Szego coordinates, quadrature domains, and double quadrature domains

TL;DR

The paper introduces Szego coordinates that transform finitely connected planar domains into quadrature domains, and explores conditions under which these become double quadrature domains, highlighting their properties and density in the domain space.

Contribution

It defines Szego coordinates for finitely connected domains and characterizes when they coincide with Bergman coordinates to form double quadrature domains, showing their density.

Findings

Szego coordinates can approximate identity and convert domains into quadrature domains.

Double quadrature domains have special properties and are dense among smooth bounded domains.

Conditions for Szego and Bergman coordinates to coincide are established.

Abstract

We define Szego coordinates on a finitely connected smoothly bounded planar domain which effect a holomorphic change of coordinates on the domain that can be as close to the identity as desired and which convert the domain to a quadrature domain with respect to boundary arc length. When these Szego coordinates coincide with Bergman coordinates, the result is a double quadrature domain with respect to both area and arc length. We enumerate a host of interesting and useful properties that such double quadrature domains possess, and we show that such domains are in fact dense in the realm of bounded finitely connected domains with smooth boundaries.