## Nihonkai Mathematical Journal

- Nihonkai Math. J.
- Volume 28, Number 2 (2017), 79-88.

### Integral Representations of Positive Definite Functions on Convex Sets of Certain Semigroups of Rational Numbers

#### Abstract

H. Glöckner proved that an operator-valued positive definite function on an open convex subset of $\boldsymbol Q^N$ is a restriction of the Laplace transform of an operator-valued measure on $\boldsymbol R^N$. We generalize this result to a function on an open convex subset of a certain subsemigroup of $\boldsymbol Q^2$.

#### Note

The author would like to express his gratitude to the referee for valuable comments.

#### Article information

**Source**

Nihonkai Math. J., Volume 28, Number 2 (2017), 79-88.

**Dates**

Received: 2 May 2016

Revised: 7 June 2017

First available in Project Euclid: 26 April 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.nihmj/1524708082

**Mathematical Reviews number (MathSciNet)**

MR3794316

**Zentralblatt MATH identifier**

06873760

**Subjects**

Primary: 43A35: Positive definite functions on groups, semigroups, etc.

Secondary: 44A60: Moment problems 47A57: Operator methods in interpolation, moment and extension problems [See also 30E05, 42A70, 42A82, 44A60]

**Keywords**

moment problem positive definite function semigroup

#### Citation

Furuta, Koji. Integral Representations of Positive Definite Functions on Convex Sets of Certain Semigroups of Rational Numbers. Nihonkai Math. J. 28 (2017), no. 2, 79--88. https://projecteuclid.org/euclid.nihmj/1524708082