On the stabilization of the inversion of some Kontorovich-Lebedev like integral transforms
type: Item
instance of: scholarly article
title: On the stabilization of the inversion of some Kontorovich-Lebedev like integral transforms
zbMATH Open document ID: 0915.44006
DOI: 10.4171/ZAA/844
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-07-08T00:00:00Z
review text: The authors deal with the Kontorovich-Lebedev transform \[ K_{ix}[f](x)=\text{cosh}(\pi x)\int^\infty_0K_{ix}(y)f(y)dy= g(x)\quad (x\geq 0)\tag{*} \] on the weighted \(L_p\)-space \(L_{\nu,p}(\mathbb{R}_+)\) \((1\leq p\leq\infty,\;\nu\in\mathbb{R})\) normed by \[ \| f\|_{\nu,p}=\left(\int^\infty_0x^{\nu p-1}\bigl| f(x)\bigr|^pdx\right)^{1/p} \] where \(K_{ix}\) is the Macdonald function. By introducing a certain regularization operator they derive a special construction of the inversion of transform \((*)\) that allows to stabilize it in the space \(L_{1+{1\over p},p}(\mathbb{R}_+)\) \((1<p<2)\). Such a construction for the inversion formula concerning the Lebedev-Skalskaya transform [see \textit{N. N. Lebedev} and \textit{I. P. Skalskaya} in: Questions of mathematical physics. Leningrad: Nauka, 68-79 (1976)] \[ \text{Re}[f](x)=\text{cosh}(\pi x)\int^\infty_0\text{Re}K_{{1\over 2}+ix}(y)f(y)dy=g(x)\;(x\geq 0) \] where \(\text{Re} K_{{1\over 2}+ix}(y)\) is the real part of the Macdonald function with the index \({1\over 2}+ix\), is also similarly treated.
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 65R10
Mathematics Subject Classification ID: 65R30
zbMATH DE Number: 1211546
zbMATH Keywords: stability estimates
zbMATH Keywords: Kontorovich-Lebedev transform
zbMATH Keywords: Macdonald function
zbMATH Keywords: regularization
zbMATH Keywords: inversion formula
zbMATH Keywords: Lebedev-Skalskaya transform
author: Hans-Jürgen Glaeske
author: S. Yakubovich
reviewed by: Nácere Hayek
MaRDI profile type: Publication
cites work: Q5824493
cites work: Q3778499
cites work: Q5844214
cites work: Q3999269
cites work: Q4288789
cites work: Q4424633
cites work: On the problem of constructing integral transforms by means of composition technique
cites work: Q4840978
cites work: Q4864688
cites work: Q4856095
cites work: Q4841241
cites work: Q4301590
full work available at URL: https://doi.org/10.4171/zaa/844
OpenAlex ID: W1971425585
description: scientific article; zbMATH DE number 1211546