An extremal problem for the Bergman kernel of orthogonal polynomials

TL;DR

This paper investigates the asymptotic behavior of probability measures minimizing the Bergman function on a curve, showing they converge to a balayage measure, with implications for orthogonal polynomials and potential theory.

Contribution

It establishes the weak-* convergence of extremal measures to the balayage of a point mass, linking Bergman kernel optimization to measures on the unit circle and Faber polynomial estimates.

Findings

Extremal measures $ u_n$ tend to the balayage measure $ ilde{ u}$ as n increases.

The convergence is shown via a connection to an optimization problem on the unit circle.

The proof utilizes estimates for Faber polynomials associated with the curve $\Gamma$.

Abstract

Let $\Gamma \subset \mathbb C$ be a curve of class $C(2,\alpha)$. For $z_{0}$ in the unbounded component of ${\mathbb C}\setminus \Gamma$, and for $n=1,2,...$, let $\nu_n$ be a probability measure with supp$(\nu_{n})\subset \Gamma$ which minimizes the Bergman function $B_{n}(\nu,z):=\sum_{k=0}^{n}|q_{k}^{\nu}(z)|^{2}$ at $z_{0}$ among all probability measures $\nu$ on $\Gamma$ (here, $\{q_{0}^{\nu},\ldots,q_{n}^{\nu}\}$ are an orthonormal basis in $L^2(\nu)$ for the holomorphic polynomials of degree at most $n$). We show that $\{\nu_{n}\}_n$ tends weak-* to $\hat\delta_{z_{0}}$, the balayage of the point mass at $z_0$ onto $\Gamma$, 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 $\Gamma$.