Frames and sampling theorem
type: Item
instance of: scholarly article
title: Frames and sampling theorem
zbMATH Open document ID: 0959.42021
DOI: 10.1007/BF02876230
published in: Science in China. Series A
publication date: 2001-03-27T00:00:00Z
review text: Let \(\{\varphi(\cdot -n)\}\) be a frame for a closed subspace \(V_0\) of \(L_2(R)\), where \(\varphi\) is a continuous function that is either locally absolutely continuous and \(\varphi^{'} \in L^2(R)\), or \( |\varphi(x)|\leq C(1+|x|)^{-1/2-\varepsilon}\) for some \(\varepsilon > 0\). The authors find a necessary and sufficient condition for the existence of a frame \(\{S(\cdot -n)\}\) for \(V_0\) such that for every \(f \in V_0\), \( f(t) = \sum_k f(k) S(t-k)\), where the convergence is in \(L^2(R)\). Among the corollaries of this result is a sampling theorem of \textit{G. G. Walter} [IEEE Trans. Inf. Theory 38, No. 2/II, 881-884 (1992; Zbl 0744.42018)]. For a smaller set of functions \(\varphi\), the authors also obtain a result on irregular sampling that yields, as a particular case, a theorem of \textit{Y. N. Liu} and \textit{G. G. Walter} [J. Fourier Anal. Appl. 2, No. 2, 181-189 (1995; Zbl 0886.42025)]. A number of examples illustrate the results obtained.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 94A12
zbMATH DE Number: 1329976
zbMATH Keywords: frames
zbMATH Keywords: sampling
zbMATH Keywords: wavelets
author: Wenchang Sun
author: Xing-Wei Zhou
reviewed by: Richard A. Zalik
MaRDI profile type: Publication
cites work: A sampling theorem for wavelet subspaces
cites work: Irregular sampling in wavelet subspaces
cites work: Ten Lectures on Wavelets
cites work: Differentiation and the Balian-Low theorem
cites work: Q4003151
full work available at URL: https://doi.org/10.1007/bf02876230
OpenAlex ID: W2002915001
description: scientific article; zbMATH DE number 1329976