The Bessel-Struve intertwining operator onℂand mean-periodic functions

A Gasmi,M Sifi

doi:10.1155/s0161171204309178

Abstract

We give a description of all transmutation operators from the Bessel-Struve operator to the second-derivative operator. Next we define and characterize the mean-periodic functions on the spaceℋof entire functions and we characterize the continuous linear mappings fromℋinto itself which commute with Bessel-Struve operator.

Highlights

Let A and B be two differential operators on a linear space X
We say that χ is a transmutation operator of A into B if χ is an isomorphism from X into itself such that Aχ = χB
In the case where A and B are two differential operators having the same order and without any singularity on the complex plan, acting on the space of entire functions on C denoted here by Ᏼ, Delsarte showed in [3] the existence of a transmutation operator between A and B and gave some applications on the theory of mean-periodic functions on C

Summary

Introduction

Let A and B be two differential operators on a linear space X. The Bessel-Struve intertwining operator χα is defined from the space Ᏼ into itself by. We study the mean-periodic functions associated with the Bessel-Struve operator and we characterize the continuous linear mappings from Ᏼ into itself which commute with α. We define the Bessel-Struve intertwining operator χα and its dual tχα; after that, we study the harmonic analysis associated with the operator α. (ii) We denote by Exp(C), the space of functions with exponential type The Bessel-Struve translation operators τz, z ∈ C, associated with the operator α, is defined on Ᏼ by τzf (w) = χα,zχα,w χα−1(f )(z + w) ∀w ∈ C.

Mean-periodic functions

The commutator of α