A class of index transforms with general kernels

Abstract

This paper deals with a class of integral transforms of the non-convolution type involving sufficiently general kernels, which depend upon two essentially independent arguments. One of them, in various particular cases, is a parameter or index of the corresponding special functions. This class of integral transforms comprises the famous Kontorovich-Lebedev and Mehler-Fock transforms. We study here the mapping properties and give also inversion theorems of the general index transforms on the space Lp(ℝ), p ≥ 1, that covers the respective measurable functions on the whole real axis with the norm (formula presented) It is shown that the images of the transforms belong to the space Lv,p(ℝ+), v ∈ ℝ, 1 ≤ p ≤ ∞ of functions normed by (formula presented) In particular, when v = 1/p we get the usual Lp(ℝ+) space. We also direct our attention to the case of the Hilbert space and give certain interesting examples of these transforms.