Two variants of fractional powers of Hankel integral transforms of arbitrary order
type: Item
instance of: scholarly article
title: Two variants of fractional powers of Hankel integral transforms of arbitrary order
zbMATH Open document ID: 1335.44002
author: Akhilesh Prasad
author: Kanailal Mahato
published in: Annali dell'Università di Ferrara. Sezione VII. Scienze Matematiche
publication date: 2015-12-07T00:00:00Z
review text: The paper introduces two variants of fractional Hankel transforms and extend them to spaces of generalized functions. Let \(0< \theta <\pi\), \(\nu\geq -1/2\) and set \[ \left({\mathcal H}^\theta_{1, \nu,\mu}\, f \right)(y) =\int_0^\infty K^\theta_1(x, y) f(x)dx, \] where \[ K^\theta_1(x,y)=\begin{cases} C_{\nu, \mu, \theta}\, e^{i(x^2+y^2) \cot \theta/2} \left(xy\csc \theta \right)^{-\mu} J_\nu (xy \csc \theta) y^{1+2\mu}, &\theta \neq n\pi, \\ (xy)^{-\mu}J_\nu (xy)y^{1+2\mu}, &\theta =\pi/2, \\ \delta(x-y,) &\theta = n\pi, \end{cases} \] and \[ \left({\mathcal H}^\theta_{2, \nu,\mu}\, g \right)(y) =\int_0^\infty K^\theta_2(x, y) g(x)dx, \] \[ K^\theta_2(x,y)=\begin{cases} C_{\nu, \mu, \theta}\, e^{i(x^2+y^2) \cot \theta/2 }\left(xy\csc \theta \right)^{-\mu} J_\nu (xy \csc \theta) x^{1+2\mu}, &\theta \neq n\pi, \\ (xy)^{-\mu}J_\nu (xy)x^{1+2\mu}, &\theta =\pi/2, \\ \delta(x-y), &\theta = n\pi, \end{cases} \] where \(C_{\nu, \mu, \theta}\) is a constant that depends on \(\mu, \nu, \theta.\) The above first and second fractional Hankel transforms reduce to the standard Hankel transform pair when \(\theta = \pi/2.\) The fractional Hankel transforms are then extended to \({\mathcal H}^\theta_{1, \nu,\mu, m}, {\mathcal H}^\theta_{2, \nu,\mu, m} ,\) where \(m\) is a natural number such that \(\nu+\mu+m \geq -1 .\) The authors then introduce two countable families of multinorms by which they construct two testing-function spaces \(\mathbf{H}^\theta_{1,\nu,\mu}(I), \mathbf{H}^\theta_{2,\nu,\mu}(I)\) as in \textit{A. H. Zemanian}'s book [Generalized integral transformations. New York etc.: Interscience Publishers, a division of John Wiley \& Sons, Inc. (1968; Zbl 0181.12701)], where \(I=(0, \infty).\) It is shown that \( {\mathcal H}^\theta_{1, \nu,\mu, m}\) is a continuous linear mapping on \(\mathbf{H}^\theta_{1,\nu,\mu}(I)\) into itself. The fractional Hankel transform \({\mathcal H}^\theta_{1, \nu,\mu, m}\) is then extended as a continuous linear mapping on a space of g…
reviewed by: Ahmed I. Zayed
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 6516562
zbMATH Keywords: Hankel transforms
zbMATH Keywords: fractional Hankel transforms
zbMATH Keywords: Zemanian spaces of distributions
MaRDI profile type: Publication
full work available at URL: s11565-015-0226-7
OpenAlex ID: W2025804256
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DOI: 10.1007/S11565-015-0226-7
description: scientific article; zbMATH DE number 6516562