The Kontorovich‐Lebedev transform and its associated pseudodifferential operator

Abstract

In this paper, we obtained some useful estimates for convolution corresponding to Kontorovich-Lebedev transform (KL-transform) in Lebesgue space. Some continuity theorems for translation, convolution, and KL-transform in test function space H(R + ) are discussed. Then an integral representation of pseudodifferential operator involving KL-transform is found out, and its estimates in Lebesgue space is obtained. At the end, some applications of KL-transform and its convolution are discussed.

Composition of wavelet transforms and wave packet transform involving Kontorovich-Lebedev transform

“…Theorem 2.4. Let ϕ, ψ ∈ L 2 (R + ; x −1 dx) be two KL-wavelets which defines the composition of KL-wavelet transform (K f )(b, a, c) and (K )(b, a, c) defined as (22), for two functions f,…”

“…The representation of KL-transform and various relations related to it like translation, convolution, Plancherel's and Parseval's relation etc. have been expressed in many ways [1,5,22,23,27,[35][36][37][38][39]. Now we consider the class of all measurable functions L p (R + ; x −1 dx), of f on R + with norm given as:…”