Convolution operators related to the Fourier cosine and Kontorovich-Lebedev transformations
type: Item
instance of: scholarly article
title: Convolution operators related to the Fourier cosine and Kontorovich-Lebedev transformations
zbMATH Open document ID: 1180.44004
author: S. Yakubovich
author: Lyubov E. Britvina
published in: Results in Mathematics
publication date: 2009-11-09T00:00:00Z
review text: The authors construct certain convolution operators, which are associated with the Fourier cosine transformation \[ (F_cf)(x)= \lim_{N\to\infty} \sqrt{{\pi\over 2}} \int^N_{1/N} f(t)\cos xt\,dt \] and the Kontorovich-Lebedev transformation \[ K_{ix}[f]= \lim_{N\to\infty} \int^N_{1/N} f(t) K_{ix}(t)\,dt, \] where the modified Bessel function \(K_{ix}(t)\) can be represented as \[ K_{ix}(t)= \int^\infty_0 e^{-t\cosh u}\cos x\,u\,du,\quad t> 0 \] and it also can be extended to the strip \(\delta\in[0,\pi/2]\) in the upper half-plane by \[ K_{ix}(t)={1\over 2} \int^{i\delta+\infty}_{i\delta-\infty} e^{-t\cosh z+ ixz}dz. \] The authors establish boundedness in a two-parametric family of Lebesgue spaces for these convolution operators related to the transformations \(F_cf\) and \(K_{ix}[f]\) and obtain some norm estimates on the weighted \(L^p\) spaces. Further more, they demonstrate some applications to the corresponding class of convolution integral equations and find necessary and sufficient conditions for the solvability of these equations in the weighted \(L^2\) spaces.
Mathematics Subject Classification ID: 44A35
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 44A05
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 45A05
zbMATH DE Number: 5627208
zbMATH Keywords: Fourier transform
zbMATH Keywords: Kontorovich-Lebedev transform
zbMATH Keywords: modified Bessel function
zbMATH Keywords: convolution integral equations
zbMATH Keywords: plancherel theorem
reviewed by: Dashan Fan
MaRDI profile type: Publication
full work available at URL: s00025-009-0393-x
OpenAlex ID: W2031627566
DOI: 10.1007/S00025-009-0393-X
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description: scientific article; zbMATH DE number 5627208