On stability of finitely generated shift-invariant systems
type: Item
instance of: scholarly article
title: On stability of finitely generated shift-invariant systems
zbMATH Open document ID: 1209.41006
published in: The Journal of Fourier Analysis and Applications
publication date: 2011-01-19T00:00:00Z
full work available at URL: R-2008-15.pdf
review text: Let \({\Psi}=\{\psi_1, \dots, \psi_N\} \subset L^2(\mathbb R^d)\), \(T({\Psi}) := \{ \psi(\cdot - k): \psi \in {\Psi}, k \in\mathbb Z^d\}\), and let \(S({\Psi})\) denote the closure of the linear span of \(T({\Psi}\)). The author focuses on the case where \(T({\Psi})\) has a unique biorthogonal system \(\{g^{\psi}\}\) in \(S({\Psi})\), defines the ``rectangular'' partial sum operators by \[ T_{\mathbf N}f:= \sum_{\psi \in {\Psi}} \sum_{k\in\mathbb Z^d: |k_i|\leq N_i} \langle f, g_k^{\psi}\rangle \psi(\cdot - k), \] for \(f \in S({\Psi})\) and \({\mathbf N}=(N_1, N_2, \dots, N_d) \in \mathbb{N}_0^d\), where \(\mathbb{N}_0\) denotes the set of nonnegative integers, and finds necessary and sufficient conditions for \(T_{\mathbf N}f\to f\), as \(\min_i N_i \to +\infty\), for all \(\;f \in S({\Psi})\).
reviewed by: Richard A. Zalik
Mathematics Subject Classification ID: 41A45
Mathematics Subject Classification ID: 42C15
zbMATH DE Number: 5838815
zbMATH Keywords: shift-invariant space
zbMATH Keywords: Schauder basis
zbMATH Keywords: integer translates
zbMATH Keywords: vector Hunt--Muckenhoupt-Wheeden theorem
zbMATH Keywords: Muckenhoupt condition
author: Morten Nielsen
MaRDI profile type: Publication
OpenAlex ID: W2094125010
cites work: The structure of shift-invariant subspaces of \(L^2(\mathbb{R}^n)\)
cites work: Inverse volume inequalities for matrix weights
cites work: Some recent developments in Fourier analysis and 𝐻^{𝑝}-theory on product domains
cites work: Ten Lectures on Wavelets
cites work: Fourier analysis of the approximation power of principal shift-invariant spaces
cites work: On the construction of multivariate (pre)wavelets
cites work: Approximation from Shift-Invariant Subspaces of L 2 (ℝ d )
cites work: The structure of finitely generated shift-invariant spaces in \(L_ 2(\mathbb{R}^ d)\)
cites work: Radial Basis Function Approximation: from Gridded Centres to Scattered Centres
cites work: Matrix-weighted Besov spaces and conditions of A_p type for 0 less than p \leq 1
cites work: Matrix \(A_p\) weights via maximal functions.
cites work: Gabor Schauder bases and the Balian-Low theorem
cites work: Q5730397
cites work: Q4896587
cites work: Weighted Norm Inequalities for the Conjugate Function and Hilbert Transform
cites work: Q5822309
cites work: Schauder bases of integer translates
cites work: Harmonic analysis on nilpotent groups and singular integrals. I: Oscillatory integrals
cites work: Q4328451
cites work: Frames and Stable Bases for Shift-Invariant Subspaces of L2(ℝd)
cites work: Matrix-weighted Besov spaces
cites work: Continuous frame decomposition and a vector Hunt-Muckenhoupt-Wheeden theorem
cites work: Wavelets and the angle between past and future
DOI: 10.1007/S00041-009-9096-7
description: scientific article; zbMATH DE number 5838815