Continuity of functions operating on characteristic functions

Content Provider	Scilit
Author	Norvidas, Saulius
Copyright Year	1996
Description	For a locally compact abelian group G let $P_{0}$(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 $P_{0}$(G) if f(φ(·)) ∈ $P_{0}$(G) whenever φ ∈ $P_{0}$(G). The natural domain for functions operating on $P_{0}$(G) is the set D(G) = {z ∈ C: z = φ(g), g ∈ G, φ ∈ $P_{0}$(G)}. It is known that every function operating on $P_{0}$(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 $P_{0}$(G) for any discrete G. We show that, for any non-discrete group G, every function operating on $P_{0}$(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).
Related Links	https://www.cambridge.org/core/services/aop-cambridge-core/content/view/D2F1B8114FBECF84C4E34B1BE57FC961/S0305004100074715a.pdf/div-class-title-continuity-of-functions-operating-on-characteristic-functions-div.pdf
Ending Page	125
Page Count	9
Starting Page	117
ISSN	03050041
e-ISSN	14698064
DOI	10.1017/s0305004100074715
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Issue Number	1
Volume Number	120
Language	English
Publisher	Cambridge University Press (CUP)
Publisher Date	1996-07-01
Access Restriction	Open
Subject Keyword	Mathematical Proceedings of the Cambridge Philosophical Society Applied Mathematics Characteristic Functions Function Operating
Content Type	Text
Resource Type	Article
Subject	Mathematics