Approximation by convolutions

This paper is concerned mainly with approximating functions on closed subsets P of a locally compact Abelian group G by absolute-convex combinations of convolutions / * g, with / and g extracted from bounded subsets of conjugate Lebesgue spaces L V (G) and L p/ (G). It is shown that the Helson subsets of G can be characterised in terms of this approximation problem, and that the solubility of this problem for P is closely related to questions concerning certain multipliers of L P (G). The final theorem shows in particular that the P. J. Cohen factorisation theorem for L^G) fails badly for L P (G) whenever G is infinite compact Abelian and p > 1.