Approximation by convolutions

Robert E. Edwards

Vol. 15 (1965), No. 1, 85–95

DOI: 10.2140/pjm.1965.15.85

Abstract

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 $f\ast g$, with $f$ and $g$ extracted from bounded subsets of conjugate Lebesgue spaces $L^p\left(G\right)$ and ${L^p}^{\prime }\left(G\right)$. 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 $L^p\left(G\right)$. The final theorem shows in particular that the P. J. Cohen factorisation theorem for $L^1\left(G\right)$ fails badly for $L^p\left(G\right)$ whenever $G$ is infinite compact Abelian and $p>1$.

Mathematical Subject Classification

Primary: 42.55

Milestones

Received: 5 February 1964

Published: 1 March 1965