], which plays the role of the impulse response [sinc] of a continuoustime invariant linear system in the standard signal processing paradigm based onShannons expansion.

Note that if is a continuous linear operator on 2

such that ()[m]() =( )[m]() for all , then Lemma 3.20 implies that for every 2

,

[]() =

=0

(1)[](0) [[m]]()

=

=0

(1)[[m]](0) []().

Since operators []() are shift invariant, such must be also shift invariant.

3.5. A geometric interpretation. For every particular value of R the map-ping of2

into 2 given by = []()N is unitary isomorphism which

maps the base of 2

, consisting of vectors () = (1)[m()], into vectors = (1)( )[m()]N. Since the first sum in (55) is independent of ,we have , = ( ), and (57) implies , = [](0). Thus, since

=0 [](0)2 < , we have

=0 [](0) 2 and

=0, =

=0 [](0) =

=0

[](0)(1)( )[m()]

N.


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Since for 2

the chromatic series of converges uniformly on R, we have[]() =

=0

[](0)(1)( )[m()]. Thus,

=0

, =

=0

[](0) = []()N = .

Thus, while the coordinates of =

[]()N in the usual base of 2

varywith , the coordinates of in the bases { }N remain the same as varies.

We now show that { }N is the moving frame of a helix : R 2.

Lemma 3.22 ([12]). Let 2

and let R vary; then () = []()Nis a continuous curve in 2.

Proof. Let 2

; then, since

=0 []()2 converges to a continuous (con-stant) function, by Dinis theorem, it converges uniformly on every finite interval

. Thus, the last two sums on the right side of inequality () ( + )2

=0([]() []( + ))2 + 2

=+1 []()2 + 2

=+1 []( + )2

can be made arbitrarily small on if is sufficiently large. Since functions

[]() have continuous derivatives, they are uniformly continuous on . Thus,=0([]() []( + ))2 can also be made arbitrarily small on by taking

sufficiently small.

Lemma 3.23 ([12]). If 2

, then lim0

() ( + ) ()

= 0; thus,

the curve () = []()N is differentiable, and ( )() = []()N.

Proof. Let be any finite interval; since 2

, for every > 0 there exists such that

=+1 [] ()2 < /8 for all . Since functions [] ()

are uniformly continuous on , there exists a > 0 such that for all 1, 2 , if1 2 < then

=0 ([](1) [](2))2 < /2. Let be an arbitrary

number such that

< ; then for every there exists a sequence of numbers

that lie between and , and such that ([]()[]())/ = []().Thus, for all ,

=0

() ( )

()2

=

=0

[]() []()]2 0. In this case

()

()d

2 2 = ( ).

By Proposition 3.15, the corresponding space L2

contains all entire functions

() which have a Fourier transform () supported in [, ] that also satisfies

2 2

()2d < . In this case the corresponding space L2

contains

functions which do not belong to 2

; the corresponding function (40) ism

() =J0() and for > 0, [m]() = (1)2 J(), where J() is the Besselfunction of the first kind of order . In the recurrence relation (26) the coefficients

are given by 0 = /

2 and = /2 for > 0.


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The chromatic expansion of a function () is the Neumann series of () (see[23]),

() = ()J0(( )) +

2

=1

[]()J(( )).

Thus, the chromatic expansions corresponding to various families of orthogonalpolynomials can be seen as generalizations of the Neumann series, while the familiesof corresponding functions {[m]()}N can be seen as generalizations and auniform representation of some familiar families of special functions.

4.3. Example 3: Hermite polynomials/Gaussian monomial functions.Let () be the Hermite polynomials; then polynomials () = (2

!)1/2()satisfy

()

() e2 d

= ( ).

The corresponding space L2

contains entire functions whose Fourier transform

() satisfies

()2 e2d < . In this case the space 2

contains non-

bandlimited signals; the corresponding function defined by (40) is m() = e2/4

and [m]() = (1)(2 !)1/2 e2

/4. The corresponding recursion coeffi-cients are given by =

( + 1)/2. The chromatic expansion of () is just the

Taylor expansion of () e2/4, multiplied by e

2/4.

4.4. Example 4: Herron family. This example is a slight modification of anexample from [9]. Let the family of orthonormal polynomials be given by therecursion 0() = 1, 1() = , and +1() = /(+1)()/(+1)1().Then

1

2

(, ) (, ) sech

2

d = (m n).

In this case m() = sech and [m]() = (1) sech tanh . The recursioncoefficients are given by = +1 for all

0. If are the Euler numbers, then

sech = =0 2 2/(2)!, with the series converging only in the disc of radius/2. Thus, in this case m() is not an entire function.

