The Bessel-Struve intertwining operator on ℂ and mean-periodic functions

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 space Ᏼ of entire functions and we characterize the continuous linear mappings from Ᏼ into itself which commute with Bessel-Struve operator. 1. Introduction. 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. This notion was introduced by Delsarte in [2] and some generalization and applications were given in [1, 3, 7, 10]. 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