Functions which operate on characteristic functions

Abstract

Let G be a locally compact abelian group and B+(G) the family of continuous, complex-valued non-negative definite functions on G. Set A complex-valued function defined on the open unit disk is said to operate on {B+1(G), B+(G)} if fϵB+(G) implies F(f)ϵB+(G), similarly for {ϕ(G),ϕ(G)}. Recently C. S. Herz has given a proof of a conjecture of W. Rudin that F operates on {B+1(G), B+(G)} if and only if for a certain class of G. We shall show by independent methods that F operates on ϕ(R1) if F is given by (*) for |z| ≦ 1 and F(1)〝 1. This answers a question posed by E. Lukacs and provides in addition an alternate proof of Herz’s theorem. © 1965 by Pacific Journal of Mathematics.