5. Weakly bounded moment functionals

5.1. To study local convergence of chromatic expansions of functions which are notin 2

we found it necessary to restrict the class of chromatic moment functionals.

The restricted class, introduced in [12], is still very broad and contains functionalsthat correspond to many classical families of orthogonal polynomials. It consists offunctionals such that the corresponding recursion coefficients > 0 appearing in(26) are such that sequences {}N and {+1/}N are bounded from belowby a positive constant, and such that the growth rate of the sequence {}N issub-linear in . For technical simplicity in the definition below these conditions are

formulated using a single constant in all of the bounds.

Definition 5.1 ([12]). Let be a moment functional such that for some > 0(26) holds.


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(1) is weakly bounded if there exist some 1, some 0 < 1 and someinteger 0, such that for all 0,

(59)1

( + ),

(60)

+1 2.

(2) is bounded if there exists some 1 such that for all 0,(61)

1

.

Since (26) is assumed to hold for some > 0, weakly bounded moment function-als are positive definite and symmetric. Every bounded functional is also weaklybounded with = 0. Functionals in our Example 1 and Example 2 are bounded.For bounded moment functionals the corresponding moment distribution ()has a finite support [2] and consequently m() is a band-limited signal. However,m() can be of infinite energy (i.e., not in 2) as is the case in our Example 2.Moment functional in Example 3 is weakly bounded but not bounded ( = 1/2);the moment functional in Example 4 is not weakly bounded ( = 1). We notethat important examples of classical orthogonal polynomials which correspond to

weakly bounded moment functionals in fact satisfy a stronger condition from thefollowing simple Lemma.

Lemma 5.2. Let be such that (26) holds for some > 0. If for some 0 < 1the sequence / converges to a finite positive limit, then is weakly bounded.

Weakly bounded moment functionals allow a useful estimation of the coefficientsin the corresponding equations (35) and (36) relating the chromatic and the stan-dard derivatives.

Lemma 5.3 ([12]). Assume that is such that for some 1, 0 and 0the corresponding recursion coefficients for all satisfy inequalities (59). Thenthe following inequalities hold for all and :

(

)[m](0)

(2)( + )!;(62) ! (0) (2).(63)

Proof. By (30), it is enough to prove (62) for all , such that . We proceedby induction on , assuming the statement holds for all . Applying (26) to[m]() we get

( +1)[m]() (+1 )[m]() + 1 (1 )[m]().Using the induction hypothesis and (30) again, we get for all + 1,

( +1) [m](0) < (( + 1 + ) + ( + ))(2)( + )!< (2)+1( + 1 + )!.

Similarly, by (37), it is enough to prove (63) for all

. This time we proceedby induction on and use (26), (59) and (60) to get+1 !

1( 1)!+ 2 1 !

.


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By induction hypothesis and using (37) again, we get that for all + 1,+1 ! (0) < (2) + 2(2)1 < (2)+1. Corollary 5.4 ([12]). Let be weakly bounded; then for every fixed

lim

( ) [m](0)/!

1/

= 0,

and the convergence is uniform in .

Proof. Let () = ( + )!/!; then () is a polynomial of degree , and, by (62),

(64)

( ) [m](0)!1/ 2()/!(1)/ < 2e1 ()/1 .

Corollary 5.5 ([12]). Letm() correspond to a weakly bounded moment functional; then

(65) lim

!

1/= lim

m

(0)

!

1/

= 0.

Thus, since (38) is satisfied with = 0, every weakly bounded moment functionalis chromatic.

Note that this and Proposition 3.5 imply that m() is an entire function. If (59)holds with = 1, then Lemma 5.3 implies only

(66) lim sup

( ) [m](0)!1/ 2.

Example 4 shows that in this case the corresponding function m() need not beentire. Thus, if we are interested in chromatic expansions of entire functions, theupper bound in (59) of the definition of a weakly bounded moment functional issharp.

Lemma 5.3 and Proposition 3.7 imply the following Corollary.

Corollary 5.6 ([12]). If is weakly bounded, then the corresponding family ofpolynomials { ()}N is a complete system in 2().

Thus, we get that the Chebyshev, Legendre, Hermite and similar classical fami-lies of orthogonal polynomials are complete in their corresponding spaces 2().

To simplify our estimates, we choose 1 such that for , and as inDefinition 5.1 for all > 0, we have

(67)(2)( + )!

!< .

The following Lemma slightly improves a result from [12].

