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Laplace Beltrami Operator in the Baran Metric and Pluripotential Equilibrium Measure: The Ball, the Simplex, and the Sphere

Laplace Beltrami Operator in the Baran Metric and Pluripotential Equilibrium Measure: The Ball,... The Baran metric $$\delta _E$$ δ E is a Finsler metric on the interior of $$E\subset {\mathbb {R}}^n$$ E ⊂ R n arising from pluripotential theory. When E is an Euclidean ball, a simplex, or a sphere, $$\delta _E$$ δ E is Riemannian. No further examples of such property are known. We prove that in these three cases, the eigenfunctions of the Laplace Beltrami operator associated with $$\delta _E$$ δ E are the orthogonal polynomials with respect to the pluripotential equilibrium measure $$\mu _E$$ μ E of E. We conjecture that this may hold in wider generality. The differential operators that we consider were introduced in the framework of orthogonal polynomials and studied in connection with certain symmetry groups. In this work, we highlight the connections between orthogonal polynomials with respect to $$\mu _E$$ μ E and the Riemannian structure naturally arising from pluripotential theory. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Laplace Beltrami Operator in the Baran Metric and Pluripotential Equilibrium Measure: The Ball, the Simplex, and the Sphere

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References (89)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-019-00286-9
Publisher site
See Article on Publisher Site

Abstract

The Baran metric $$\delta _E$$ δ E is a Finsler metric on the interior of $$E\subset {\mathbb {R}}^n$$ E ⊂ R n arising from pluripotential theory. When E is an Euclidean ball, a simplex, or a sphere, $$\delta _E$$ δ E is Riemannian. No further examples of such property are known. We prove that in these three cases, the eigenfunctions of the Laplace Beltrami operator associated with $$\delta _E$$ δ E are the orthogonal polynomials with respect to the pluripotential equilibrium measure $$\mu _E$$ μ E of E. We conjecture that this may hold in wider generality. The differential operators that we consider were introduced in the framework of orthogonal polynomials and studied in connection with certain symmetry groups. In this work, we highlight the connections between orthogonal polynomials with respect to $$\mu _E$$ μ E and the Riemannian structure naturally arising from pluripotential theory.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Sep 7, 2019

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