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.