Differentiation operator in the Beurling space of ultradifferentiable functions of normal type on an interval

TL;DR

This paper investigates differentiation-invariant subspaces within ultradifferentiable functions, introduces a spectral synthesis approach accounting for non-trivial subspaces without exponential monomials, and characterizes solutions to convolution equations.

Contribution

It develops a new spectral synthesis framework for ultradifferentiable functions invariant under differentiation, addressing cases with no exponential monomials and applying it to convolution equations.

Findings

Characterization of differentiation-invariant subspaces in ultradifferentiable functions.

Introduction of a spectral synthesis method for non-trivial subspaces.

Solution descriptions for systems of convolution equations.

Abstract

In this paper we study closed subspaces of ultradifferentiable functions which are invariant under the differentiation operator. We propose a version of spectral synthesis which takes into account the presence of non-trivial differentiation invariant subspaces containing no exponential monomials. As an application, we describe the sets of solutions of finite and infinite systems of <<local>> homogeneous convolution equations.