Criteria of Pólya type for radial positive definite functions

Abstract

This article presents sufficient conditions for the positive definiteness of radial functions f ( x ) = φ ( ‖ x ‖ ) f(x) = \varphi (\|x\|) , x ∈ R n x \in \mathbb {R}^n , in terms of the derivatives of φ \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 ( λ ) k_n(\lambda ) which gives the minimal value of κ \kappa such that the truncated power function ( 1 − ‖ x ‖ λ ) + κ (1-\|x\|^\lambda )_+^\kappa , x ∈ R n x \in \mathbb {R}^n , is positive definite. Analogous problems and criteria of Pólya type for ‖ ⋅ ‖ α \|\cdot \|_\alpha -dependent functions, α > 0 \alpha > 0 , are also considered.