Criteria of Polya type for radial positive definite functions

This article presents sufficient conditions for the positive definiteness o f radial functions f(x) =phi(parallel tox parallel to), x is an element of R-n, in terms of the derivatives of phi. The criterion extends and unifies the previous analogues of Polya'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 parallel tox parallel to (lambda))(+)(kappa), x is an element of R-n, is positive definite. Analogous problems and criteri a of Polya type for parallel to.parallel to (alpha)-dependent functions, al pha >0, are also considered.