A structure theorem for the space \((G^{\alpha}_{\alpha})^{\prime}\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1292.46024
DOI: 10.1142/S0219025713800027
published in: Infinite Dimensional Analysis, Quantum Probability and Related Topics
publication date: 2013-10-17T00:00:00Z
Mathematics Subject Classification ID: 46F05
zbMATH DE Number: 6215976
zbMATH Keywords: Laguerre function
zbMATH Keywords: Laguerre expansion
zbMATH Keywords: structure theorem
zbMATH Keywords: space \((G_{\alpha}^{\alpha})'\)
author: Bishnu Prasad Dhungana
MaRDI profile type: Publication
cites work: Laguerre expansions of tempered distributions and generalized functions
cites work: Gelfand-Shilov spaces for the Hankel transform
cites work: Laguerre expansions of Gel'fand-Shilov spaces
cites work: Q3137751
title: A structure theorem for the space \((G^{\alpha}_{\alpha})^{\prime}\)
review text: Spaces of type \(G_{\alpha}^{\beta}\) (\(\alpha,\beta\geq 1\)) were introduced by \textit{A. J. Duran} [Indag. Math., New Ser. 3, No. 2, 137--151 (1992; Zbl 0777.46022)]. The spaces \(G_{\alpha}^{\alpha}\) (\(\alpha\geq 1\)) are invariant under the Hankel-Clifford transform and their elements can be characterized in terms of their Fourier-Laguerre coefficients, see [\textit{A. J. Duran}, J. Approximation Theory 74, No. 3, 280--300 (1993; Zbl 0789.41018)].NEWLINENEWLINEThe author of this paper gives a structure theorem for the dual space \((G_{\alpha}^{\alpha})'\), \(\alpha\geq 1\), as follows:NEWLINENEWLINE\( \mathcal{F}\) is an element of \((G_{\alpha}^{\alpha})'\) if and only if there exist a sequence \(\{b_m\}\subset (0,\infty)\) and a continuous bounded function \(f\) on \((0,\infty)\) such that, for every \(d>0\), there exists a constant \(C>0\) satisfying NEWLINE\[NEWLINE \mathcal{F}= \sum_{m=0}^{\infty} b_m (-xD^2-D+\frac{x}{4})^{m}f \quad \text{and} \quad |b_m|\leq\frac{C d^m}{(m!)^{\alpha}}. NEWLINE\]
reviewed by: María Carmen Gómez-Collado
description: scientific article; zbMATH DE number 6215976