Contributions to harmonic analysis

Abstract

Let G be a locally compact abelian group, G its dual. For any ideal I of L'(G) let cosp(I) denote the cospectrum of I; this is defined as the closed set in G formed by the common zeros of the Fourier transforms of all functions in I. Consider two closed ideals I, I2 with disjoint cospectra. Then the algebraic sum I, + I2, which is again an ideal, has empty cospectrum. But II + 12 may not be closed in L'(G); this can be shown by examples. In case it is closed, we have, by Wiener's theorem, I, + I2 = L'(G). This suggests the following definition. Let two (non-empty) disjoint subsets A1, A2 of G be given. We say that the pair (A1, A2) has the closure property with respect to L'(G) if all pairs of closed ideals I, I2 c L'(G) such that