Two Universality Results for Polynomial Reproducing Kernels

TL;DR

This paper establishes two new universality results for polynomial reproducing kernels associated with specific measures, expanding understanding of kernel behavior in complex measure settings.

Contribution

It introduces two novel universality theorems for polynomial reproducing kernels on measures with singularities and on lemniscates, using Lubinsky's methods.

Findings

Universality for measures on the unit circle with singularities.

Universality for area-type measures on polynomial lemniscates.

Application of Lubinsky's methods to new measure classes.

Abstract

We prove two new universality results for polynomial reproducing kernels of compactly supported measures. The first applies to measures on the unit circle with a jump and a singularity in the weight at $1$ and the second applies to area-type measures on a certain disconnected polynomial lemniscate. In both cases, we apply methods developed by Lubinsky to obtain our results.