Intrinsic interpolation, near-circularity and maximal convergence

Abstract

Let $E$ be compact and connected with $\mathrm{cap}E>0$ and connected complement $\Omega =\overline{ℂ}\setminus E$, let $g_{\Omega }(z,\infty )$ be the Green’s function of $\Omega$ with pole at infinity and let $E_{\sigma }≔\{z\in \Omega :g_{\Omega }(z,\infty )<log\sigma \}\cup E,1<\sigma <\infty ,$ be the Green domains with boundaries ${\Gamma }_{\sigma }$. Let $f$ be holomorphic on $E$ and let $\rho \left(f\right)$ denote the maximal parameter of holomorphy of $f$ and let ${\left(p_n\right)}_{n\in N}$ be a sequence of polynomials converging maximally to $f$ on $E$. If $\sigma$, $1<\sigma <\rho \left(f\right)<\infty$, is fixed and if $m_n\left(\sigma \right)$ denotes the number of interpolation points of $p_n$ to $f$ in $E_{\sigma }$ with normalized counting measure ${\mu }_{\sigma ,n}$, then there exists a subset $\Lambda \subset N$ such that $m_n\left(\sigma \right)=n+o\left(n\right)\text{as}n\in \Lambda ,n\to \infty ,$ $\widehat{{{\mu }_{\sigma ,n}}_{{|}_E}}+{{\mu }_{\sigma ,n}}_{{|}_{\Omega }}\stackrel{\ast }{⟶}{\mu }_E\text{as}n\in \Lambda ,n\to \infty ,$ where ${\mu }_{\sigma ,n}={{\mu }_{\sigma ,n}}_{{|}_E}+{{\mu }_{\sigma ,n}}_{{|}_{\Omega }}$, $\widehat{{{\mu }_{\sigma ,n}}_{{|}_E}}$ denotes the balayage measure of ${{\mu }_{\sigma ,n}}_{{|}_E}$ onto the boundary of $E$ and ${\mu }_E$ is the equilibrium measure of $E$. Moreover, there exists a sequence ${\left({\sigma }_n\right)}_{n\in \Lambda }$ converging to $\sigma$ such that the closed curves ${\gamma }_n=(f-p_n)\left({\Gamma }_{{\sigma }_n}\right)$ do not pass through the point 0 and the winding numbers ${\text{Ind}}_{{\gamma }_n}\left(0\right)$ satisfy ${\text{Ind}}_{{\gamma }_n}\left(0\right)=m_n\left({\sigma }_n\right)=n+o\left(n\right)\text{as}n\in \Lambda ,n\to \infty .$