Jackson's Theorem for finite products and homomorphic images of locally compact abelian groups

Abstract

Let G be a Hausdorff locally compact abelian group. The author has shown (Bull. Austral. Math. Soc. 10 (1974), 59–66) that, given ε > 0 and a certain base {Vi}i∈I of symmetric open neighbourhoods of zero, the algebra L1(G) admits a bounded positive approximate unit {ki}i∈I such that for every p–th integrable function f on G,where ω(p; f; Vi) denotes the mean modulus of continuity with exponent p of f. The purpose of this paper is to obtain {ki}i∈I (as above) with a simple dependence of supp on {ki}i∈I on Vi; this is achieved for finite products and homomorphic images of groups for which such a simple dependence is known. The results obtained are used to give a simplified proof of the classical Jackson's Theorem for the circle group, and an analogue of this theorem for the a-adic solenoid.