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 → a|E from L1(R) into C(E) is onto. Helson proved that then, for μ ∈ M(E), lim|y|→∞ | μ(y)| = 0 is possible only if μ = 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. © 2009 Elsevier Inc. All rights reserved