Subspaces of $C^\infty$ invariant under the differentiation

Abstract

Let $L$ be a proper differentiation invariant subspace of $C^\infty(a,b)$ such that the restriction operator $\frac{d}{dx}\bigl{|}_L$ has a discrete spectrum $\Lambda$ (counting with multiplicities). We prove that $L$ is spanned by functions vanishing outside some closed interval $I\subset(a,b)$ and monomial exponentials $x^ke^{\lambda x}$ corresponding to $\Lambda$ if its density does not exceed the critical value $\frac{|I|}{2\pi}$, and moreover, we show that the result is not necessarily true when the density of $\Lambda$ equals the critical value. This answers a question posed by the first author and B. Korenblum. Finally, if the residual part of $L$ is trivial, then $L$ is spanned by the monomial exponentials it contains.