Remarks concerning completeness of translates in function spaces
type: Item
instance of: scholarly article
title: Remarks concerning completeness of translates in function spaces
zbMATH Open document ID: 0944.46023
published in: Journal of Approximation Theory
publication date: 2000-09-20T00:00:00Z
review text: In his previous papers [Sem. Math. V. A. Steklov, Math. Inst. Leningrad 19, 132-136 (1970; Zbl 0258.47030)] and [``Selected problems of weighted approximation and spectral analysis'', Trudy Math. Inst. Steklova 120 (1974) (in Russian), Proc. Steklov Math. Inst. 120 (1974; Zbl 0342.41028) (Engl. transl.)], the author considered certain translation invariant sequence spaces \(\ell(\mathbb{Z}, E)\) where \(E\) is an arbitrary separable Banach space. For example, \(\ell\) might denote \(\ell^p\) or \(c_0\). A vector \(x\in \ell(\mathbb{Z}, E)\) is called 1-cyclic (respectively 2-cyclic) if the closed linear span of its integer (respectively positive integer) translates is the whole space. In the second cited paper, he proved that \(\ell(\mathbb{Z},E)\) has a 1-cyclic (respectively 2-cyclic) vector if and only if the same is true of \(\ell(\mathbb{Z}, \mathbb{C}^2)\). Moreover, \(\ell(\mathbb{Z}, \mathbb{C}^2)\) has no 2-cyclic vector if its space of Fourier transforms is continuously embedded in the space of measurable functions on the circle. In the present paper, the author shows how these results translate readily into the language of completeness theorems for integer translates for functions on \(\mathbb{R}\). In particular, there exist functions \(f\) whose positive integer translates are complete in the spaces \(C^k_0(\mathbb{R})\), \(L^p(\mathbb{R})\), \(W^k_p(\mathbb{R})\) as well as in their weighted and vector-valued analogues.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1309603
zbMATH Keywords: cyclic vector
zbMATH Keywords: simple spectrum
zbMATH Keywords: translation invariant sequence spaces
zbMATH Keywords: space of Fourier transforms
zbMATH Keywords: space of measurable functions on the circle
author: N. K. Nikol'skij
reviewed by: Alan J. Pryde
MaRDI profile type: Publication
full work available at URL: jath.1999.3365
OpenAlex ID: W2016819272
cites work: Completeness of integer translates in function spaces on \(\mathbb{R}\)
cites work: Q3965130
cites work: Q5638775
cites work: Q5673972
cites work: Q4110074
cites work: Q4205957
cites work: Q4147137
cites work: Q3793222
DOI: 10.1006/JATH.1999.3365
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description: scientific article; zbMATH DE number 1309603