An Extremal Problem for the Bergman Kernel of Orthogonal Polynomials

Let \(\Gamma \subset \mathbb {C}\) $\mathrm{\Gamma }\subset \mathbb{C}$ be a curve of class \(C(1,\alpha )\) $C(1,\alpha )$ . For \(z_{0}\) $z_0$ in the unbounded component of \(\mathbb {C}\setminus \Gamma \) $\mathbb{C}\backslash \mathrm{\Gamma }$ , and for \(n=1,2,...\) $n=1,2,...$ , let \(\nu _n\) ${\nu }_n$ be a probability measure with \(\mathop {\textrm{supp}}\nolimits (\nu _{n})\subset \Gamma \) $\text{supp}({\nu }_n)\subset \mathrm{\Gamma }$ which minimizes the Bergman function \(B_{n}(\nu ,z):=\sum _{k=0}^{n}|q_{k}^{\nu }(z)|^{2}\) $B_n(\nu ,z):={\sum }_{k=0}^n{|q_k^{\nu }(z)|}^2$ at \(z_{0}\) $z_0$ among all probability measures \(\nu \) $\nu$ on \(\Gamma \) $\mathrm{\Gamma }$ (here, \(\{q_{0}^{\nu },\ldots ,q_{n}^{\nu }\}\) $\{q_0^{\nu },\dots ,q_n^{\nu }\}$ are an orthonormal basis in \(L^2(\nu )\) $L^2(\nu )$ for the holomorphic polynomials of degree at most n). We show that \(\{\nu _{n}\}_n\) ${\{{\nu }_n\}}_n$ tends weak-* to \({{\widehat{\delta }}}_{z_{0}}\) ${\widehat{\delta }}_{z_0}$ , the balayage of the point mass at \(z_0\) $z_0$ onto \(\Gamma \) $\mathrm{\Gamma }$ , 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 \(\Gamma \) $\mathrm{\Gamma }$ .