A new characterization of the bounded operators commuting with Hankel translation
type: Item
instance of: scholarly article
title: A new characterization of the bounded operators commuting with Hankel translation
zbMATH Open document ID: 0915.46033
published in: Archiv der Mathematik
publication date: 1999-06-28T00:00:00Z
review text: A new characterization for the linear bounded operator \(T\) on Zemanian's space \({\mathcal H}_\mu\), \(\mu>-1/2\), commuting with Hankel translation \[ (\tau_x \psi)(y) = \int_0^\infty D_\mu (x,y,z)\psi(z) dz,\quad x,y\in (0,\infty), \] where \[ D_\mu (x,y,z) = \int_0^\infty t^{-\mu -1/2} (xt)^{1/2} J_\mu(xt) (yt)^{1/2}J_\mu (yt) (zt)^{1/2} J_\mu (zt) dt,\;x,y,z \in (0,\infty) \] and \(J_\mu\) denotes the Bessel function of the first kind and order \(\mu\) is given. Namely, it is shown that the operator \(T\) commutes with the Hankel translation if, and only if, it commutes with the Bessel operator \(S_\mu = x^{-\mu -1/2}D x^{2\mu +1} D x^{-\mu -1/2}\).
reviewed by: Yuri Luchko
Mathematics Subject Classification ID: 46F12
zbMATH DE Number: 1145770
zbMATH Keywords: Zemanian space
zbMATH Keywords: Hankel transform
zbMATH Keywords: Hankel convolution
zbMATH Keywords: Hankel translation
zbMATH Keywords: Bessel operator
author: Jorge J. Betancor
MaRDI profile type: Publication
full work available at URL: s000130050138
OpenAlex ID: W1988563936
DOI: 10.1007/S000130050138
description: scientific article; zbMATH DE number 1145770