Polyharmonicity and algebraic support of measures

Our main result states that two signed 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 NP where the natural number NP is explicitly computed by the properties of the polynomial P (x). The method of proof depends on a definition of a multivariate Markov transform which is 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 to those in the one-dimensional case. Acknowledgement: Both authors have been sponsored by the Institutes partnership project at the Alexander von–Humboldt Foundation; the first was sponsored by a Greek–Bulgarian S&T Cooperation project. 2000 Mathematics Subject Classification: Primary: 44A15, Secondary 35D55, 42C05: