Criteria of Pólya type for radial positive definite functions

Gneiting, Tilmann

doi:10.1090/S0002-9939-01-05839-7

Criteria of Pólya type for radial positive definite functions
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Proc. Amer. Math. Soc. 129 (2001), 2309-2318 Request permission

Abstract:

This article presents sufficient conditions for the positive definiteness of radial functions $f(x) = \varphi (\|x\|)$, $x \in \mathbb {R}^n$, 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 $(1-\|x\|^\lambda )_+^\kappa$, $x \in \mathbb {R}^n$, is positive definite. Analogous problems and criteria of Pólya type for $\|\cdot \|_\alpha$-dependent functions, $\alpha > 0$, are also considered.

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