Laplace Beltrami operator in the Baran metric and pluripotential equilibrium measure: the ball, the simplex and the sphere

The Baran metric \(\delta_E\) is a Finsler metric on the interior of \(E\subset \R^n\) arising from Pluripotential Theory. We consider the few instances, namely \(E\) being the ball, the simplex, or the sphere, where \(\delta_E\) is known to be Riemaniann and we prove that the eigenfunctions of the...

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Bibliographic Details
Published in:arXiv.org 2017-04
Main Author: Piazzon, Federico
Format: Article
Language:English
Subjects:
Boundary conditions
Differential equations
Eigenvectors
Operators (mathematics)
Polynomials
Online Access:Get full text
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Summary:The Baran metric \(\delta_E\) is a Finsler metric on the interior of \(E\subset \R^n\) arising from Pluripotential Theory. We consider the few instances, namely \(E\) being the ball, the simplex, or the sphere, where \(\delta_E\) is known to be Riemaniann and we prove that the eigenfunctions of the associated Laplace Beltrami operator (with no boundary conditions) are the orthogonal polynomials with respect to the pluripotential equilibrium measure \(\mu_E\) of \(E.\) We conjecture that this may hold in a wider generality. The considered differential operators have been already introduced in the framework of orthogonal polynomials and studied in connection with certain symmetry groups. In this work instead we highlight the relationships between orthogonal polynomials with respect to \(\mu_E\) and the Riemaniann structure naturally arising from Pluripotential Theory
ISSN:2331-8422