Universal sampling and interpolation of band-limited signals
type: Item
instance of: scholarly article
title: Universal sampling and interpolation of band-limited signals
zbMATH Open document ID: 1169.42014
author: Alexander Ulanovskii
author: Alexander Olevskiĭ
published in: Geometric and Functional Analysis. GAFA
publication date: 2009-06-17T00:00:00Z
review text: Let \(S\subset {\mathbb R}\) be a bounded (measurable) set and let \(PW_S\) denote the Paley--Wiener space of all functions \(f\in L^2({\mathbb R})\) which are the Fourier transforms of functions from \(L^2(S)\). Given a sequence of distinct real numbers \(\Lambda=\{\lambda_j, j\in {\mathbb Z}\}\), one associates with each \(f \in PW_S\) its ``samples'' \(f\rightarrow \{f(\lambda)\}\), \(\lambda\in \Lambda\). One can raise the following three problems: A) When can the signal \(f\) be recovered (i.e. uniquely defined) from the samples? B) When do the sample provide a ``stable reconstruction'' of the signal, that is, the frame condition \(A \|f\|_{L^2({\mathbb R})}\leq (\sum_{\lambda\in\Lambda} | f(\lambda) |^2)^{1/2}\leq B \|f\|_{L^2({\mathbb R})}\) holds with some positive constants \(A\) and \(B\) for every \(f\) in \(PW_S\)? C) For every data \(\{c_j, j\in {\mathbb Z}\}\in \ell^2({\mathbb Z})\) when is there a signal \(f\in PW_S\) satisfying \(f(\lambda_j)=c_j\), \(j\in {\mathbb Z}\), i.e. \(\Lambda\) is the set of interpolation for \(PW_S\)? In this paper, the authors are interested in the following version of problems A) and B): Is it possible to define a ``univeral'' sampling sequence \(\Lambda\) of a given density which provides reconstruction or stable reconstruction of any signal whose spectrum \(S\) has a sufficiently small Lebegue measure? A similar variant of C) is: Does there exist \(\Lambda\) of positive density which is a set of interpolation for every space \(PW_S\) with sufficiently large mes \(S\)? It is proved that the universal uniqueness can be achieved in the general situation. On the other hand, for stable reconstruction it is crucial to know whether the spectrum is compact or dense. A similar result is true for the interpolation. Use is made of Beurling's and Kahane's densities.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 42C30
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 94A20
zbMATH DE Number: 5565915
zbMATH Keywords: band-limited signal
zbMATH Keywords: universal sampling pattern
zbMATH Keywords: universal interpolation
zbMATH Keywords: universal completeness of exponentials
zbMATH Keywords: frame
zbMATH Keywords: Riesz basis
MaRDI profile type: Publication
full work available at URL: s00039-008-0674-7
OpenAlex ID: W2014453432
DOI: 10.1007/S00039-008-0674-7
description: scientific article; zbMATH DE number 5565915