# Reproducing kernel of the space $R^t(K,\mu)$

## TL;DR

This paper investigates the behavior of the reproducing kernel in the space of rational functions on a compact set in the complex plane, focusing on boundary behavior near the unit circle under certain measure conditions.

## Contribution

It provides new insights into the boundary behavior of the reproducing kernel of $R^t(K, )$ spaces, especially near the boundary of analytic bounded point evaluations.

## Key findings

- Reproducing kernel behavior is characterized near boundary points.
- Boundary measure positivity influences kernel boundary behavior.
- Results extend understanding of rational function spaces on complex sets.

## Abstract

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