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Criteria of Pólya type for radial positive definite functions

Authors:	Tilmann Gneiting

Institution:	Department of Statistics, Box 354322, University of Washington, Seattle, Washington 98195

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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