On the Representation of Pn-Positive definite Functions and Applications.

Positive definite functions arise in various areas in pure and applied mathematics, such as orthogonal polynomials, numerical integration, and time series analysis. In these applications, the notion of positive definiteness is depending on an underlying group or semigroup structure. We extend some central results on positive definite functions to more general algebraic structures, which are induced by polynomial sequences. In particular, we show that every positive definite function of this type is the transform of a positive finite Borel measure on the reals, and find conditions which yield more information on the character of the support of this measure. For illustration and application of our results, we consider stationary sequences and certain non-autonomous linear Volterra difference equations. In the latter case, statements on the existence of unbounded solutions are obtained