Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0401264

Quadrature domains and kernel function zipping

Steven R. Bell
View original source ↗Read source document ↗Export metadata ↓

Abstract

It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact, they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain. Following this string of ideas leads to the discovery that the Bergman kernel can be zipped down to a strikingly small data set. It is also proved that the kernel functions associated to a quadrature domain must be algebraic.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Steven R. Bell. 2004-01-20. Quadrature domains and kernel function zipping. https://doi.org/10.1007/bf02384780

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pre-Schwarzian and Schwarzian norm estimates for harmonic functions with fixed analytic part

In the present article, we discuss about the estimate of the pre-Schwarzian and Schwarzian norms for locally univalent harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$. First, we prove a general result for the estimate of the pre-Schwarzian norm which rectify few earlier flawed results. We also consider a new class $\mathcal{F}_0$ consisting of all harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that ${\rm Re\,}\left(1+z\frac{h''(z)}{h'(z)}\right)>0$ for $z\in\mathbb{D}$ with dilatation $ω_f(z)\in Aut(\mathbb{D})$ and obtain best possible estimates of the pre-Schwarzian and Schwarzian norms for functions in the class $\mathcal{F}_0$. Moreover, we obtain the distortion and coefficient estimates of the co-analytic function $g$ when $f=h+\overline{g}\in\mathcal{F}_0$.

math.CV↗
arXiv2026

The Reciprocal Problem on Weighted Bergman Spaces

The reciprocal problem on weighted Bergman spaces has been posed as an open problem. In this paper, we establish several sufficient conditions for the reciprocal property and clarify the parameter ranges in which the available methods are applicable. In particular, we prove that functions in $A_α^p\cap H^\infty$ enjoy the reciprocal property in the parameter ranges where the required analytic Besov composition theorem is available. In addition, using Hardy boundary estimates, we solve the reciprocal problem in the Drury--Arveson space $H_d^2$ when the dimension is $d=3$, and give an equivalent condition for the reciprocal problem in the four-dimensional Drury--Arveson space.

math.CV↗
arXiv2026

Möbius Maps, Reflections and Lipschitz Constants

We introduce the chordal isometric circle of a Möbius map, and use this to give a factorization of any Möbius map as the composition of a chordal isometry and either a reflection, or a rotary reflection, across a circle. We then use this to find the chordal, and spherical, Lipschitz constants of a Möbius map, and compare this with related results in the literature.

math.CV↗