Pré-Publication, Document De Travail Année : 2024

An Extremal Problem for the Bergman Kernel of Orthogonal Polynomials

Résumé

Let Γ ⊂ C be a curve of class C(2, α). For z 0 in the unbounded component of C \ Γ, and for n = 1, 2, ..., let ν n be a probability measure with supp(ν n ) ⊂ Γ which minimizes the Bergman function B n (ν, z) := n k=0 |q ν k (z)| 2 at z 0 among all probability measures ν on Γ (here, {q ν 0 , . . . , q ν n } are an orthonormal basis in L 2 (ν) for the holomorphic polynomials of degree at most n). We show that {ν n } n tends weak-* to δ z 0 , the balayage of the point mass at z 0 onto Γ, by relating this to an optimization problem for probability measures on the unit circle. Our proof makes use of estimates for Faber polynomials associated to Γ.

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Dates et versions

hal-04949357 , version 1 (15-02-2025)

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S. Charpentier, N. Levenberg, Franck Wielonsky. An Extremal Problem for the Bergman Kernel of Orthogonal Polynomials. 2025. ⟨hal-04949357⟩
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