## Hiroshima Mathematical Journal

### Polyharmonicity and algebraic support of measures

#### Abstract

Our main result states that two signed 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 explicitly computed by the properties of the polynomial $P\left( x\right)$. 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.

#### Article information

Source
Hiroshima Math. J., Volume 37, Number 1 (2007), 25-44.

Dates
First available in Project Euclid: 11 April 2007

https://projecteuclid.org/euclid.hmj/1176324093

Digital Object Identifier
doi:10.32917/hmj/1176324093

Mathematical Reviews number (MathSciNet)
MR2308522

Zentralblatt MATH identifier
1124.44003

#### Citation

Kounchev, Ognyan; Render, Hermann. Polyharmonicity and algebraic support of measures. Hiroshima Math. J. 37 (2007), no. 1, 25--44. doi:10.32917/hmj/1176324093. https://projecteuclid.org/euclid.hmj/1176324093