Related papers: Error analysis of the Bergman kernel method with s…
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…
Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the "native" Hilbert space $\calh$ in which they are reproducing. Continuous kernels on compact domains have an expansion into…
The main result of the present article is a (practically optimal) criterium for the pseudoeffectivity of the twisted relative canonical bundles of surjective projective maps. Our theorem has several applications in algebraic geometry; to…
Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is…
We consider a bounded domain $\Omega \subseteq \mathbb C^d$ which is a $G$-space for a finite complex reflection group $G$. For each one-dimensional representation of the group $G,$ the relative invariant subspace of the weighted Bergman…
The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial, the kernel polynomial, and using properties of the curve, the kernel curve,…
Let G be a bounded Jordan domain in the complex plane and consider the infinite upper Hessenberg matrix M associated with the Bergman orthogonal polynomials of G. This matrix represents the Bergman shift operator of G. The main purpose of…
Very recently one has started to study Bergman and Szeg\"o kernels in the setting of octonionic monogenic functions. In particular, explicit formulas for the Bergman kernel for the octonionic unit ball and for the octonionic right…
We analyse the convergence of sampling algorithms for functions in reproducing kernel Hilbert spaces (RKHS). To this end, we discuss approximation properties of kernel regression under minimalistic assumptions on both the kernel and the…
In this paper, weighted Bergman spaces on the unit ball in C^n are investigated. A characterization of the Carleson embeddings is established. Pointwise and norm estimates on the reproducing kernel function of weighted Bergman spaces on the…
We give an extensive study on the Bergman kernel expansions and the random zeros associated with the high tensor powers of a semipositive line bundle on a complete punctured Riemann surface. We prove several results for the zeros of…
Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…
We give a complete description of the boundary behaviour of the Poisson kernel and the harmonic Bergman kernel of a bounded domain with smooth boundary, which in some sense is an analogue of the similar description for the usual Bergman…
Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry. Particularly in the context of machine learning, they are central to clustering, exponential families, parameter estimation and…
We provide improved error bounds for kernel-based numerical differentiation in terms of growth functions when kernels are of a finite smoothness, such as polyharmonic splines, thin plate splines or Wendland kernels. In contrast to existing…
We prove the boundedness of Bergman type projections in two different analytic function spaces in bounded strongly pseudoconvex domains with the smooth boundary. Our results were previously well-known in the case of the unit disk.
We give the parameter version of localization theorem for Bergman metric near the boundary points of strictly pseudoconvex domains. The approximation theorem for square integrable holomorphic functions on such domains in the spirit of…
Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we~obtain expansions for the Szeg\"o and the weighted Bergman kernels of $M$-harmonic functions, i.e.~functions annihilated by the invariant…
This paper introduces an approach for detecting differences in the first-order structures of spatial point patterns. The proposed approach leverages the kernel mean embedding in a novel way by introducing its approximate version tailored to…
For a planar domain $\Omega$, we consider the Dirichlet spaces with respect to a base point $\zeta\in\Omega$ and the corresponding kernel functions. It is not known how these kernel functions behave as we vary the base point. In this note,…