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A general framework for Shannon samplingon graphs and manifolds

Isaac Z. Pesenson

Department of Mathematics, Temple University, Philadelphia, PA 19122

GRAPH SIGNAL PROCESSING WORKSHOP, UPENN,May 25, 2016

Introduction

During the last years a number of methods from the classicalsignal and image processing where developed in the setting ofcombinatorial graphs.

Below are names of the people whose work I familiar with andwho are greatly responsible for development of different partsof this theory:

A. Ortega, J. Kovacevic, D. Shuman, P. Vandergheynst, J.Moura, S. Narang, A. Gadde, R. Frossard, A. Anis, S. Chen,R. Varma, A. Singh.

Introduction

Some of these new results are using ideas from the classicalShannon sampling theory by developing methods to reducelarge sets of vertices to smaller subsets.

Introduction

One does not have to be very smart to understand that it isalways possible to sacrifice some number of vertices withoutloosing much of information.

However, it is challenging and important problem to findquantitative relations between amount of information that canbe lost and the size of a sampling set.

To describe some of such quantitative relations is one of themain objectives of my talk.

Introduction

When talking about discrete graphs Shannon sampling makesthem even "more discrete".

On the other hand, the same Shannon sampling gives a way todecrsetize continuous objects such as manifolds and functionson them.

Recently, the Shannon sampling theory was extended tosmooth manifolds and quantum graphs and functions on them.Sampling results on such manifolds as unit spheres S2, S3, unitballs B2, B3, group of rotations SO(3) already found manyimportant applications.

Introduction

I’m particularly proud of my recent publication with a group ofpeople which includes a physicist, a statistician, and anumerical analysis expertClaudio Durastanti ("Tor Vergata" University, Roma, Italy),Yabebal T. Fantaye, (University of Oslo, Norway), Frode K.Hansen, (University of Oslo, Norway), Domenico Marinucci,("Tor Vergata" University, Roma, Italy), Isaac Z. Pesenson,(Temple University, Philadelphia, USA)

"A Simple Proposal for Radial 3D Needlets", PhysicalReview D 90, 103532-Published 26 November 2014.

In this paper Shannon sampling was used to construct localizedframes (wavelets) on the unit ball B3 and to apply them to a realdata to make a map of the Cosmic Microwave Backgroundtemperature distribution.

Introduction

In may talk I will start with the classical Shannon Theorem andwill show how this theorem can be formulated and proved forfunctions defined on combinatorial graphs.

It will be demonstrated that some of our results are sharp.

I will also try to include the spline interpolation on graphs andto discuss sampling of non-stationary signals which satisfySchrodinger-type equation on graphs.

Introduction

In the second part of my talk I will present a unified frameworkfor an abstract version of the Shannon Theorem which willinclude sampling of graphs and manifolds.

If time will permit the unified framework will be illustrated bydeveloping sampling on on quantum graphs and on manifolds.

Classical Shannon-Nyquist sampling Theorem

Let us remind that the Classical Shannon-Nyquist samplingTheorem states that for all Paley-Wiener functions of a fixedbandwidth defined on Euclidean space one can find "not verydense" sampling sets which can be used to represent allrelevant Paley-Wiener functions.

In some sense it allows to reduce the set of all points ofEuclidean space to a countable set of points.


A function f ∈ L2(R) is called ω-bandlimited if its L2-Fouriertransform

f̂ (t) =

∫ +∞

−∞f (x)e−2πixtdx

has support in [−ω, ω].

The Paley-Wiener theorem states that f ∈ L2(R) isω-bandlimited if and only if f is an entire function ofexponential type not exceeding 2πω.

ω-bandlimited functions form the Paley-Wiener class PWω andoften called Paley-Wiener functions.


The classical sampling theorem says, that if f is ω-bandlimitedthen f is completely determined by its values at pointsj/2ω, j ∈ Z, and can be reconstructed in a stable way from thesamples f (j/2ω), i.e.

f (x) =∑j∈Z

f(

j2ω

)sin(2πω(x − j/2ω))

2πω(x − j/2ω), (1)

where convergence is understood in the L2-sense.


This surprising result is a direct consequence of anotherremarkable fact that the following equality between "continuous"and "discrete" norms holds true for functions in PWω:

(∫ +∞

−∞|f (x)|2dt

)1/2

=

12ω

∑j∈Z|f (j/2ω)|2

1/2

. (2)

This equality follows from the fact that the functions e2πit(j/2ω)

form an orthonormal basis in L2[−ω, ω].

The formulas (1.1) and (1.2) involve regularly spaced pointsj/2ω, j ∈ Z.

