Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces
type: Item
instance of: scholarly article
title: Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces
zbMATH Open document ID: 1016.42022
DOI: 10.1016/S1063-5203(02)00503-1
published in: Applied and Computational Harmonic Analysis
publication date: 2003-04-02T00:00:00Z
review text: The paper concerns non-uniform sampling, and the setup is as follows. Assume that the continuous function \(\phi: \mathbb{R}^d\to C\) (belonging to the Wiener space) generates a Riesz basis for the shift-invariant space \[ V^2=\Bigl \{ \sum_{k\in Z^d} c_k \phi(\cdot -k): \{c_k\}\in \ell^2\Bigr\}. \] Consider furthermore some points \(\{x_j\}\subset \mathbb{R}^d\) and associated bounded functions \(\psi_{x_j}\) such that supp \(\psi_{x_j}\subset x_j+ [-a,a]^d\) for some \(a>0\), and such that for some constants \(c,C>0\), \[ c||f||^2 \leq \sum_j |\langle f, \psi_{x_j}\rangle |^2 \leq C||f||^2, \;\forall f\in V^2. \] Note that this is different from a frame condition because the functions \(\psi_{x_j}\) are not assumed to belong to \(V^2\). The paper presents an iterative algorithm to recover a function \(f\in V^2\) from the data \(\{\langle f, \psi_{x_j}\rangle \}\), provided that the points \(\{x_j\}\) are sufficiently dense and the support of \(\psi_{x_j}\) is sufficiently small. The convergence is geometric, and the rate of convergence is estimated.
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1891397
zbMATH Keywords: wavelet space
zbMATH Keywords: non-uniform sampling
zbMATH Keywords: Riesz basis
zbMATH Keywords: shift-invariant space
author: Akram Aldroubi
MaRDI profile type: Publication
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full work available at URL: s1063-5203(02)00503-1
OpenAlex ID: W2100836068
description: scientific article; zbMATH DE number 1891397