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On the Derivatives of Radial Positive Definite Functions

✍ Scribed by Tilmann Gneiting


Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
73 KB
Volume
236
Category
Article
ISSN
0022-247X

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Radial Positive Definite Functions Gener
✍ Tilmann Gneiting πŸ“‚ Article πŸ“… 1999 πŸ› Elsevier Science 🌐 English βš– 305 KB

Radial positive definite functions are of importance both as the characteristic functions of spherically symmetric probability distributions, and as the correlation functions of isotropic random fields. The Euclid's hat function h n (&x&), x # R n , is the self-convolution of an indicator function s

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Let f be a positive definite function on a locally compact abelian group G. In [3] we showed that measurability of 1 on an open neighbourhood of the zero implies measurability of f on G. As a main tool we used a result about the support of f [3, Th. I]. The aim of this note is to simplify the proof

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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 gh implies the existence of a character y of Gd (the discrete version of G) such that yg and y h are continuous. As corollary we get a special

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Let \ be a nonnegative homogeneous function on R n . General structure of the set of numerical pairs ($, \*), for which the function (1&\ \* (x)) $ + is positive definite on R n is investigated; a criterion for positive definiteness of this function is given in terms of completely monotonic function