Isaac Z. Pesenson

Sampling in Paley-Wiener spaces on combinatorial graphs

A notion of Paley-Wiener spaces on combinatorial graphs is introduced. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such uniqueness sets are described in terms of Poincare-Wirtinger-type inequalities. A reconstruction algorithm of Paley-Wiener functions from uniqueness sets which uses the idea of frames in Hilbert spaces is developed. Special consideration is given to the n-dimensional lattice, homogeneous trees, and eigenvalue and eigenfunction problems on finite graphs.

A sampling theorem on homogeneous manifolds

We consider a generalization of entire functions of spherical exponential type and Lagrangian splines on manifolds. An analog of the PaleyWiener theorem is given. We also show that every spectral entire functionl on a manifold is uniquely determined by its values on some discrete sets of points. The main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using Lagrangian splines.

Poincaré-Type inequalities and reconstruction of Paley-Wiener functions on manifolds

The main goal of the article is to show that Paley-Wiener functions ƒ ∈ L2(M) of a fixed band width to on a Riemannian manifold of bounded geometry M completely determined and can be reconstructed from a set of numbers Φi (ƒ), i ∈ ℕwhere Φiis a countable sequence of weighted integrals over a collection of “small” and “densely” distributed compact subsets. In particular, Φi, i ∈ ℕ,can be a sequence of weighted Dirac measures δxi, xi ∈M.It is shown that Paley-Wiener functions on M can be reconstructed as uniform limits of certain variational average spline functions.To obtain these results we establish certain inequalities which are generalizations of the Poincaré-Wirtingen and Plancherel-Polya inequalities.Our approach to the problem and most of our results are new even in the one-dimensional case.

Sampling of Paley-Wiener functions on stratified groups

We consider a generalization of entire functions of spherical exponential type on stratified groups. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a stratified group is uniquely determined by its values on some discrete subgroups. The main result of the article is reconstruction formula of spectral entire functions from their values on discrete subgroups using Lagrangian splines.

Bernstein--Nikolskii inequalities and Riesz interpolation formula on compact homogeneous manifolds

Bernstein-Nikolskii inequalities and Riesz interpolation formula are established for eigenfunctions of Laplace operators and polynomials on compact homogeneous manifolds.

A reconstruction formula for band limited functions in

It is shown that a band limited function from L2(R) can be reconstructed from irregularly sampled values as a limit of spline functions. The assumption about the sampling sequence is that it should be dense enough. It is known that band limited functions, i.e. functions whose Fourier transform have compact support, are uniquely determined and can be recovered from their values on specific discrete set of points xn. The classical Whittaker-Shannon Theorem gives the sharp answer in the case of equally spaced points in R. The irregularly spaced sets of sampling points were considered by Paley and Wiener [6]. More recent results and extensive lists of references can be found in surveys of Benedetto [1] and Feichtinger and Gröchening [2]. We prove that a band limited function from L2(R) can be recovered from its values on specific irregularly distributed sampling points as the limit of d-dimensional polyharmonic spline functions. The assumption about the sampling sequence is that it should be dense enough. Our reconstruction formula is very stable due to the fact that fundamental splines have exponential decay. Moreover, we prove that convergence takes place not only in the space Lp(R), 2 ≤ p ≤ ∞, but also in the C norm for any k > 0. In particular, it shows the way to reconstruct all derivatives of a band limited function using samples of the function. Splines as a tool for reconstruction in the one-dimensional case were used by Schoenberg [9] for equally spaced knots and recently by Lyubarskii and Madych [3] under the assumption that corresponding exponential functions form a Riesz basis in L2([−π, π]). Technically our paper is close to [5]. Our proofs rely explicitly on the observation that certain constants in inequalities from Lemmas 1 and 2 below have a very special form: they depend exponentially of the order of derivatives involved in inequalities. This allows us to control the convergence when the order of derivatives goes to infinity. This technique is very flexible and can be used to solve the reconstruction problem in new situations [7], [8]. A new application of this technique is also given at the end of this paper. Let X(λ) denote a countable set {xγ} ∈ R such that the rectangles D(xν , λ) of diameter ≤ λ each containing exactly one point xν form a disjoint cover of R. Received by the editors August 23, 1997 and, in revised form, February 17, 1998. 1991 Mathematics Subject Classification. Primary 42A65; Secondary 42C15. c ©1999 American Mathematical Society

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We present here a simple construction of a wavelet system for the three-dimensional ball, which we label \emph{Radial 3D Needlets}. The construction envisages a data collection environment where an observer located at the centre of the ball is surrounded by concentric spheres with the same pixelization at different radial distances, for any given resolution. The system is then obtained by weighting the projector operator built on the corresponding set of eigenfunctions, and performing a discretization step which turns out to be computationally very convenient. The resulting wavelets can be shown to have very good localization properties in the real and harmonic domain; their implementation is computationally very convenient, and they allow for exact reconstruction as they form a tight frame systems. Our theoretical results are supported by an extensive numerical analysis.

Simple proposal for radial 3D needlets

We prove Poincare and Plancherel--Polya inequalities for weighted

Poincaré and Plancherel--Polya Inequalities in Harmonic Analysis on Weighted Combinatorial Graphs

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Band limited functions on quantum graphs

-spaces on weighted graphs in which the constants are explicitly expressed in terms of some geometric characteristics of a graph. We use a Poincare-type inequality to obtain some new relations between geometric and spectral properties of the combinatorial Laplace operator. Several well-known graphs are considered to demonstrate that our results are reasonably sharp. The Plancherel--Polya inequalities allow for application of the frame algorithm as a method for reconstruction of Paley--Wiener functions on weighted graphs from a set of samples. The results are illustrated by developing Shannon-type sampling in the case of a line graph.