Completeness of integer translates in function spaces on \(\mathbb{R}\)
type: Item
instance of: scholarly article
title: Completeness of integer translates in function spaces on \(\mathbb{R}\)
zbMATH Open document ID: 0872.46021
published in: Journal of Approximation Theory
publication date: 1997-10-12T00:00:00Z
review text: The following problem of completeness is considered. Let \({\mathbf B}\) be a Banach space of functions (or distributions) on \(\mathbb{R}\) that is translation invariant, that is, \(T_sf\in{\mathbf B}\) when \(f\in{\mathbf B}\) and \(s\in\mathbb{R}\); \(T_sf(x)= f(x-s)\) stands for the translation operator. The problem is, does \({\mathbf B}\) contain a function \(f\in{\mathbf B}\) whose integer translates span the whole space, where span means the closed linear hull: \({\mathbf B}=\text{span} (T_nf:n\in\mathbb{Z})\)? Such functions are called generators or \(\mathbb{Z}\)-cyclic elements; functions \(f\) satisfying \({\mathbf B}=\text{span} (T_nf:n\geq 0)\) are called \(\mathbb{Z}_+\)-cyclic. The problem of completeness of all translates \(T_sf\), \(s\in\mathbb{R}\) is classical and well studied. For instance, the Wiener Tauberian theorem for the space \(L^1 (\mathbb{R})\), or generally, for the group algebra \(L^1(G)\) on a locally compact abelian group \(G\), describes all generators with respect to the group action in terms of their Fourier transforms. The existence in itself of such group generators is almost obvious for most, but not for all, translation invariant spaces. For integer translates, the situation is completely different, and, at first sight, the very fact of the existence of \(\mathbb{Z}\)-cyclic elements seems quite striking. For example, the basic spectral theory of unitary operators implies that there is no finite subset \(F\subset L^2(\mathbb{R})\) which would be \(\mathbb{Z}\)-cyclic in \(L^2(\mathbb{R})\). It is an easy exercise to see that the same is true for the space \(L^1(\mathbb{R})\). However, for many other natural function spaces, \(\mathbb{Z}\)-cyclic, and even \(\mathbb{Z}_+\)-cyclic, elements exist. For instance, the main result of the paper under review (theorem 2.1) says that each of the Banach spaces \(C_0^k(\mathbb{R})\), \(H^{p,k}(\mathbb{R})\), \(2<p<\infty\), \(k=0,1,\dots\) and \(L^2(\mathbb{R}, \rho(x)dx)\), where \(\rho…
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 980675
zbMATH Keywords: integer translates
zbMATH Keywords: uniqueness sets
zbMATH Keywords: weighted function spaces
zbMATH Keywords: problem of completeness
zbMATH Keywords: generators
zbMATH Keywords: \(\mathbb{Z}\)-cyclic elements
zbMATH Keywords: Wiener Tauberian theorem
zbMATH Keywords: composition operator
zbMATH Keywords: Orlicz
zbMATH Keywords: Lorentz
zbMATH Keywords: Marcinkiewicz
zbMATH Keywords: rearrangement invariant spaces
author: Aharon Atzmon
author: Alexander Olevskiĭ
MaRDI profile type: Publication
full work available at URL: jath.1996.0106
OpenAlex ID: W2065065158
DOI: 10.1006/JATH.1996.0106
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description: scientific article; zbMATH DE number 980675