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Advances in Operator Theory
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Volume Volume 3 (2018)
Issue Issue 2
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Hassairi, A., Lajmi, S. (2018). Singular Riesz measures on symmetric cones. Advances in Operator Theory, 3(2), 337-350. doi: 10.15352/aot.1706-1183
Abdelhamid Hassairi; Sallouha Lajmi. "Singular Riesz measures on symmetric cones". Advances in Operator Theory, 3, 2, 2018, 337-350. doi: 10.15352/aot.1706-1183
Hassairi, A., Lajmi, S. (2018). 'Singular Riesz measures on symmetric cones', Advances in Operator Theory, 3(2), pp. 337-350. doi: 10.15352/aot.1706-1183
Hassairi, A., Lajmi, S. Singular Riesz measures on symmetric cones. Advances in Operator Theory, 2018; 3(2): 337-350. doi: 10.15352/aot.1706-1183

Singular Riesz measures on symmetric cones

Article 2, Volume 3, Issue 2 - Serial Number 8, Spring 2018, Page 337-350  XML PDF (169 K)
Document Type: Original Article
DOI: 10.15352/aot.1706-1183
Authors
Abdelhamid Hassairi 1; Sallouha Lajmi2
1Sfax university
2Sfax University
Abstract
‎A fondamental theorem due to Gindikin says that the‎ ‎generalized power $\Delta_{s}(-\theta^{-1})$ defined on a symmetric‎ ‎cone is the Laplace transform of a positive measure $R_{s}$ if and ‎only if $s$ is in a given subset $\Xi$ of $\Bbb{R}^{r}$‎, ‎where $r$‎ ‎is the rank of the cone‎. ‎When $s$ is in a well defined part of‎ ‎$\Xi$‎, ‎the measure $R_{s}$ is absolutely continuous with respect to‎ ‎Lebesgue measure and has a known expression‎. ‎For the other elements‎ ‎$s$ of $\Xi$‎, ‎the measure $R_{s}$ is concentrated on the boundary of‎ ‎the cone and it has never been explicitly determined‎. ‎The aim of the‎ ‎present paper is to give an explicit description of the measure‎ ‎$R_{s}$ for all $s$ in $\Xi$‎. ‎The work is motivated by the‎ ‎importance of these measures in probability theory and in statistics‎ ‎since they represent a generalization of the class of measures‎ ‎generating the famous Wishart probability distributions‎.
Keywords
Jordan algebra; symmetric cone; generalized power; Laplace transform; Riesz measure
Main Subjects
42Bxx: Harmonic analysis in several variables; 46Gxx: Measures, integration, derivative, holomorphy; 60Bxx: Probability theory on algebraic and topological structures
Statistics
Article View: 209
PDF Download: 148
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