Polyharmonicity and algebraic support of measures

Abstract

We introduce a multivariate Markov transform which generalizes the well-known one-dimensional Stieltjes transform from the Moment problem and Spectral theory. Our main result states that two measures μ and ν with bounded support contained in the zero set of a polynomial P(x) are equal if they coincide on the subspace of all polynomials of polyharmonic degree N_{P} where the natural number N_{P} is explictly computed by the properties of the polynomial P(x). The method of proof depends on a definition of a multivariate Markov transform which another major objective of the present paper. The classical notion of orthogonal polynomial of second kind is generalized to the multivariate setting: it is a polyharmonic function which has similar features as in the one-dimensional case.