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

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.