Subspaces of C\infty invariant under the differentiation

Abstract

We prove that a proper differentiation invariant subspace of C\infty such that the restriction of the differentiation operator has a discrete spectrum is spanned by functions vanishing outside some closed interval and monomial exponentials corresponding to the spectrum if its density does not exceed the critical value. In addition, we prove that the result is not necessarily true when the density equals the critical value. This answers a question posed by the first author and B. Korenblum.