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., k I k dx 1+|x| = ∞.