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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Functions operating on positive definite matrices and a theorem of Schoenberg


Authors: Jens Peter Reus Christensen and Paul Ressel
Journal: Trans. Amer. Math. Soc. 243 (1978), 89-95
MSC: Primary 15A48; Secondary 42A63
MathSciNet review: 502895
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Abstract: We prove that the set of all functions $ f:\,[ - 1,\,1] \to [ - 1,\,1]$ operating on real positive definite matrices and normalized such that $ f(1)\, = \,1$, is a Bauer simplex, and we identify its extreme points. As an application we obtain Schoenberg's theorem characterising positive definite kernels on the infinite dimensional Hilbert sphere.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0502895-2
PII: S 0002-9947(1978)0502895-2
Article copyright: © Copyright 1978 American Mathematical Society