Irregular sampling Theorem

If one would like to consider irregular sampling at a sequenceof points {xj} and still have a stable reconstruction from thesamples f (xj) then instead of equality (1.2) the followingPlancherel-Polya inequality should hold for functions in PWω

C1∑j∈Z|f (xj)|2 ≤

∫ +∞

−∞|f (x)|2dx ≤ C2

∑j∈Z|f (xj)|2. (3)

Such inequalities are also known as the frame inequalities.

Irregular sampling Theorem

There is a remarkable result of Duffin and Schaeffer,Transactions of AMS, 1952, that the inequalities (1.3) implyexistence of ω-bandlimited functions θj such that anyω-bandlimited function can be reconstructed according to thefollowing formula

f (x) =∑j∈Z

f (xj)θj(x), (4)

which is a far going generalization of the Shannon formula (1.1).

In this paper they introduced such notion as Hilbert framewhich became very popular today.

Hilbert frames

A set of vectors {ψv} in a Hilbert space H is called a frame ifthere exist constants A,B > 0 such that for all f ∈ H

A∑

v

|〈f , ψv 〉|2 ≤ ‖f‖22 ≤ B∑

v

|〈f , ψv 〉|2 (5)

The largest A and smallest B are called lower and upper framebounds.The set of scalars {〈f , ψv 〉} represents a set of measurementsof a signal f . To synthesize signal f from this set ofmeasurements one has to find another (dual) frame {Ψv} andthen a reconstruction formula is

f =∑

v

〈f , ψv 〉Ψv . (6)

Parseval Hilbert frames

Dual frame is not unique in general. Moreover it is difficult tofind a dual frame.

However, for frames with A = B the decomposition andsynthesis of functions can be done with the same frame. Inother words

f =∑

v

〈f , ψv 〉ψv . (7)

Such frames are known as tight or Parseval frames.

An example of such frame is the so-called Mecedes-Benzframe in R2 which is formed by three vectors of length

√2/3

and angles π/3 between them.

One of the goals of my talk is to show that analysis of lowerfrequencies on combinatorial graph can be performed on asmaller subgraph.

Note that in many situations lower frequencies are moreinformative while higher frequencies are usually associatedwith noise.

Another goal is to show how sampling methods can be used toconstruct efficient ways for representing bandlimitedfunctions on quantum (continuous) graphs and onmanifolds.

Combinatorial graphs

We consider a finite graph G = (V ,E), where V = V (G) is theset of |V | vertices or nodes and E = E(G) is the set of edges orlinks connecting these vertices.

The weight of the edge connecting two nodes u and v isdenoted by w(u, v) ≥ 0. We consider undirected graphs whichmeans that w(u, v) = w(v ,u).

Combinatorial graphs

The degree µ(v) of the vertex v is the sum of weights of alleges incident to v :

µ(v) =∑u∼v

w(u, v).

The adjacency matrix W of the graph is an |V | × |V | matrixsuch that W (u, v) = w(u, v).

Combinatorial Laplace operator

The Hilbert space L2(G) is the set of all complex valuedfunctions f on V (G)

f : V → C,

with the following inner product

〈f ,g〉L2(G) = 〈f ,g〉 =∑

v∈V (G)

f (v)g(v)µ(v). (8)

For such graph the weighted Laplace operator ∆ isintroduced via

(∆f )(v) =∑

u∈V (G)

(f (v)− f (u))w(v ,u) . (9)

The graph Laplacian is a well-studied object; it is known to be apositive-semidefinite self-adjoint bounded operator. Thus, ∆has |V | real and nonnegative eigenvalues.

Moreover, since ∆1 = 0, where 1 = (1, 1, . . . , 1), is the all 1constant function, zero is an eigenvalue of ∆ corresponding tothe eigenfunction e0 = 0. It is of multiplicity zero as long as G isa connected graph.

Fourier analysis

Let 0 = λ0 < λ1 ≤ · · · ≤ λ|V |−1 be the set of eigenvalues of ∆and let eλ0 , ....,eλ|V |−1 be an orthonormal complete set ofeigenfunctions.

For a function f ∈ L2(G) it’s Fourier coefficients cj(f ) aredefined as usual

cj(f ) =∑

v∈V (G)

f (v)eλj (v).

and then one has the Fourier representation

f (v) =∑

j

cj(f )eλj (v), v ∈ V (G).

Bandlimited functions

DefinitionA function (signal) f on a finite weighted graph G is said to beω-bandlimited if it has expansion

f =∑λj≤ω

cjeλj .

One can also say that a function f ∈ L2(G) is ω-bandlimited ifits Graph Fourier transform has support in [0, ω].

The space of ω-bandlimited signals is also called thePaley-Wiener space and is denoted by PWω(G).

