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

Published online by Cambridge University Press:  17 April 2009

Walter R. Bloom
Walter R. Bloom
Affiliation:
Department of Mathematics, University of Tasmania, Hobart, Tasmania;
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Abstract

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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 12 , Issue 2 , April 1975 , pp. 301 - 309
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Bloom, Walter R., “Jackson's Theorem for locally compact abelian groups”, Bull. Austral. Math. Soc. 10 (1974), 59–66.CrossRefGoogle Scholar
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