## Hiroshima Mathematical Journal

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

### Polyharmonicity and algebraic support of measures

Ognyan Kounchev and Hermann Render

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.32917/hmj/1176324093

**Mathematical Reviews number (MathSciNet)**

MR2308522

**Zentralblatt MATH identifier**

1124.44003

**Subjects**

Primary: 44A15: Special transforms (Legendre, Hilbert, etc.)

Secondary: 35D55 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

**Keywords**

Markov function Stieltjes transform polynomial of second kind polyharmonic function

#### 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