The Hankel-Clifford transformation on certain spaces of ultradistributions
type: Item
instance of: scholarly article
title: The Hankel-Clifford transformation on certain spaces of ultradistributions
zbMATH Open document ID: 0691.46030
author: Jorge J. Betancor
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1989-01-01T00:00:00Z
review text: This work deals with the generalized Hankel-Clifford transformation on certain spaces of ultradistributions. Following mainly the works of \textit{J. M. Méndez} [La transformation integral de Hankel-Clifford, Secretariado de Publicaciones de la Universidad de La Laguna, Coleccion Monografias \(n^ 08\) (1981)] and \textit{J. M. González} [Ph. D. Thesis, Faculty of Mathematics, University of La Laguna, (1985)] the author defines testing function spaces \({}^ pS^{\beta}_{\mu,\alpha}\) and he calls the elements of its dual as ultradistributions. Méndez called a integral transform given by the following pair \[ F(y)=h_{\mu}\{f(x)\}(y)=y^{\mu}\int^{\infty}_{0}C_{\mu}(xy)f(x)d x \] \[ f(x)=h_{\mu}\{F(y)\}(x)=x^{\mu}\int^{\infty}_{0}C_{\mu}(xy)f(y)d y \] as the Hankel-Clifford transformation. The kernel of this transform \(C_{\mu}\) is the Bessel Clifford function of the first kind and order \(\mu\). The author maintains that the Hankel-Clifford transform \(h_{\mu}\) (\(\mu\geq 0)\) is a continuous linear operator in spaces of \({}^ pS^{\beta}_{\mu,\alpha}\). The generalized Hankel-Clifford transform \(h'_{\mu}\) is then defined as a continuous linear mapping between the dual spaces. In the end, the developed theory is applied to find classes of existence of classical and generalized solutions for a Cauchy problem containing the Bessel-type operator \(B_{\mu}=Dx^{\mu +1}Dx^{-\mu}\).
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 44A20
Mathematics Subject Classification ID: 33E99
zbMATH DE Number: 4131034
zbMATH Keywords: generalized Hankel-Clifford transformation on certain spaces of ultradistributions
zbMATH Keywords: testing function spaces
zbMATH Keywords: kernel
zbMATH Keywords: Bessel Clifford function of the first kind
zbMATH Keywords: classical and generalized solutions for a Cauchy problem containing the Bessel-type operator
reviewed by: Gandikota L. N. Rao
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4131034