Note, that f ∈ PWω(G) if and only if the followingBernstein-type inequality holds

‖∆sf‖ ≤ ωs‖f‖

for every s > 0.

Uniqueness Sets and removable sets

DefinitionFor a given ω > 0 a subset of vertices Uω ⊂ V (G) is auniqueness set (also known as a sampling set) for the spacePWω(G), ω > 0, if for any two signals from PWω(G), the factthat they coincide on Uω implies that they coincide on V (G).

In general, every space PWω has many uniqueness sets.

If Ucω is the complement of a uniqueness set Uω for PWω we call

it a removable set for PWω.

Plancherel-Polya-type inequalities

Theorem(Pesenson I., 2008)For a given ω > 0 a set of vertices Sω is a uniqueness set forthe subspace PWω if and only if there exist positive constants1 ≤ C1 = C1(Sω) < C2 = C2(Sω) such that

C1∑

u∈Sω

|f (u)|2µ(u) ≤∑

u∈V (G)

|f (u)|2µ(u) ≤ C2∑

u∈Sω

|f (u)|2µ(u),

orC1‖f |Sω

‖2 ≤ ‖f‖2 ≤ C2‖f |Sω‖2. (10)

Note, that the left hand side here is always satisfied withC1 = 1.

It is a kind of surprise that one can have C1 > 1

*****************************************

Poincare constant as a measure of complexity

For a subset of nodes S ⊂ V (G) let L2(S) be the subspace offunctions in L2(G) which are identical to zero on thecompliment Sc = V (G) \ U.

DefinitionFor a given Λ > 0 we say that U ⊂ V (G) is a Λ-set if anyf ∈ L2(Uc) the following inequality holds

‖f |Uc‖ ≤ Λ‖∆f‖, f ∈ L2(Uc). (11)

The best constant Λ = Λ(U) in this inequality is called thePoincare constant.

It is important to realize the the constant Λ in (11) depends onlyon the way U is connected to Uc .


The following Theorem is of primary importance.

Theorem(Pesenson I., 2008)If a set U ⊂ V (G) is a Λ = Λ(U)-set then U is a uniqueness setfor any space PWω(G) with 0 < ω < Λ−1.

An upper bound for such constant Λ(U) for any set U can beobtained by using only the geometry of a graph.

Relative degree

Given any subset S ⊂ V (G), we define the following relativedegree of a vertex v ∈ V (G) with respect to S:

wS(v) =∑u∈S

w(u, v), v ∈ V (G),

Measures of connectivity

and then introduce the following measures of connectivitybetween S and its compliment Sc :

D = D(S) = DS→Sc = supv∈S

wSc (v),

K = K (S) = KS←Sc = infv∈Sc

wS(v).

Estimating the Poincare constant

The following Theorem follows from a more general estimateswhich were proved in

I. Pesenson, M. Pesenson ,Sampling, filtering and sparse approximations on graphs,

JFAA, 2010, Vol.16, no. 6, 921-942,

Führ, Hartmut; Pesenson, Isaac Z. ,SIAM J. Discrete Math. 27 (2013), no. 4, 2007-2028.

Theorem

For any S ⊂ V (G) for which

K (S) = KS←Sc = infv∈Sc

wS(v)

is not zero the following inequality holds for any f ∈ L2(G)

‖f‖ ≤ K (S)−1/2‖∆1/2f‖ +

(D(S)

K (S)

)1/2

‖f |S‖. (12)


The previous inequality implies that for any f ∈ L2(Sc)

‖f |Sc‖ ≤ K−1(S)‖∆f‖, f ∈ L2(Sc), (13)

if K (S) ≥ 1.

Since a Poincare constant in (11) is the best possible, oneimmediately obtains the following statement.

TheoremThe following holds true

1 If for some s ⊂ V (G) one has K (S) ≥ 1 then the followingupper estimate holds

Λ(S) ≤ K−1(S),

where K (S) = infv∈Sc wS(v).2 S is a uniqueness set for every space PWω(G) withω < K (S) ≤ Λ−1(S).

The following result shows that any randomly selected set ofvertices can be used to obtain some meaningfulinformation about spectrum of the Laplace operator.

In what follows the notation N [a, b] is used for the number ofeigenvalues of ∆ in [a,b].

TheoremIf S ⊂ V (G) is a Λ-set then S is a uniqeness set for PWΛ−1 andthe following inequalities hold

1 N[0, Λ−1) ≤ |S|;

2 N[Λ−1, λ|V |−1

]≥ |Sc |;

3 λ|S| ≥ Λ−1 .

Example: complete bipartite graph

I will use a complete bipartite non-weighted graph G todemonstrate that in this case all the statements of our lasttheorem are exact.