Lemma 5.7. Let be weakly bounded and < 1 and 1 such that (59) and(67) hold. Let also be an integer such that

1/(1

). Then there exists a

polynomial () of degree 1 such that for every and every C,

(68) [m]() <

!1() e .


25/32


Proof. Using the Taylor series for [m](), (30), (62) and (67), we get that for such that 1,

[m]() 0 be any number such that lim sup ()()/!11/ < ;then there exists 1 such that ()() !1 for all . Using (36)


26/32


and (63) we get

[]()

=0

!

(0)

()() (2) =0

!1

< (2)

=0

(+1)(1)

.

Summation of the last series shows that []() < 2(2 )!1 for suffi-ciently large .

Corollary 5.10. Let be weakly bounded, < 1 as in (59), () a functionanalytic on a domain C and .

(1) If the sequence ()()/!11/ is bounded, then the chromatic expansionCE[, ]() of() converges uniformly to () on a disc centeredat .

(2) In particular, if ()()/!11/ converges to zero, then the chromaticexpansion CE[, ]() of() converges for all and the convergenceis uniform on every closed disc around , contained in .

Corollary 5.11 ([12]). If is bounded, then for every entire function and all, C, the chromatic expansion [, ]() converges to () for all , and theconvergence is uniform on every disc around of finite radius.

Proof. If() is entire, then for every , lim ()()/!1/ = 0. The Corollarynow follows from Corollary 5.10 with = 0.

Proposition 5.12. Assume is weakly bounded and let 0 < 1 be such that(59) holds, and such that 1/(1 ). Then there exists , > 0 such that()

e

for all() L2

.

Proof. Since () L2

, the chromatic expansion of () and (68) yield

() =0

[](0)2 =0

[m]()21/2

()e

=0

2!2(1)

1/2,

which, using the method from the proof of Lemma 5.7, can easily be shown to implyour claim.

Note that for bounded moment functionals, such as those corresponding to theLegendre or the Chebyshev polynomials, we have = 0; thus, Proposition 5.12 im-plies that functions which are in L2

are of exponential type. For corresponding

to the Hermite polynomials p=1/2 (see Example 3); thus, we get that there exists, > 0 such that ()

e2

for all L2

. It would be interestingto establish when the reverse implication is true and thus obtain a generalization

of the Paley-Wiener Theorem for functions satisfying () < e for > 1.


27/32


5.3. Generalizations of some classical equalities for the Bessel functions.Corollaries 5.11 and 5.10 generalize the classic result that every entire function canbe expressed as a Neumann series of Bessel functions [23], by replacing the Neu-mann series with a chromatic expansion that corresponds to any (weakly) boundedmoment functional. Thus, many classical results on Bessel functions from [23]immediately follow from Corollary 5.11, and, using Corollary 5.10, generalize tofunctions

[m]() corresponding to any weakly bounded moment functional

.

Below we give a few illustrative examples.

Corollary 5.13. Let () be the orthonormal polynomials associated with aweakly bounded moment functional ; then for every C,

(69) ei =

=0

i () [m]().

Proof. If < 1 then

lim

dd ei =0

1= lim

1

= 0,

and the claim follows from Proposition 5.10 and (27).

Corollary 5.13 generalizes the well known equality for the Chebyshev polynomials() and the Bessel functions J(), i.e.,

ei = J0() + 2

=1

i()J().

In Example 3, 5.13 becomes the equality for the Hermite polynomials ():

ei =

=0

()

!

i

2

e

2

4 .

By applying Corollary 5.10 to the constant function () 1, we get that itschromatic expansion yields that for all C

m() +

=1

=1

2221

2[m]() = 1,with the recursion coefficients from (17). This equality generalizes the equality

J0() + 2

=1

J2() = 1.

Using Proposition 3.16 to expand m( + ) L2

into chromatic series around = 0, we get that for all , C

m( + ) =

=0

(1)[m]()[m](),

which generalizes the equality

J0( + ) = J0()J0() + 2

=1

(1)J()J().


28/32


6. Some Non-Separable Spaces

6.1. Let be weakly bounded; then periodic functions do not belong to 2

be-cause

=0 []()2 diverges. We now introduce some nonseparable inner product

spaces in which pure harmonic oscillations have finite norm and are pairwise or-thogonal.

Note: In the remainder of this paper we consider only weakly bounded moment functionals and real functions which are restrictions of entire functions.

