On positive definiteness of some functions

Let rho be a nonnegative homogeneous function on R-n. General structure of the set of numerical pails (delta, lambda), for which the function (1 - rho(lambda)(x))(+)(delta) is positive definite on R-n is investigated: a criterion for positive definiteness of this Function is given in terms of completely monotonic Functions: a connection of this problem with the Schoenberg problem on positive definiteness of the function exp( -p(lambda)(si) is found. We also obtain a general sufficient condition of Polya type for a function f(rho(x)) to be positive definite on R-n. (C) 2000 Academic Press.