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Strictly Positive Definite Functions

✍ Scribed by Kuei-Fang Chang

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 490 KB
Volume: 87
Category: Article
ISSN: 0021-9045
DOI: 10.1006/jath.1996.0097

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✦ Synopsis

We give a complete characterization of the strictly positive definite functions on the real line. By Bochner's theorem, this is equivalent to proving that if the separated sequence of real numbers [a n ] describes the points of discontinuity of a distribution function, there exists an almost periodic polynomial with the zeros [a n ]. We prove a useful necessary condition that every strictly normalized, positive definite function f satisfies | f (x)| <1 for all x{0. It is a sufficient condition for strictly positive definiteness that if the carrier of a nonzero finite Borel measure on R is not a discrete set, then the Fourier Stieltjes transform +^of + is strictly positive definite.

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