J Austral Math Soc Ser A 58 pp387--403, 1995.

The Class L(log L)^a and some Lacunary Sets

Sanjiv Kumar Gupta, Shobha Madan and U. B. Tewari

Abstract

A well-known result of Zygmund states that if f Î L(log⁺ L)^1/2 on the circle group T and E is a Hadamard set of integers, then Ùf |_E Î l₂ (E). In this paper we investigate similar results for the classes B_a = L(log⁺ L)^a , a > 0 on an arbitrary infinite compact abelian group G and Sidon subsets E of the dual G. These results are obtained as special cases of more general results concerning a new class of lacunary sets S_a,b, 0 < a £ b, where a subset E of G is an S_a,b set if ÙB_a |_E Í l_2b/a (E). We prove partial results on the distinctness of the S_a,b sets in the index b.

1991 AMS Subject Classification: 42A55, 42A05

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Authors

Sanjiv Kumar Gupta

Shobha Madan

U. B. Tewari

Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur, 208016, India.