Vol. 15, No. 1, 1965

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Approximation by convolutions

Robert E. Edwards

Vol. 15 (1965), No. 1, 85–95

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 f g, with f and g extracted from bounded subsets of conjugate Lebesgue spaces Lp(G) and Lp(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 Lp(G). The final theorem shows in particular that the P. J. Cohen factorisation theorem for L1(G) fails badly for Lp(G) whenever G is infinite compact Abelian and p > 1.

Mathematical Subject Classification
Primary: 42.55
Received: 5 February 1964
Published: 1 March 1965
Robert E. Edwards