An abstract form of a theorem of Helson and applications to sets of synthesis and sets of uniqueness

Let E be a compact perfect subset of the real line R such that the restriction of the Fourier transform a bar right arrow (a) over cap vertical bar(E) from L(1)(R) into C(E) is onto. Helson proved that then, for mu is an element of M(E), lim(vertical bar y vertical bar ->infinity vertical bar) (u) over cap (y)vertical bar = 0 is possible only if mu = 0. In this paper we present an abstract version of this theorem of Helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness. (C) 2009 Elsevier Inc. All rights reserved.