Reproducing kernel of the space $R^t(K,\hat{I}¼)$

Abstract

For $1\le t<\infty ,$ a compact subset $K$ of the complex plane $\mathbb{C},$ and a finite positive measure $\hat{I}¼$ supported on $K,$ $R^t(K,\hat{I}¼)$ denotes the closure in $L^t(\hat{I}¼)$ of rational functions with poles off $K$. Let $\hat{I}\text{c◯}$ be a connected component of the set of analytic bounded point evaluations for $R^t(K,\hat{I}¼)$. In this paper, we examine the behavior of the reproducing kernel of $R^t(K,\hat{I}¼)$ near the boundary $\partial \hat{I}\text{c◯}\cap \mathbb{T}$, assuming that $\hat{I}¼(\partial \hat{I}\text{c◯}\cap \mathbb{T})>0$, where $\mathbb{T}$ is the unit circle.

To appear in the Proceedings of IWOTA 2018