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Salinger and Varopoulos [5] give a condition for a closed subset of metrizable LCA group to be a set of analyticity. In this note we shall prove that every set satisfying their condition contains a set of nonsynthesis. ThiP will be done by first improving one of their results in [5] and then applying E theorem of Varopoulos [6]. We shall also give improvements of the mair results of Rago [3] and Graham [1]. Let G be a nondiscrete LCA group, and A (G) = (L'(G))' the Fouriei algebra of G. A closed subset E of G is called a set of synthesis if there exists only one closed ideal I in A (G) such that n { f I(o): f E I} = E. If every function defined on the interval [ 1, 1] that operates in A (E) = A (G)IE can be extended to an analytic function on some neighborhood of [1, 1] in the complex plane, then E is called a set of analyticity. Finally we let M(G) denote the convolution measure algebra of G, and MC(G) the ideal of all continuous measures in M(G).

Convolutions of continuous measures and sets of nonsynthesis