Thus as long as we consider general graphs our geometricestimate of a capacity constant of a sebset of vertices can notbe improved.

A non-weighted complete bipartite graph G consists of twodisjoint sets of vertices (components) S and Sc where

1 every vertex in one component is connected to everyvertex in another;

2 no edges inside of components;3 every edge has weight one.

Let |S| = N, |Sc | = M, and N > M.

Example: complete bipartite graph

One can show that in this case the Laplacian ∆ on graph G haseigenvalues λ0, ..., λM+N−1, where

λ0 = 0 <

< λ1 = ... = λN−1 = M <

< λN = ... = λN+M−2 = N <

< λN+M−1 = N + M.

For the Poincare constant of Sc we have K = |S| = N and thisvalue is sharp.

Let’s return to the case of a non-weighted complete bipartitegraph G with components S, |S| = N, andSc , |Sc | = M, N > M.In this situation all the statements of the Corollary are sharpand we have:

1 for the Poincare constant of Sc we haveΛ−1 = K = |S| = N;

2 the set S is the uniqueness set for the span of the first Neigenfunctions e0, ...,eN−1, where ∆ej = λjej ;

3 N [0, N) = N = |S|;4 N [N, λN+M−1] = M = |Sc |;5 λN = N = |S| .

In particular, the following holds:

1 Any function in the span of the correspondingeigenfunctions u0,u1, ...,uN−1 is completely determined byits values on the set S with |S| = N.

2 If N � M then "almost half" of frequencies are determinedby their values on "almost a half" of a graph .

The previous results give an upper bound for the capacityconstant for any set S by using only the geometry of a graph.

Later on a deferent method estimating the same capacity wasobtained by

A.Anis, A.Gadde, A.Ortega in Acoustics, Speech and SignalProcessing (ICASSP), 2014 IEEE International Conference.

Their method is algebraic:

TheoremLet S ⊂ V (G) and Sc = V (G) \ S. Let D be a matrix which isobtained from the matrix of ∆ by replacing by zero columns androws corresponding to the set S. Then S is a uniqueness setfor all signals f ∈ PWω(G) with ω < σ where σ is the smallestpositive eigenvalue of D.

Our example with bipartite graph is consistent with results ofS.Narang and A. Ortega in Acoustics, Speech and SignalProcessing (ICASSP), 2011 IEEE International Conference

and a recent result of R. Strichartz in Fourier Analysis andApplications, 2016.

Example: Cycle graph

As another example let us consider a cycle graphC100 = {1,2, ...,100} on 100 vertices and suppose we aregoing to determine all eigenvalues which are not greater thanω = 0.002. Note that the space PW0.002(C100) is the span of alleigenfunctions whose eigenvalues are not greater than 0.002.


According to our results a sampling set for the spacePW0.002(C100) can be constructed as a compliment of a setS =

⋃j Sj such that Sj = Sj ∪ bSj (here bSj is the vertex

boundary of Sj ) are disjoint and

|Sj | <π

2 arcsin√

0.0022

− 1 > 49− 1 = 48.

Thus we can take |Sj | = 48 and it means that one of possibleuniqueness sets U will contain four vertices with numbers 1, 2,51, and 52. Thus we can conclude that there are at most foureigenvalues of the Laplace operator which are not greater than0.002. In fact there are three such eigenvalues λ0 = 0, and adouble eigenvalue λ1 = 1− cos(2π/100) ≈ 0.001973.


Similar calculations show that in the case when ω = 0.008 thedimension of a uniqueness set U can be taken equal eight andthere are five eigenvalues which are less than 0.008: λ0 = 0and two double eigenvalues λ1 ≈ 0.001973, andλ2 = 1− cos(4π/100) ≈ 0.007885.

Bandlimited localized frames on combinatorial graphs

Given a sequence of subsets of vertices

U1 ⊂ U2 ⊂ .... ⊂ UJ ⊂ V (G)

one can estimate a corresponding sequence of cut-offfrequencies

0 ≤ ω1 ≤ ω2 ≤ ... ≤ ωJ ≤ λ|V (G)|−1

and the corresponding sequence of subspaces

PW[ωj−1,ωj ], j = 1, ..., J


Let Pj , j = 1, ..., J, represents orthogonal projection

Pj : L2(G)→ PW[ωj−1,ωj ]

Every f ∈ L2(G) can be represented as

f =∑

j

Pj f


Note, that Plancherel-Polya inequality (10) for everyuniqueness set Uj can be rewritten as

C1∑u∈Uj

|⟨Pj f , δu

⟩|2 ≤ ‖Pj f‖2 ≤ C2

∑u∈Uj

|⟨Pj f , δu

⟩|2, (14)

where δu is a Dirac measure at a vertex u.

