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Continuity of functions operating on characteristic functions

Published online by Cambridge University Press:  24 October 2008

Saulius Norvidas
Saulius Norvidas
Affiliation:
Institute of Mathematics and Informatics, and Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania
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Abstract

For a locally compact abelian group G let P0(G) denote the set of all characteristic functions on G (i.e., continuous positive definite functions φ with φ(0) = 1). A complex-valued function f is said to operate on P0(G) if f(φ(·)) ∈ P0(G) whenever φ ∈ P0(G). The natural domain for functions operating on P0(G) is the set D(G) = {z ∈ C: z = φ(g), g ∈ G, φ ∈ P0(G)}. It is known that every function operating on P0(G) is continuous on the interior of D(G) for each infinite group G. On the other hand, functions discontinuous on D(G) operate on P0(G) for any discrete G. We show that, for any non-discrete group G, every function operating on P0(G) is continuous on D(G). Together with some earlier results, this statement allows us to obtain a constructive description of operating functions on the whole set D(G).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 1 , July 1996 , pp. 117 - 125
Copyright
Copyright © Cambridge Philosophical Society 1996

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