Criteria of Pólya type for radial positive definite functions

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.