Using self-adjointness of operators Pj and summing over all jwe obtain a wavelet-type frame in the space L2(G).


C1∑

j

∑u∈Uj

∣∣∣⟨f ,Φuj

⟩∣∣∣2 ≤ ‖f‖2 ≤ C2∑

j

∑u∈Uj

∣∣∣⟨f ,Φuj

⟩∣∣∣2 ,where Φu

j = Pjδu, and δu is a Dirac measure at a vertex u.

It is important that every Φuj is

1 bandlimited to interval [ωj−1, ωj ];2 gets essentially localized around vertex u when j is

increasing.

Variational splines on combinatorial graphs (Pesenson I., 2009)

Given a subset of vertices U = {u} ⊂ V (G), a functionf ∈ L2(G), a natural k , we consider the following variationalproblem:Find a function Y from the space L2(G) which has the followingproperties:

1 Y U,fk (u) = f (u),u ∈ U,

2 Y U,fk minimizes functional Y → ‖∆k/2Y‖.

We prove that this problem has a unique solution.

The function Y U,fk is called a variational interpolating spline of

order k . The set of all variational splines for a fixed U ⊂ V (G)and fixed k will be denoted as Y(U, k).The following representation completely describes structure ofall variational splines

Y U,fk =

∑u∈U

αuEu2k , αu = αu(Y U,f

k ) (15)

where every Eu2k is a fundamental solution in the sense that

∆kEu2k = δu, (16)

where δu is a Dirac measure at u ∈ V (G).

Another representation of splines is given by the formula

Y U,fk =

∑u∈U

f (u)Luk , (17)

where Luk is the so-called Lagrangian spline which is the

solution to the same variational problem with constraints:Lu

k (u) = 1, u ∈ U, and Luk (v) = 0 for all other points v in U. The

formula shows that Lagrangian splines {Luk}u∈U form a basis in

the space of all splines Y(U, k).These variational interpolating splines are used for recovery ofeigenfunctions on graphs. Namely, the following ApproximationTheorem holds true.

TheoremFor any 1/Λ-removable set S every function f ∈ Eω(∆) where

0 < ω <1Λ

(18)

is uniquely determined by its values on U = V (G) \ S and canbe reconstructed from these values as the following limit

f = limk→∞

Y U,fk , k ∈ N,

where Y U,fk is a unique spline in Y(U, k) interpolating f on the

set U = V (G) \ S. Furthermore, the following error estimateholds true

‖f − Y U,fk ‖ ≤ 2γk‖f‖, γ = Λω < 1, k ∈ N. (19)

Eigenvalue and eigenfunction approximation on combinatorialgraphs. The Rayleigh-Ritz method (Pesenson I., 2009)

To explain the main result let us remind, that according to themin-max principle for a self-adjoint positive definite operator ∆in the Hilbert space L2(G) the j-th eigenvalue λj can becalculated by the formula

λj = infF⊂L2(G)supf∈F‖∆1/2f‖2

‖f‖2, f 6= 0, (20)

where inf is taken over all j-dimensional subspaces F of L2(G).

This formula shows that in order to determine an eigenvalue λjone has to search over all j-dimensional subspaces.

The idea is to use subspaces of variational interpolating splinesto find approximations to eigenvalues.

Note that originally, the idea to use spaces of piecewise linearfunctions (which are splines) as trial subspaces for theRayleigh-Ritz method for one-dimensional Sturm-Liouvilleboundary value problems belongs to Courant. We extend theseideas to combinatorial graphs.

Now we return to the variational definition of eigenvalues anddefine them as limits of eigenvalues of certain matrices inspaces of interpolating splines. We introduce the numbersλ

(k)j (U) by the formula

λ(k)j (U) = infF⊂Y(U,k)supf∈F

‖∆1/2f‖2

‖f‖2, f 6= 0, (21)

where inf is taken over all j-dimensional subspaces of Y(U, k).

As a consequence of the min-max principle we obtain that thenumbers λ(k)

j (U) are the eigenvalues of the matrixD(k) = D(k)(U) with entries

dγ,νk =∑

v∈V (G)

(∆Lµk )(v)Lνk (v), µ, ν ∈ U. (22)

Now we can formulate our main result which shows thateigenvalues of matrices D(k) approximate eigenvalues of theLaplace-Beltrami operator and the rate of convergence isexponential.

TheoremIf S is a 1/Λ-removable set and U = V (G) \ S then for any0 < ω < 1/Λ, and all sufficiently large k every eigenvalueλj ≤ ω can be approximated by the following double inequality

λ(k)j (U)− 2ωγk ≤ λj ≤ λ

(k)j (U), (23)

whereγ = Λω < 1.

