On solvability of convolution equations in spaces of generalized distributions with restricted growth
type: Item
instance of: scholarly article
title: On solvability of convolution equations in spaces of generalized distributions with restricted growth
zbMATH Open document ID: 0927.46024
author: Saleh Abdullah
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-05-24T00:00:00Z
review text: In mathematics, analysis is endowed with an immensely wide variety of texts containing articles on solvability in spaces of generalized distributions for convolution equations. For the restricted growth, the author has textured the text in a much lucid and impressive form. The present article examines, in a systematic manner, some of the recent works and has employed them in proving the necessary and sufficient conditions for the solvability of convolution equations in spaces of generalized distributions with restricted growth. The main result of this paper is the theorem, which under certain suitable prescribed conditions, states that, if \(u\in{\mathcal O}_c'\) (\(K_{M,w}'\); \(K_{M,w}'\)) and \(s*u\in K_{M,w}\), then \(u\in K_{M,w}\) and other conditions are equivalent for a generalized distribution \(S\). Using the Paley-Wiener theorem, applied to the Fourier transform and to the element of \(K_{M,w}\), the said theorem is established. The importance of this paper lies in the fact, that, it involves the topology of the space of convolution operators \(K_{M,w}'\) with restricted exponential growth.
reviewed by: P. K. Banerji
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 45P05
zbMATH DE Number: 1289963
zbMATH Keywords: solvability in spaces of generalized distributions
zbMATH Keywords: convolution equations
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: Fourier transform
zbMATH Keywords: space of convolution operators
zbMATH Keywords: exponential growth
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6301
OpenAlex ID: W2076109669
cites work: Solvability of convolution equations in \(K_ M'\)
cites work: Linear partial differential operators and generalized distributions
cites work: La dualité dans les espaces \((\mathcal F)\) et \((\mathcal{LF})\)
cites work: Solvability of convolution equations in spaces of generalized distributions with restricted growth
cites work: Relation between solvability and a regularity of convolution operators in \(\mathcal{K}'_ p\), \(p>1\)
DOI: 10.1006/JMAA.1999.6301
description: scientific article; zbMATH DE number 1289963