相关论文: Quadrature domains and kernel function zipping
A streamlined proof that the Bergman kernel associated to a quadrature domain in the plane must be algebraic will be given. A byproduct of the proof will be that the Bergman kernel is a rational function of z and one other explicit function…
We make use of the Bergman kernel function to study quadrature domains for square-integrable holomorphic functions of several variables. Emphasis is given to generalizing biholomorphic mapping properties of planar quadrature domains to the…
Given a finite family of compact subsets of the complex plane we propose a certificate of mutual non-overlapping with respect to area measure. The criterion is stated as a couple of positivity conditions imposed on a four argument…
We prove that the Bergman kernel function associated to a finitely connected domain in the plane is given as a rational combination of only three basic functions of one complex variable: an Alhfors map, its derivative, and one other…
We make a systematic investigation of quadrature properties for quadrics, namely integration of holomorphic functions over planar domains bounded by second degree curves. A full understanding requires extending traditional settings by…
We prove two density theorems for quadrature domains in $\mathbb{C}^n$, $n \geq 2$. It is shown that quadrature domains are dense in the class of all product domains of the form $D \times \Omega$, where $D \subset \mathbb{C}^{n-1}$ is a…
We show that the Bergman, Szego, and Poisson kernels associated to a finitely connected domain in the plane are all composed of finitely many easily computed functions of one variable. The new formulas give rise to new methods for computing…
We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains…
It is well known that, in the plane, the boundary of any quadrature domain (in the classical sense) coincides with the zero set of a polynomial. We show, by explicitly constructing some four-dimensional examples, that this is not always the…
The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in…
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…
We highlight an intrinsic connection between classical quadrature domains and the well-studied theme of removable singularities of analytic sets in several complex variables. Exploiting this connection provides a new framework to recover…
We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the…
We study the Bergman kernel of certain domains in $\mathbb{C}^n$, called elementary Reinhardt domains, generalizing the classical Hartogs triangle. For some elementary Reinhardt domains, we explicitly compute the kernel, which is a rational…
A standard objective in computer experiments is to approximate the behaviour of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable:…
We introduce multi-sheeted versions of algebraic domains and quadrature domains, allowing them to be branched covering surfaces over the Riemann sphere. The two classes of domains turn out to be the same, and the main result states that the…
It is known that any target function is realized in a sufficiently small neighborhood of any randomly connected deep network, provided the width (the number of neurons in a layer) is sufficiently large. There are sophisticated theories and…
We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in C^3…
We present a framework for reconstructing any simply connected, bounded or unbounded, quadrature domain $\Omega$ from its quadrature function $h$. Using the Faber transform, we derive formulae directly relating $h$ to the Riemann map for…
We contruct two classes of Zalcman-type domains, on which the Bergman distance functions have certain pre-described boundary behaviors. Such examples also lead to generalizations of uniformly perfectness in the sense of Pommerenke. These…