On a testing-function space for distributions associated with the Kontorovich--Lebedev transform
type: Item
instance of: scholarly article
title: On a testing-function space for distributions associated with the Kontorovich--Lebedev transform
zbMATH Open document ID: 1116.46028
published in: Collectanea Mathematica
publication date: 2006-12-13T00:00:00Z
full work available at URL: https://eudml.org/doc/41775
review text: Let \(A_x\) denote the differential operator \(A_x=x^2-x\frac{d}{dx}\left[x\frac{d}{dx}\right]\) and \(A_x^k\) its \(k\)-th iterate. For any \(2\leq p < \infty\) and \(\nu > 0\), the author considers the space \({\mathcal A}_{\nu,p}\) consisting of those complex-valued smooth function on \((0,+\infty)\) such that \[ \alpha_{k,\nu,p}(\varphi):=\left(\int_0^{\infty}| A_x^k\varphi| ^px^{\nu p -1}\,dx\right)^{\frac{1}{p}}<\infty \] for every \(k\in{\mathbb N}\). Then the Kontorovich-Lebedev transform of \(f\in {\mathcal A}^\prime_{\nu,p}\) is defined as \[ {\mathcal KL}[f](s):=\left<f, K_s(\cdot)\right>,\;s\in{\mathbb C}, \] where \(K_s\) is the modified Bessel function of complex index \(s\). This extends the classical Kantorovich--Lebedev transform, defined only for appropriate functions and for a purely imaginary index. The author proves that \({\mathcal KL}[f]\) is an analytic function on a vertical strip and obtains an inversion formula for the above transformation. As an application, a Dirichlet problem for a wedge is considered.
reviewed by: Antonio Galbis
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 5079948
zbMATH Keywords: distributions
zbMATH Keywords: Kontorovich-Lebedev transform
zbMATH Keywords: modified Bessel functions
zbMATH Keywords: Dirichlet problem
author: S. Yakubovich
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5079948