# The Kontorovich-Lebedev Transform

## Abstract

In the previous Chapters 1–5, we considered the so-called Mellin convolution type integral transforms and some of their important particular cases. However, the family of one-dimensional integral transforms is very different. Namely, there are representations of arbitrary functions, where integration has been realized with respect to an index of special function of hypergeometric type from the kernels. The basic examples of such transforms are the Kontorovich-Lebedev and Mehler-Fock transforms (see M.I. Kontorovich and N.N. Lebedev (1938), F.G. Mehler (1881) and V. A. Fock (1943)). Our hypergeometric approach makes it possible not only to investigate these known integral transforms from the new point of view, but, according to Wimp (see J. Wimp (1964)), also to construct some generalizations and inversions. For instance, an inversion of the Wimp transform with respect to the index of Meijer’s *G*-function has been simplified by the first author in 1983 using the theory of generalized *H*-transform (4.1) (see S.B. Yakubovich (1985)). Thus, these so-called index transform classes are very interesting and very important in applications.

## Keywords

Compact Support Inversion Formula Uniform Boundedness Poisson Kernel Previous Chapter## Preview

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