Distributional convolutors for Fourier transform
type: Item
instance of: scholarly article
title: Distributional convolutors for Fourier transform
zbMATH Open document ID: 1126.46028
author: Claudio Jerez
author: Lourdes Rodríguez-Mesa
author: Jorge J. Betancor
author: Sandra Monica Molina
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: A distributional Fourier analysis was developed in a series of six papers by \textit{K.\,B.\thinspace Howell} [J.~Math.\ Anal.\ Appl.\ 168, No.\,2, 342--350 (1992; Zbl 0776.42012); ibid.\ 173, No.\,2, 419--429 (1993; Zbl 0776.42013); ibid.\ 175, No.\,1, 257--267 (1993; Zbl 0776.42014); ibid.\ 180, No.\,1, 79--92 (1993; Zbl 0811.42004); ibid.\ 187, No.\,2, 567--582 (1994; Zbl 0819.46028); ibid.\ 193, No.\,3, 832--838 (1995; Zbl 0869.46017)]. The work done by the authors in the present paper complements this work, i.e., it fills up some gaps in the work done by Howell. This is exemplified by their Theorem 3.1, amongst many other theorems stated and proved; it states the following. Let \(m\in \mathbb Z\), \(m<0\), and \(T\in {\mathcal O}'_{c,G,m}\). Then \[ T(x) = \lim_{r\to\infty} \int^{+r}_{-r} F(T)(z)e^{-ixz}\,dz \] in the sense of convergence in \(G'\). Here, \(G\) and \({\mathcal O}_{c,G,m}\), are Fréchet spaces and \({\mathcal O}'_{c,G,m}\) and \(G'\) are their corresponding dual spaces, respectively.
reviewed by: J. N. Pandey
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 42A85
zbMATH DE Number: 5078938
zbMATH Keywords: Fourier analysis
zbMATH Keywords: convolution operators
zbMATH Keywords: distribution spaces
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.01.066
OpenAlex ID: W2027165933
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DOI: 10.1016/J.JMAA.2006.01.066
description: scientific article; zbMATH DE number 5078938