Chui's conjecture in Bergman spaces

TL;DR

This paper proves Chui's conjecture in weighted Bergman spaces, showing that equispaced poles minimize the norm of simplest fractions and providing sharp asymptotics and closure descriptions.

Contribution

It establishes the minimal norm configuration for simplest fractions in weighted Bergman spaces and characterizes their closure, solving a longstanding conjecture.

Findings

Poles equispaced on the circle minimize the norm

Sharp asymptotics of simplest fractions norms

Closure of simplest fractions characterized via $L^2$ approximation

Abstract

We solve Chui's conjecture on the simplest fractions (i.e., sums of Cauchy kernels with unit coefficients) in weighted (Hilbert) Bergman spaces. Namely, for a wide class of weights, we prove that for every $N$, the simplest fractions with $N$ poles on the unit circle have minimal norm if and only if the poles are equispaced on the circle. We find sharp asymptotics of these norms. Furthermore, we describe the closure of the simplest fractions in weighted Bergman spaces, using an $L^2$ version of Thompson's theorem on dominated approximation by simplest fractions.