This Theorem shows that a way to determine eigenvalues froma small interval [0, ω] is by keeping a set U = V (G) \ S withΛ(S) < 1/ω fixed and by letting degree of smoothness k togo to infinity.

One should realize that this way of convergence is quitedifferent from a traditional way of approximation by splines onRn when the distance between interpolation points goes tozero.

Note, that the dimension of the spline space Y(U, k) is exactlythe cardinality of the set U. In our applications to theeigenvalue problem U is always a uniqueness set of the formU = V (G) \ S for an appropriate removable set S. Thus, byreducing cardinality of U (≡ making removable set S bigger) wereduce cardinality of the space Y(U, k) in which we seekapproximations to eigenfunctions and eigenvalues by using theRayleigh-Ritz method.

Sampling of non-stationary signals. Schrodinger equation ongraphs. (Pesenson I., 2014)

We consider the following Cauchy problem for aSchrodinger-type equation

dg(t , v)

dt= i∆g(t , v), g(0, v) = f (v) ∈ L2(G), (24)

where v ∈ V (G), t ∈ R.The unique solution to this problem is given by the formulag(v , t) = eit∆f (v), −∞ < t <∞, v ∈ V (G), where eit∆ is agroup of unitary operators in L2(G).

A regular sampling theorem

The next theorem is a generalization of what is known as theValiron-Tschakaloff sampling/interpolation formula.

It gives an explicit formula for a solution g(t , ·) = eit∆f (·) to (24)in terms of its samples g

(kπ‖∆‖ , ·

), where ‖∆‖ is the norm of the

operator ∆.

Let us remind that sinc(t) is defined as sinπtπt , if t 6= 0, and 1, if

t = 0.

A regular sampling theorem

Theorem

For every f ∈ L2(G) we have for all t ∈ R

g(t , v) = it sinc(‖∆‖tπ

)∆f (v) + sinc

(‖∆‖tπ

)f (v)+

∑k∈Z, k 6=0

‖∆‖tkπ

sinc(‖∆‖tπ− k

)g(

kπ‖∆‖

, v), (25)

where g(t , v) = eit∆f (v) and convergence is in the space ofabstract functions L2 ((−∞,∞), L2(G)) with the regularLebesgue measure.

Theorem

For f ∈ Eω(G), ω > 0, we have for all t ∈ R

g(t , v) = it sinc(ωtπ

)∆f (v) + sinc

(ωtπ

)f (v)+

∑k∈Z, k 6=0

ωtkπ

sinc(ωtπ− k

)g(

kπω, v), (26)

where g(t , v) = eit∆f (v) and convergence is in the space ofabstract functions L2 ((−∞,∞), L2(G)) with the regularLebesgue measure.

Sampling of Paley-Wiener vectors in Hilbert spaces

The goal of this section is to develop an abstract "irregular"sampling theory for Paley-Wiener vectors associated with aself-adjoint operator ∆ in a Hilbert space H.

In what follows the notation Dk , k ∈ N is used for the domain ofthe operator ∆k with the graph norm

‖f‖k = ‖f‖+ ‖∆k f‖.


We assume that on the space D1 a set of continuousfunctionals {Φν} is given and they satisfy the following twoinequalities. There exist constants λ,C, c ≥ 0, such that forevery f ∈ D1

c

(∑ν

|Φν(f )|2)1/2

≤ ‖f‖ ≤

C

(∑ν

|Φν(f )|2)1/2

+ λ‖∆f‖. (27)

Let Z0 be the intersection of all kernels Ker Φν . We say that aset M is a uniqueness set for the set of functionals {Φν} ifthe intersection of M and Z0 is trivial.


Let’s assume ( for simplicity only) that ∆ has a discretespectrum

0 ≤ λ1 ≤ λ2 ≤ ....

and a corresponding set of eigenvectors{

eλj

}∆eλj = λjeλj

which form an orthonormal basis of H.


One can prove the following uniqueness theorem.

TheoremIf the right-hand side of the inequality (27) holds true thenPWω(∆) is the uniqueness set for the set of functionals {Φν} aslong as

ωλ < 1. (28)


Now we are going to introduce a reconstruction method thatuses the idea of Hilbert frames. Since by assumption thefunctionals Φν are continuous on a Hilbert space Dk , the RieszTheorem about continuous functionals implies the existence ofvectors ψν ∈ Dk such that for any f ∈ Dk ,

Φν(f ) = 〈f , ψν〉 .


