A Paley-Wiener theorem for the Hankel transform of Colombeau type generalized functions
type: Item
instance of: scholarly article
title: A Paley-Wiener theorem for the Hankel transform of Colombeau type generalized functions
zbMATH Open document ID: 0928.46024
author: Jorge J. Betancor
author: Lourdes Rodríguez-Mesa
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-01-09T00:00:00Z
review text: The discovery of generalized functions is one of the most distinguished and interesting fields, but Colombeau has developed a new theory of generalized functions by generalizing onto the transformations. The properties, such as, convolution, convolutors, tempered generalized functions of the Hankel transform for Colombeau type of generalized function were introduced in the recent years. In the present paper, after defining some of the above properties in a systematic manner, the authors derive a Paley-Wiener theorem for the Hankel transformation \[ h_\mu(\varphi)(y)= \int^\infty_0 (xy)^{1/2} J_\mu(xy)\varphi(x) dx,\quad y\in(0,\infty), \] where \(J_\mu\) denotes the Bessel function of the first kind and of order \(\mu>-1/2\) on generalized function of the Colombeau type spaces \({\mathcal H}_{\mu,\tau,h}\) and \({\mathcal H}_{\mu,\tau,c}\), which are subspaces of \({\mathcal H}_{\mu,\tau}\). For a Paley-Wiener theorem, few lemmas have been proved and few properties for the Colombeau type spaces are justified for the Hankel transformation. The main theme and importance of this paper is, that, a Paley-Wiener type theorem is established for the Hankel transform in the setting of the tempered generalized functions of Colombeau type.
reviewed by: P. K. Banerji
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 46F30
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 1271510
zbMATH Keywords: Colombeau type of generalized function
zbMATH Keywords: Hankel transform
zbMATH Keywords: Paley-Wiener theorem
zbMATH Keywords: Bessel function of the first kind
zbMATH Keywords: Colombeau type spaces
zbMATH Keywords: tempered generalized functions of Colombeau type
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6145
OpenAlex ID: W1972087896
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DOI: 10.1006/JMAA.1998.6145
description: scientific article; zbMATH DE number 1271510