Solvability of convolution equations in spaces of generalized distributions with restricted growth
type: Item
instance of: scholarly article
title: Solvability of convolution equations in spaces of generalized distributions with restricted growth
zbMATH Open document ID: 0836.46030
author: J. Ibbotson
author: Z. Zielezny
published in: Hokkaido Mathematical Journal
publication date: 1992-12-15T00:00:00Z
review text: \textit{L. Ehrenpreis} [Am. J. Math. 82, 522-588 (1960; Zbl 0098.084)]\ established necessary and sufficient conditions on the Fourier transform of a distribution \(S\in {\mathcal E} (\mathbb{R}^n)\) in order that \(S*{\mathcal D}' (\mathbb{R}^n) \supset {\mathcal D}' (\mathbb{R}^n)\). This holds if and only if there are positive constants \(A_1\), \(A_2\) and \(A_3\) such that for every \(\xi\in \mathbb{R}^n\) there exists an \(\eta\in \mathbb{R}^n\) satisfying the condition \[ |\xi- \eta|\leq A_1\log (2+|\xi|) \qquad \text{and} \qquad |\widehat {S} (\eta)|\geq (A_2+ |\xi |)^{-A_3}. \] In this case \(S\) is called invertible. Later on \textit{S. Abdullah} [Hokkaido Math. J. 17, No. 2, 197-209 (1988; Zbl 0661.46033)], \textit{Ch.-Ch. Chou} [`La transformation de Fourier complexe et l'equation de convolution', Lect. Notes Math. 325 (1973; Zbl 0257.46037)]\ used other versions of the invertibility conditions to prove the existence of solutions of convolution equations in various spaces of distributions and generalized distributions. In this paper the authors have studied convolution equations in the spaces of generalized distributions of \textit{G. Björck} with restricted growth [Ark. Mat. 6, 351-407 (1966; Zbl 0166.365)]. They have characterized the convolution operators \(S\) in \({\mathcal K}_{M,\omega}'\) for which \(S*{\mathcal K}_{M, \omega}' \supset {\mathcal K}_{M, \omega}'\). Incidentally, Abdullah's (referred above) result comes out as a particular case of the result obtained in this paper. The Section 3 of the paper deals with the solvability of convolution equations in \({\mathcal K}_{M, \omega}'\).
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 47A50
zbMATH DE Number: 78118
zbMATH Keywords: Fourier transform
zbMATH Keywords: invertibility
zbMATH Keywords: convolution equations
reviewed by: R. K. Bose
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381413689
OpenAlex ID: W2056152654
DOI: 10.14492/HOKMJ/1381413689
description: scientific article; zbMATH DE number 78118