Definition 6.1. Let let0 < 1 be as in (59). We denote by the vector spaceof functions such that the sequence

(70) () =1

( + 1)1

=0

[]()2

converges uniformly on every finite interval R.Proposition 6.2. Let, and

(71) () =1

( + 1)1

=0 []()

[]();

then the sequence { ()}N converges to a constant function. In particular,{()}N also converges to a constant function.Proof. Since () and

() given by (70) converge uniformly on every finite in-

terval, the same holds for the sequence (). Consequently, it is enough to showthat for all , the derivative ()

of () satisfies lim ()

= 0. Let

() = []()2 + +1[]()2 + []()2 + +1[]()2;then, since , , the sequence 1/( + 1)1=0 () converges everywhereto some (). We now show that if is such that () > 0, then there are infinitelymany such that () < 2(). Assume opposite, and let be such that() 2() for all . Then, since

=

>+1

d =( + 1)1 1

1 ,

we would have that for all > ,= ()

( + 1)1 2()

=

( + 1)1>

2()(( + 1)1 1)( + 1)1(1 ) .

However, since 0 < 1, this would imply =0 ()/( + 1)1 > () forall sufficiently large , which contradicts the definition of (). Consequently, forinfinitely many all four summands in () must be smaller than 2() . Forthose values of we have

+1[]() []() + []() +1[]() < 4().

Since is weakly bounded, (31) and (59) imply that for some 1 and aninteger , () < ( + )( + 1)1 (+1[]() []() + []() +1[]()).


29/32


Thus, for infinitely many we have () < 4( + ) ()( + 1)1 .Consequently, lim inf

() = 0 and since lim () exists, it must beequal to zero.

Corollary 6.3. Let 0 be the vector space consisting of functions () such thatlim () = 0; then in the quotient space 2 = /0 we can introduce ascalar product by the following formula whose right hand side is independent of :

, = lim

1

( + 1)1

=0

[]() []().(72)

The corresponding norm on 2 is denoted by . Clearly, all real valuedfunctions from 2

belong to 0 .

Proposition 6.4. If 2 , then the chromatic expansion of () converges to() for every and the convergence is uniform on every finite interval.

Proof. Since 1/( + 1)1=0

[]()2 converges to 0 < (

)2 0 functions () = sin and () = cos form

an orthogonal system in 2 , and = = e2/2/ 4

2.

Proof. For all and for ,

() ( + 1)

1

2

22 ( 2 + 1)

e2

2 cos

2 + 1 2

0;

see, for example, 8.22.8 in [18]. Using the Stirling formula we get

(74) ()

2

14

1

4 e2

2 cos

2 + 1 2

0.

This fact and (18) are easily seen to imply that , = 0 for all distinct, > 0, while (74) and (19) imply that

(75) lim

=0

2()2

2 + 1

1

=0

2+1()2

2 + 1

= 0.

Since () = 2 1(), we have

() =

2 1(). Using this, (74)

and (20) one can verify that

lim

=0 ()2 + 1

=2

e2

,

which, together with (75), implies that = e2/2/ 4

2.


31/32


Note that in this case, unlike the case of the family associated with the Chebyshevpolynomials, the norm of a pure harmonic oscillation of unit amplitude depends onits frequency.

One can verify that propositions similar to Proposition 6.5 and Proposition 6.6hold for other classical families of orthogonal polynomials, such as the Legendrepolynomials. Our numerical tests indicate that the following conjecture is true.7

Conjecture 6.7. Assume that for some < 1 the recursion coefficients in (17)are such that/ converges to a finite positive limit. Then, for the corresponding

family of orthogonal polynomials we have

(76) 0 < lim

1

( + 1)1

=0

()2 <

for all in the support () of the corresponding moment distribution function(). Thus, in the corresponding space 2 all pure harmonic oscillations withpositive frequencies belonging to the support of the moment distribution ()have finite positive norm and are mutually orthogonal.

Note that (20) implies that (76) is equivalent to

0 < lim

+1() () +1() ()( + 1)12

< .

7. Remarks

The special case of the chromatic derivatives presented in Example 2 were firstintroduced in [11]; the corresponding chromatic expansions were subsequently in-troduced in [14]. These concepts emerged in the course of the authors design of apulse width modulation power amplifier. The research team of the authors startup,Kromos Technology Inc., extended these notions to various systems correspondingto several classical families of orthogonal polynomials [5, 9]. We also designed andimplemented a channel equalizer [8] and a digital transceiver (unpublished), basedon chromatic expansions. A novel image compression method motivated by chro-matic expansions was developed in [3,4]. In [6] chromatic expansions were relatedto the work of Papoulis [17] and Vaidyanathan [19]. In [15] and [20] the theory wascast in the framework commonly used in signal processing. Chromatic expansionswere also studied in [5], [1] and and [22]. Local convergence of chromatic expan-sions was studied in [12]; local approximations based on trigonometric functionswere introduced in [13]. A generalization of chromatic derivatives, with the prolatespheroidal wave functions replacing orthogonal polynomials, was introduced in [21];the theory was also extended to the prolate spheroidal wavelet series that combinechromatic series with sampling series.