If f ∈ PWω(∆) and the assumption (28) is satisfied, theinequalities (27) along with the Bernstein inequality show thatthere exist constants A,B > 0 such that the following frameinequality

A

(∑ν

|〈f , ϕν〉|2)1/2

≤ ‖f‖ ≤ B

(∑ν

|〈f , ϕν〉|2)1/2

(29)

holds where ϕν is the orthogonal projection of ψν on the spacePWω(∆). Thus, by using the classical ideas of Duffin andSchaeffer about dual frames we obtain the followingreconstruction formula.


TheoremThere exists a frame {θν} in the space Dk such that thefollowing reconstruction formula holds

f =∑ν

〈f , ϕν〉 θν =∑ν

Φν(f )θν , (30)

for every f ∈ PWω(∆).

Variational splines in Hilbert spaces

Now we introduce another reconstruction algorithm that usesthe idea of variational splines in Hilbert spaces. To formulatethis reconstruction algorithm we have to introduce the followingdefinition of variational splines associated with a self-adjointoperator.

For the given sequence a = {aν} ∈ l2 the set of all vectors fromDk such that Φν(f ) = aν will be denoted by Za(Dk ).

DefinitionA variational spline interpolating vector f ∈ ∆k , k ∈ N, isdenoted by sk (f ) and it is a vector in Za(Dk ),a = {Φν(f )},which minimizes the functional u → ‖∆ku‖,u ∈ Dk .

Reconstruction in Hilbert spaces using splines

The following result holds:

TheoremUnder the above assumptions the optimization problem has aunique solution for every k = 2l , l ∈ N. Moreover, if thefunctionals Φν satisfy (27), then any f ∈ PWω(∆) can bereconstructed through the formula

f = limk→∞

sk (f ) (31)

and the error estimate is

‖f − sk (f )‖ ≤ 2(λω)k‖f‖, k = 2l , l = 0,1,2, ....

Quantum graphs

Quantum graphs found numerous applications in physics,chemistry, engineering and quantum computing. They serve asmodels in many situations when one deals with waves thatpropagate in "thin" media.

By a quantum graph we understand a pair (Γ,∆), where Γ is ametric graph and ∆ is a Hamiltonian on Γ, which acts on eachedge as the second derivative and whose domain is describedin terms of the Neumann(Kirchhoff) compatibility conditions atvertices, which link the edges together.

Quantum graphs

A metric graph Γ is a set of vertices V = {vi} and edgesE = {ei} each of length |ei | ∈ (0,∞]. We identify every edge ewith a segment [0, |e|] of R1 and use coordinate xe along it.

We consider graphs with finite number of edges of finite length.Graph Γ can be equipped with a natural metric and theLebesgue measure dx .

Quantum graphs

The space L2(Γ) is defined as the direct sum of spacesL2(e),e ∈ E , with the scalar product

< f ,g >=∑e∈E

∫e

f gdx , f ,g ∈ L2(Γ), (32)

and the norm

‖f‖L2(Γ) =

(∑e∈E

∫e|f |2dx

)1/2

. (33)

Quantum graphs

The Sobolev space H1(Γ) consists of all continuous functionson Γ that belong to H1(e) on every edge and we will always usethe following norms

‖f‖H1(e) =

(∫e

(|f |2 +

∣∣∣∣ dfdx

∣∣∣∣2)

dx

)1/2

, (34)

and

‖f‖H1(Γ) =

(∑e∈E

∫e

(|f |2 +

∣∣∣∣ dfdx

∣∣∣∣2)

dx

)1/2

. (35)

Quantum graphs

The continuity assumption means that for every vertex v andany two edges e1,e2 containing v the following boundarycondition holds true

limx→v ,x∈e1

f (x) = limx→v ,x∈e2

f (x) = f (v). (36)

There are many ways to introduce a self-adjoint operator on Γwhich is called a Hamiltonian. The following definition gives aprecise description of the operator we are dealing with.

DefinitionThe Hamiltonian ∆ is defined by the formula

− d2

dx2 (37)

on each edge e ∈ E and its domain D(∆) consists of allfunctions f from L2(Γ) such that 1) f belongs to the Sobolevspace H2(e) on each edge e ∈ Γ, 2) f is continuous on Γ, 3) atevery vertex v of degree d every f ∈ D(∆) satisfies the socalled Neumann (Kirchhoff) conditions∑

e∈Ev

dfdx

(v) = 0, (38)

where Ev is the set of all edges containing v as a vertex andthe derivatives are taken in the directions away from the vertex.

The operator ∆ is a self-adjoint non-negative operator and weintroduce the scale of Sobolev spaces H2k (Γ) associated withthe Hamiltonian ∆ as the domains of the powers ∆k with thegraph norm.

