The algebra of functions with Fourier transforms in 𝐿^{𝑝}

Abstract

The purpose of this note is to prove the following Theorem. Let G be a nondiscrete locally compact Abelian group with dual group G. Let AP(G)= {fEL\G):jEL(G)}, which for p^l is a Banach algebra under convolution, with norm IWI'-IWIi + IWU Let AP(G)2 denote the ideal generated by the products f * g, where f, gEAp(G). Then