Polyharmonicity and algebraic support of measures

TL;DR

This paper introduces a multivariate Markov transform extending the classical Stieltjes transform, and establishes measure uniqueness based on polyharmonic polynomial support, generalizing orthogonal polynomials to multivariate polyharmonic functions.

Contribution

It defines a multivariate Markov transform and proves measure equality criteria using polyharmonic polynomial support, extending classical univariate results to multivariate settings.

Findings

Measures with bounded support in polynomial zero sets are equal if they agree on polyharmonic polynomials.

Introduces a multivariate Markov transform generalizing the classical one-dimensional transform.

Generalizes the orthogonal polynomial of second kind to multivariate polyharmonic functions.

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 {\mu} and {\nu} 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.