High degree simple partial fractions in the Bergman space: Approximation and Optimization

TL;DR

This paper investigates simple partial fractions in weighted Bergman spaces, showing conditions for minimal norm configurations, asymptotic behaviors, and the structure of their closures, revealing new phenomena related to pole distribution.

Contribution

It characterizes minimal norm simple partial fractions with poles on the unit circle and explores their asymptotics and closure properties in Bergman spaces, highlighting new phenomena.

Findings

Minimal norm fractions are equidistributed on the circle under certain conditions.

Asymptotic formulas for norms of these fractions are established.

The closure of these fractions in Bergman spaces is described.

Abstract

We consider the class of standard weighted Bergman spaces $A^2_{\alpha}(\mathbb{D})$ and the set $SF^N(\mathbb{T})$ of simple partial fractions of degree $N$ with poles on the unit circle. We prove that under certain conditions, the simple partial fractions of order $N$, with $n$ poles on the unit circle attain minimal norm if and only if the points are equidistributed on the unit circle. We show that this is not the case if the conditions we impose are not met, exhibiting a new interesting phenomenon. We find sharp asymptotics for these norms. Additionally we describe the closure of these fractions in the standard weighted Bergman spaces.