Analysis of Log-Weighted Quadrature Domains

TL;DR

This paper explores log-weighted quadrature domains (LQDs), revealing unique phenomena due to the logarithmic singularity and providing explicit characterizations and examples within the classical and singular frameworks.

Contribution

It introduces a generalized Schwarz function for LQDs, characterizes them via Riemann maps, and extends classical quadrature domain theory to include singular weights.

Findings

Quadrature data are non-unique when the domain contains the origin.

A domain is an LQD iff the outer factor of its Riemann map extends to the exponential of a rational function.

Explicit formulas relate the quadrature function and Riemann map via the Faber transform.

Abstract

This paper studies plane domains satisfying a quadrature identity with respect to the singular weight $\rho_0(w)=|w|^{-2}$. These are referred to as log-weighted quadrature domains (LQDs). The logarithmic singularity at $w=0$ leads to phenomena not present in the classical theory: in particular, when the domain contains the origin, the associated quadrature data are no longer unique, but are determined only up to a point charge at $0$. A generalized Schwarz function characterization of LQDs is established together with a natural formulation of the inverse problem in the singular setting. In the simply connected case, it is shown that a domain is an LQD if and only if the outer factor of its Riemann map extends to the exponential of a rational function. This characterization yields explicit formulae relating the quadrature function and the Riemann map via the Faber transform, thereby extending earlier formulae from the non-singular theory. Several basic classes of LQDs are also covered, and explicit examples are computed.