Principal shift-invariant spaces with extra invariance nearest to observed data
type: Item
instance of: scholarly article
title: Principal shift-invariant spaces with extra invariance nearest to observed data
zbMATH Open document ID: 1278.46026
author: Romain Tessera
author: Akram Aldroubi
author: Ilya A. Krishtal
author: Haichao Wang
published in: Collectanea Mathematica
publication date: 2013-06-13T00:00:00Z
review text: The principal shift-invariant (PSI) space generated by \(\phi \in L^2(\mathbb R)\) is the space \[ V(\phi) = \{ \sum_{k \in \mathbb Z} c_k \, \phi(\cdot -k): (c_k) \in \ell^2\}. \] Shift-invariant spaces are widely used in approximation theory, numerical analysis, wavelet theory, etc. Of particular interest are those PSI spaces with an orthonormal generator which also satisfy some additional invariance property, such as translation invariance or invariance with respect to \(\frac 1n {\mathbb Z}\) translations, for some \(n \in \mathbb N\). In a prior work, the authors have shown that, given a set of functions \(F =\{f_1, \dots. f_n\} \subset L^2(\mathbb R)\), there exists a PSI space \(V(\phi) \) with an orthonormal generator which is the nearest to \(F\) in the least squares sense, that is, \[ V(\phi) = \text{argmin}_{\, V' \in \mathcal V} \sum_{i=1}^m || f_i - P_{V'} f_i||^2, \] where \({\mathcal V}\) is the set of all PSI spaces of \(L^2(\mathbb R)\) with an orthonormal generator and \(P_{V'}\) is the orthogonal projection on \(V'\). The main result of this paper is the following refinement of the above observation. Namely, given a set of functions \(F =\{f_1, \dots. f_n\} \subset L^2(\mathbb R)\), the authors show how to construct a PSI space \(V(\phi)\) with an orthonormal generator which is the nearest to \(F\) in the least squares sense, where \(V(\phi)\) also satisfies translation invariance or invariance with respect to \(\frac 1n {\mathbb Z}\) translations, for some fixed \(n \in \mathbb N\).
reviewed by: Demetrio Labate
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 6175357
zbMATH Keywords: shift-invariant spaces
zbMATH Keywords: \(\frac 1n \mathbb Z\) invariance
zbMATH Keywords: least squares
zbMATH Keywords: translation invariance
MaRDI profile type: Publication
full work available at URL: s13348-011-0047-7
OpenAlex ID: W2061223304
cites work: Optimal shift invariant spaces and their Parseval frame generators
cites work: Invariance of a shift-invariant space
cites work: Nonuniform sampling and reconstruction in shift-invariant spaces
cites work: Convolution, average sampling, and a Calderon resolution of the identity for shift-invariant spaces
cites work: Uncertainty principles and Balian-Low type theorems in principal shift-invariant spaces
cites work: On the existence of optimal unions of subspaces for data modeling and clustering
cites work: Extra invariance of shift-invariant spaces on LCA groups
cites work: Parseval frame wavelets with \(E_{n}^{(2)}\)-dilations
cites work: The structure of shift-invariant subspaces of \(L^2(\mathbb{R}^n)\)
cites work: An introduction to frames and Riesz bases
cites work: Tight frame oversampling and its equivalence to shift-invariance of affine frame operators
cites work: Affine frame decompositions and shift-invariant spaces
cites work: Sampling expansions for functions having values in a Banach space
cites work: Non-translation-invariance and the synchronization problem in wavelet sampling
cites work: Q4896587
cites work: Fractional Splines and Wavelets
cites work: On the translation invariance of wavelet subspaces
DOI: 10.1007/S13348-011-0047-7
description: scientific article; zbMATH DE number 6175357