In the case of a finite graph Γ the spectrum of the Hamiltonian∆ is discrete, non-negative and goes to infinity.

We will use the notation PWω(Γ) for the linear span of alleigenfunctions of the Hamiltonian ∆ whose correspondingeigenvalues are not greater than a positive ω.

Given two numbers

0 < α ≤ β ≤ mine∈E|e|,

we say that a set Iα,β of open and pairwise disjoint intervals Ij isan admissible (α, β)-cover of Γ if:

1 for every jα ≤ |Ij | ≤ β;

2 the union of open intervals Ij does not contain vertices of Γ;3 closures of the intervals Ij cover the graph Γ.

An (α, β)-lattice Xα,β is a set of points {xj} where every xjbelongs to an open interval Ij from an admissible (α, β)-coverIα,β.

Note that the second condition implies that every interval Ijbelongs to the interior of an edge.

One can show that there are two absolute constantsC1 > 0,C2 > 0, such that for any (α, β)-lattice Xα,β the followinginequalities holds true for all m = 2l , l = 0,1, ..., f ∈ H2m(Γ)

C1β1/2

∑j

|f (xj)|21/2

≤ ‖f‖L2(Γ) ≤

C2

β1/2

∑j

|f (xj)|21/2

+ β2m‖∆mf‖L2(Γ)

. (39)

The last two inequalities imply the Plancherel-Polya inequalitiesfor functions in PWβ−1/2(Γ)

C1β1/2

∑j

|f (xj)|21/2

≤ ‖f‖L2(Γ) ≤ C2β1/2

∑j

|f (xj)|21/2

.

(40)

SAMPLING ON MANIFOLDS

In the last decade, methods based on various kinds of waveletbases on the unit sphere S2 and on the rotation group SO(3)have found applications in virtually all areas where analysis ofspherical data is required, including cosmology, weatherprediction, geodesy, crystallography, and even biology.

In this talk I will discuss Shannon-type sampling of bandlimitedfunctions on compact Riemannian manifolds.

Bandlimideness

A Riemannian manifold M is a differentiable manifold which isequipped with a notion of distance (=metric).

In the case of a Riemannian manifold there always exists asecond-order differential elliptic operator L on M which isnaturally attached to this metric. This operator is known as theLaplace-Beltrami operator. It is self-adjoint and non-negativedefinite.

EXAMPLE. If M is the standard unit sphere Sd−1:x2

1 + ...+ x2d = 1 in Rd then Laplace-Beltrami operator on Sd−1

is a restriction of the regular Laplace operator in Rd .

In this case a function f ∈ L2(M) is ω-bandlimited if it is a linearcombination of eigenfunctions uk of L whose correspondingeigenvalues are not greater ω.

The following Plancherel-Polya-type Theorem shows that inspaces of bandlimited functions the regular L2(M) norm isequivalent to a discrete one.

It is again, a frame-type inequality.

A sampling theorem on manifolds

TheoremThere exist constant c0 = c0(M) such that for any 0 < δ < 1 anyω > 0, every metric ρ-lattice Mρ = {xk} with ρ = c0ω

−1/2 thefollowing inequalities hold true

(1− δ)

(∑k

|f (xk )|2)1/2

≤

ρ−d/2‖f‖L2(M) ≤

(1 + δ)

(∑k

|f (xk )|2)1/2

, d = dim M, (41)

for all f ∈ PWω(L).

Pointwise Sampling Theorem on Riemannian manifolds

It implies the following analog of the Shannon SamplingTheorem holds on compact and non-compact manifolds ofbounded geometry.

Theorem(Pesenson I., 2000)ω-bandlimited functions are completely determined by theirvalues on a metric lattice Mρ = {xk} of points xk ∈ M"uniformly" distributed over M with a spacing ρ comparable to

c0(M)√ω∼ ρ, (42)

where c0(M) > 0 depends only on M. Every f ∈ PWω(L) can bereconstructed from the values {f (xk )}, xk ∈ Mρ in a stable way.

The formula (42) is very important since it gives a specificrelation between bandwidth and a rate of sampling.

It turns out that this formula is essentially optimal.

Indeed, according to the Weyl’s asymptotic formula one has

dim PWω(L) ∼ C Vol(M)ωd/2, d = dim M, (43)

where d = dim M and C is an absolute constant.

Optimality of sampling and Weyl’s formula

The formula (42) shows that in the case of a compact manifoldthe dimension of PWω(L) is comparable to the number of pointsin any c0(M)ω−1/2 lattice.But the number of points in an "optimal" lattice can beapproximately estimated as

Vol(M)

c′0ω−d/2

= cVol(M)ωd/2, d = dim M.

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