#### Vol. 15, No. 1, 1965

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Approximation by convolutions

### Robert E. Edwards

Vol. 15 (1965), No. 1, 85–95
##### 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 $p>1$.

Primary: 42.55