Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces
type: Item
instance of: scholarly article
title: Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces
zbMATH Open document ID: 0964.42020
DOI: 10.1007/BF02510120
published in: The Journal of Fourier Analysis and Applications
publication date: 2001-07-09T00:00:00Z
full work available at URL: https://eudml.org/doc/59633
review text: In classical sampling theorems it is shown that a function can be recovered from its samples only if it is band limited. For irregular sampling, one could assume that the function belongs to some shift invariant space \(V^p(\phi)=\{f(\cdot)=\sum_{k\in{\mathbb Z}} c_k\phi(\cdot -k): (c_k)\in\ell^p({\mathbb Z})\}\) where \(\phi\) is some generating function (like in a multiresolution analysis). For example, if \(\phi\) is the indicator function \(\chi[0,1]\), then \(V^p(\phi)\) is the space of piecewise constant functions in \(L^p({\mathbb R})\). It is sufficient that there is one sample in each interval to reconstruct the function. The set of sample points \(X\) has to have a density \(D_\phi(X)=\inf\#(X\cap[k,k+1))\geq 1\). Moreover, it is important for practical computations that the reconstruction is stable, meaning that \(\|f\|_p\leq C(\sum_{x_j\in X}|f(x_j)|^p)^{1/p}\). This paper discusses this kind of results for \(\phi\) equal to the spline function \(\beta^n=\chi[0,1]*\chi[0,1]*\cdots*\chi[0,1]\) or when it is compactly supported. A general sufficient condition is given for stable reconstruction from irregular sampling in \(V^p(\beta^n)\) and necessary density conditions are given for general shift invariant subspaces \(V^p(\phi)\) when \(\phi\) is compactly supported.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 42C30
zbMATH DE Number: 1445628
zbMATH Keywords: irregular sampling
zbMATH Keywords: spline
zbMATH Keywords: shift invariant space
zbMATH Keywords: reproducing kernel Hilbert spaces
zbMATH Keywords: wavelets
author: Akram Aldroubi
author: Karlheinz Gröchenig
MaRDI profile type: Publication
cites work: Portraits of Frames
cites work: Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The 𝐿^{𝑝}-theory
cites work: Q4027567
cites work: Sampling procedures in function spaces and asymptotic equivalence with shannon's sampling theory
cites work: The collected works of Arne Beurling. Volume 1: Complex analysis. Volume 2: Harmonic analysis. Ed. by Lennart Carleson, Paul Malliavin, John Neuberger, John Wermer
cites work: Q3037302
cites work: Irregular sampling theorems for wavelet subspaces
cites work: Q4003152
cites work: A Class of Nonharmonic Fourier Series
cites work: Generalized Amalgams, With Applications to Fourier Transform
cites work: Q4027016
cites work: Iterative Reconstruction of Multivariate Band-Limited Functions from Irregular Sampling Values
cites work: Q4365116
cites work: Amalgams of 𝐿^{𝑝} and 𝑙^{𝑞}
cites work: Reconstruction Algorithms in Irregular Sampling
cites work: On Landau's Necessary Density Conditions for Sampling and Interpolation of Band-Limited Functions
cites work: The Zak transform and sampling theorems for wavelet subspaces
cites work: The Shannon sampling theorem—Its various extensions and applications: A tutorial review
cites work: Q3975831
cites work: Necessary density conditions for sampling an interpolation of certain entire functions
cites work: Irregular sampling for spline wavelet subspaces
cites work: Irregular sampling in wavelet subspaces
cites work: Multiresolution Approximations and Wavelet Orthonormal Bases of L 2 (R)
cites work: Q3995454
cites work: Q3895980
cites work: Nonperiodic sampling of bandlimited functions on unions of rectangular lattices.
cites work: A sampling theorem for wavelet subspaces
cites work: On sampling theorem, wavelets, and wavelet transforms
cites work: Applications of reproducing kernel Hilbert spaces–bandlimited signal models
OpenAlex ID: W2027349577
description: scientific article; zbMATH DE number 1445628