Representation of multipliers on spaces of real analytic functions

Abstract

We consider multipliers on the spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We prove representation theorems in terms of analytic functionals and in terms of holomorphic functions. We characterize Euler differential operators among multipliers. Then we characterize when such operators are surjective or have a continuous linear right inverse on the space of real analytic functions over an interval not containing zero. In particular we solve the problem when Euler differential equation of infinite order has a solution in the space of real analytic functions on an interval not containing zero.