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 Ik. 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 Ik dx 1+|x| = ∞.