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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Criteria of Pólya type for radial positive definite functions


Author: Tilmann Gneiting
Journal: Proc. Amer. Math. Soc. 129 (2001), 2309-2318
MSC (2000): Primary 42B10, 60E10, 42A82
Published electronically: January 17, 2001
MathSciNet review: 1823914
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Abstract:

This article presents sufficient conditions for the positive definiteness of radial functions $f(x) = \varphi(\Vert x\Vert)$, $x \in \mathbb{R}^n$, in terms of the derivatives of $\varphi$. The criterion extends and unifies the previous analogues of Pólya's theorem and applies to arbitrarily smooth functions. In particular, it provides upper bounds on the Kuttner-Golubov function $k_n(\lambda)$ which gives the minimal value of $\kappa$ such that the truncated power function $(1-\Vert x\Vert^\lambda)_+^\kappa$, $x \in \mathbb{R}^n$, is positive definite. Analogous problems and criteria of Pólya type for $\Vert\cdot\Vert _\alpha$-dependent functions, $\alpha > 0$, are also considered.


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Additional Information

Tilmann Gneiting
Affiliation: Department of Statistics, Box 354322, University of Washington, Seattle, Washington 98195
Email: tilmann@stat.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05839-7
PII: S 0002-9939(01)05839-7
Received by editor(s): November 29, 1999
Published electronically: January 17, 2001
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society