Geometry of Finite Dimensional Moment Spaces and Applications to Orthogonal Polynomials

Abstract

Various geometrical properties of the finite dimensional moment spaces generated by normalized distribution functions over [0,∞) and (-∞,∞) are investigated. The moment spaces are found to be dual to the polynomial spaces. The structure of the latter is studied by means of this duality and of a representation theorem for positive polynomials. The extreme points of the polynomial spaces are associated with polynomials orthogonal with respect to the distributions generating the moment spaces. This correspondence is used in order to derive several properties of orthogonal polynomials.