Interpolation of holomorphic functions and surjectivity of Taylor coefficient multipliers

Abstract

We prove criteria for global analytic solvability for some Euler type partial differential equations — a class of linear partial differential equations with variable (polynomial) coefficients. This is based on a very general Mellin type theory developed in the present paper which is valid for analytic functionals (in particular, distributions of compact support) substituting the classical Mellin transform. As a consequence, we get a characterization of moment sequences of analytic functionals by interpolation of holomorphic functions satisfying some restrictive growth conditions. This allows to study the so-called Hadamard type multipliers on real analytic functions of several variables, i.e., operators for which monomials are eigenvectors. We characterize multiplier sequences (i.e., sequences of eigenvalues of multipliers) by interpolation and describe the image of some multipliers, especially, the image of the principal examples of Hadamard multipliers: Euler type partial differential operators. We also get some results showing that the image remains unchanged after some perturbation of the original Hadamard type operator.