## Nihonkai Mathematical Journal

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

Koji Furuta

#### 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
Revised: 7 June 2017
First available in Project Euclid: 26 April 2018

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

Mathematical Reviews number (MathSciNet)
MR3794316

Zentralblatt MATH identifier
06873760

#### 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

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