On a generalized Hankel type convolution of generalized functions
type: Item
instance of: scholarly article
title: On a generalized Hankel type convolution of generalized functions
zbMATH Open document ID: 1005.46018
DOI: 10.1007/BF02829619
author: S. P. Malgonde
author: G. S. Gaikawad
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2002-11-28T00:00:00Z
review text: The authors consider the Hankel-type transformation given by \[ F(\tau)=(h_{\mu,\nu} f)(\tau)=\nu\tau^{-1}\int_0^{\infty}(x\tau)^{\nu}J_{\mu}[(x\tau)^{\nu}]f(x)dx \qquad (\tau>0). \] For \(\nu>0\) and \(\mu>-1/2\), they define a space \(H_{\mu,\nu}\) of test functions, analogous to that introduced by \textit{A. H. Zemanian} [SIAM J. Appl. Math. 14, 561-576 (1966; Zbl 0154.13803)], where \(h_{\mu,\nu}\) is an automorphism. Motivated by the Parseval relation \[ \int_0^{\infty}xf(x)g(x)dx=\int_0^{\infty}\tau F(\tau)G(\tau)d\tau, \] they extend \(h_{\mu,\nu}\) by transposition to the dual \(H_{\mu,\nu}^{* '}\) of \(H_{\mu,\nu}^{*}\), the space \(H_{\mu,\nu}\) skewed by the factor \(x\). The classical Hankel convolution introduced by \textit{I. I. Hirschman} [J. Anal. Math. 8, 307-336 (1961; Zbl 0099.31301)], \textit{D. T. Haimo} [Trans. Am. Math. Soc. 116, 330-375 (1965; Zbl 0135.33502)] and \textit{F. M. Cholewinski} [Mem. Am. Math. Soc. 58 (1965; Zbl 0137.30901)] is then extended to a generalized convolution, defined by \[ \langle x^{\nu-1}f*\sigma(x),x\phi(x) \rangle = \langle f(x),x^{\nu}\sigma*\phi(x)\rangle \] whenever \(f\in H_{\mu,\nu}^{* '}\), \(\phi\in H_{\mu,\nu}\) and \(\sigma\) is a distribution of compact support in \((0,\infty)\). This convolution is shown to satisfy the exchange formula with respect to the generalized \(h_{\mu,\nu}\)-transformation. Algebraic and differentiability properties of this generalized convolution are established. The existence of an identity element is discussed. The paper under review is a straightforward extension of one by \textit{J. de Sousa Pinto} [SIAM J. Math. Anal. 16, 1335-1346 (1985; Zbl 0592.46038)] where the same results are obtained for \(\nu=1\) and \(\mu=0\). There is an extensive literature by J.J. Betancor and the reviewer concerning the definition and study of a generalized Hankel convolution in distribution spaces wider than that considered by de Sousa Pinto.
reviewed by: Isabel Marrero
Mathematics Subject Classification ID: 46F12
zbMATH DE Number: 1740621
zbMATH Keywords: Hankel-type transformation
zbMATH Keywords: Hankel-type convolution
zbMATH Keywords: Delsarte translation
zbMATH Keywords: generalized functions
MaRDI profile type: Publication
cites work: Q4087679
cites work: A Hankel convolution complex inversion theory
cites work: Une extension nouvelle de la théorie des fonctions presque-périodiques de Bohr
cites work: Variation diminishing Hankel transforms
cites work: A mixed Parseval equation and the generalized Hankel type transformations
cites work: The \(n\)-dimensional generalized cut Hankel type transformation of generalized functions
cites work: On Functions Positive Definite Relative to the Orthogonal Group and the Representation of Functions as Hankel-Stieltjes Transforms
cites work: A Generalised Hankel Convolution
cites work: Q3871989
cites work: Q5846799
cites work: Q5571178
full work available at URL: https://doi.org/10.1007/bf02829619
OpenAlex ID: W2073819272
description: scientific article; zbMATH DE number 1740621