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The algebra of functions with Fourier transforms in a given function space

Published online by Cambridge University Press:  17 April 2009

U.B. Tewari  and
A.K. Gupta
U.B. Tewari
Affiliation:
Department of Mathematics, Indian Institute of Technology, Kanpur, India.
A.K. Gupta
Affiliation:
Department of Mathematics, Indian Institute of Technology, Kanpur, India.
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Abstract

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Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 1 , August 1973 , pp. 73 - 82
Copyright
Copyright © Australian Mathematical Society 1973

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