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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.
Computational Methods and Function Theory – Springer Journals
Published: Sep 7, 2019
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