Note: Some Kromos technical reports and some manuscripts can be found at theauthors web site http://www.cse.unsw.edu.au/ignjat/diff/.

7We have tested this Conjecture numerically, by setting = for several values of < 1, andin all cases a finite limit appeared to exist. Paul Nevai has informed us that the special case of

this Conjecture for = 0 is known as Nevai-Totik Conjecture, and is still open.


32/32


References

[1] J. Byrnes. Local signal reconstruction via chromatic differentiation filter banks. In Proc. 35thAsilomar Conference on Signals, Systems, and Computers, Monterey, California, November

2001.[2] T. S. Chihara. An introduction to Orthogonal Polynomials. Gordon and Breach, 1978.[3] M. Cushman. Image compression. Technical report, Kromos Technology, Los Altos, Califor-

nia, 2001.

[4] M. Cushman. A method for approximate reconstruction from filterbanks. In Proc. SIAMConference on Linear Algebra in Signals, Systems and Control, Boston, August 2001.[5] M. Cushman and T. Herron. The general theory of chromatic derivatives. Technical report,

Kromos Technology, Los Altos, California, 2001.

[6] M. Cushman, M. J. Narasimha, and P. P. Vaidyanathan. Finite-channel chromatic derivativefilter banks. IEEE Signal Processing Letters, 10(1), 2003.

[7] G. Freud. Orthogonal Polynomials. Pergamon Press, 1971.[8] T. Herron. Towards a new transform domain adaptive filtering process using differential

operators and their associated splines. In Proc. ISPACS, Nashville, November 2001.

[9] T. Herron and J. Byrnes. Families of orthogonal differential operators for signal processing.Technical report, Kromos Technology, Los Altos, California, 2001.

[10] F. B. Hildebrand. Introduction to Numerical Analysis. Dover Publications, 1974.[11] A. Ignjatovic. Signal processor with local signal behavior, 2000. US Patent 6115726. Provi-

sional Patent Disclosure this patent was filled October 3, 1997. Patent application this patent

was filled May 28, 1998. The patent was issued September 5, 2000.[12] A. Ignjatovic. Local approximations based on orthogonal differential operators. Journal of

Fourier Analysis and Applications, 13(3), 2007.[13] A. Ignjatovic. Chromatic derivatives and local approximations. IEEE Transactions on Signal

Processing, to appear.

[14] A. Ignjatovic and N. Carlin. Method and a system of acquiring local signal behavior param-

eters for representing and processing a signal, 2001. US Patent 6313778. Provisional PatentDisclosure this patent was filled July 9, 1999; patent application was this patent filled July

9, 2000. The patent was issued November 6, 2001.[15] M. J. Narasimha, A. Ignjatovic, and P. P. Vaidyanathan. Chromatic derivative filter banks.

IEEE Signal Processing Letters, 9(7), 2002.[16] A. Oppenheim and A. Schafer. DiscreteTime Signal Processing. Prentice Hall, 1999.[17] A. Papulis. Generalized sampling expansion. IEEE Transactions on Circuits and Systems,

24(11), 1977.[18] G. Szego. Orthogonal Polynomials. American Mathematical Society, 1939.

[19] P. P. Vaidyanathan. Multirate Systems and Filter Banks. Prentice-Hall, 1993.[20] P. P. Vaidyanathan, A. Ignjatovic, and S. Narasimha. New sampling expansions of band

limited signals based on chromatic derivatives. In Proc. 35th Asilomar Conference on Signals,Systems, and Computers, Monterey, California, November 2001.

[21] G. Walter. Chromatic Series and Prolate Spheroidal Wave Functions. Journal of IntegralEquations and Applications, 20(2), 2006.

[22] G. Walter and X. Shen. A sampling expansion for non bandlimited signals in chromaticderivatives. IEEE Transactions on Signal Processing, 53, 2005.

[23] G. N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press,1966.

School of Computer Science and Engineering, University of New South Wales and

National ICT Australia (NICTA), Sydney, NSW 2052, Australia

E-mail address: [email protected]

URL: www.cse.unsw.edu.au/ignjat

chromatic derivatives

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