Vol. 15, No. 4, 1965

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Functions which operate on characteristic functions

Alan G. Konheim and Benjamin Weiss

Vol. 15 (1965), No. 4, 1279–1293
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

B+1 (G ) = {f ∈ B+ (G) : f(0) < 1}

Φ (G ) = {f ∈ B+(G ) : f(0) = 1}

A complex-valued function defined on the open unit disk is said to operate on {B1+(G),B+(G)} if f B1+(G) implies F(f) B+(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 {B1+(G),B+(G)} if and only if

        ∞
F(z) = ∑   cmnzmzn, cmn ≧ 0,|z| < 1.
m,n=0
(*)

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.

Mathematical Subject Classification
Primary: 42.54
Secondary: 60.08
Milestones
Received: 20 June 1964
Revised: 23 August 1964
Published: 1 December 1965
Authors
Alan G. Konheim
Benjamin Weiss