Spectral synthesis for exponentials and logarithmic length

Abstract

We study hereditary completeness of systems of exponentials on an interval such that the corresponding generating function $G$ is small outside of a lacunary sequence of intervals $I_k$. We show that, under some technical conditions, an exponential system is hereditarily complete if and only if the logarithmic length of the union of these intervals is infinite, i.e., ${\sum }_k{\int }_{I_k}\frac{dx}{1+|x|}=\infty$.

21 pages