On the index integral transformation with Nicholson's function as the kernel
type: Item
instance of: scholarly article
title: On the index integral transformation with Nicholson's function as the kernel
zbMATH Open document ID: 1003.44003
DOI: 10.1016/S0022-247X(02)00043-4
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-09-08T00:00:00Z
review text: This paper contains a new and interesting study of the Nicholson transformation of the form \[ g(x)= {\pi\over 4} \int^\infty_0 \bigl[ J^2_\alpha (\sqrt x)+ Y^2_\alpha (\sqrt x)\bigr] \tau f(\tau) d\tau, \] where \(\alpha= |{i \tau\over 2} |\) and integration is with respect to an index of the sum of squares of Bessel functions of the first and the second kind. Several properties of this index integral transformation including the composition properties, with other well-known integral transformation, the inversion theorem, Parseval's type relation are proved. Composition representation and relationships with the Meijer K-transform, the Mellin transform, the Kontorovich-Lebedev transform and the sine Fourier transform are discussed. It is shown that the Nicholson transformation is a bounded operator given by the formula \[ [JYf](x)=-{x\pi^{5/2}\over 2} \int^\infty_0 {d\over dx} \bigl[J^2_\alpha (\sqrt x)+ Y^2_\alpha(\sqrt x)\bigr] {\tau f(\tau) d\tau\over \cosh({\pi \tau\over 2})}. \] Although no particular application of the Nicholson transformation is given, it seems to be useful for solving problems in applied mathematics and physical sciences.
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 1797503
zbMATH Keywords: Parseval relation
zbMATH Keywords: Nicholson transformation
zbMATH Keywords: Bessel functions
zbMATH Keywords: index integral transformation
zbMATH Keywords: inversion theorem
zbMATH Keywords: Meijer K-transform
zbMATH Keywords: Mellin transform
zbMATH Keywords: Kontorovich-Lebedev transform
zbMATH Keywords: sine Fourier transform
author: S. Yakubovich
reviewed by: Lokenath Debnath
MaRDI profile type: Publication
cites work: Q5824493
cites work: Index integral transformations of Titchmarsh type
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full work available at URL: s0022-247x(02)00043-4
OpenAlex ID: W2090321970
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description: scientific article; zbMATH DE number 1797503