  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.
