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Jackson's Theorem for locally compact abelian groups

Published online by Cambridge University Press:  17 April 2009

Walter R. Bloom
Walter R. Bloom
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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If f is a p–th integrable function on the circle group and ω(p; f; δ) is its mean modulus of continuity with exponent p then an extended version of the classical theorem of Jackson states the for each positive integer n, there exists a trigonometric polynomial tn of degree at most n for which

‖f-tn‖p ≤(p; f; 1/n).

In this paper it will be shewn that for G a Hausdorff locally compact abelian group, the algebra L1(G) admits a certain bounded positive approximate unit which, in turn, will be used to prove an analogue of the above result for Lp(G).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 1 , February 1974 , pp. 59 - 66
Copyright
Copyright © Australian Mathematical Society 1974

References

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