Polyharmonicity and algebraic support of measures

Details Polyharmonicity and algebraic support of measures Polyharmonicity and algebraic support of measures

2011-12-07

arxiv.org/abs/1112.1511v1

Hiroshima Mathematical Journal, Volume 37, Number 1 (2007), 25-44

Mathematics

Complex Variables

20 pages

Scientific paper

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.

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