Synthesable differentiation-invariant subspaces

TL;DR

This paper characterizes differentiation-invariant subspaces of smooth functions that allow spectral synthesis, linking the problem to polynomial approximation and de Branges spaces, thus providing a comprehensive solution to a longstanding question.

Contribution

It offers a complete description of such subspaces and reveals their connection to classical approximation problems and de Branges space theory.

Findings

Characterization of differentiation-invariant subspaces with spectral synthesis

Connection established between spectral synthesis and polynomial approximation

Linkage to the theory of de Branges spaces

Abstract

We describe differentiation-invariant subspaces of $C^\infty(a,b)$ which admit spectral synthesis. This gives a complete answer to a question posed by A.~Aleman and B.~Korenblum. It turns out that this problem is related to a classical problem of approximation by polynomials on the real line. We will depict an intriguing connection between these problems and the theory of de Branges spaces.