Two problems about completeness of exponential systems

We consider two, at first glance essentially different, completeness problems for exponential systems. The first of them is the problem about completeness, in $L^2$ on an interval, of a mixed system which consists of shifts of some fixed function and of exponentials corresponding to the zeros of its Fourier transform. The second problem, posed in 2007 by A. Aleman and B. Korenblum, is related to the spectral synthesis problem for the subspaces of the space $C^\infty(a,b)$ invariant with respect to the differentiation: is it true that any such subspace is generated by its spectral part (exponentials and exponential monomials) and residual part (functions which vanish identically on some subinterval)? The solution of the two above-mentioned problems is based on a recent theorem due to Yu. S. Belov.