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On Continuity and Decomposition of Positive Definite Functions
Author(s) -
Sasvári Zoltán
Publication year - 1989
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19891400110
Subject(s) - mathematics , positive definite matrix , corollary , pure mathematics , neighbourhood (mathematics) , zero (linguistics) , commutative property , mathematical analysis , linguistics , eigenvalues and eigenvectors , physics , philosophy , quantum mechanics
Let G be a locally compact commutative group and let g and h be positive definite functions on G , which are not identically zero. We show that continuity of gh̄ implies the existence of a character γ of G d (the discrete version of G ) such that γ g and γ h are continuous. As corollary we get a special case of a result of K. de Leeuw and I. Glicksberg concerning almost continuous group representations. In the second part of the paper we prove decomposition theorems for positive definite functions defined on a neighbourhood of the zero.