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 $x^k{e}^{\lambda x}$ 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.

17 pages

References in corpus (1)

• A restricted shift completeness problem