Designing local orthogonal bases on finite groups. I: Abelian case
type: Item
instance of: scholarly article
title: Designing local orthogonal bases on finite groups. I: Abelian case
zbMATH Open document ID: 0956.43004
DOI: 10.1007/BF02510115
author: R. Bernardini
author: Jelena Kovačević
published in: The Journal of Fourier Analysis and Applications
publication date: 2001-03-07T00:00:00Z
full work available at URL: https://eudml.org/doc/59628
review text: Starting from the well-known equivalence property for the orthogonality of integer translates of functions \(f\in L_2\) \[ \langle f,f(\cdot-n)\rangle= \delta_{0n}\Leftrightarrow \sum_{K\in\mathbb{Z}} |\widehat f(\cdot+ 2\pi k)|^2= 1\quad\text{a.e.}\tag{1} \] the authors investigate a similar relation for general finite Abelian groups of unitary transforms. More precisely, they prove a similar orthogonality condition as in (1) for the sets \(\{Ub:U\in \Gamma\}\), \((b\in V)\), where \(V\) is a vector space with inner product and \(\Gamma\) is a finite Abelian group of unitary transforms on \(V\). The authors use their examinations to design an orthogonalization procedure for \(\{Ub: U\in\Gamma\}\). Further, they deal with several generating vectors \(b_1,\dots, b_L\). The non-Abelian case is addressed in Part II of this paper [ibid. 6, 207-231 (2000; see the following review)].
Mathematics Subject Classification ID: 43A70
Mathematics Subject Classification ID: 65T60
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 1445623
zbMATH Keywords: integer translates
zbMATH Keywords: finite Abelian groups
zbMATH Keywords: unitary transforms
zbMATH Keywords: orthogonalization procedure
reviewed by: Gabriele Steidl
MaRDI profile type: Publication
cites work: Local orthogonal bases. I: Construction
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OpenAlex ID: W2090858256
description: scientific article; zbMATH DE number 1445623