Related papers: The Bergman Kernel on some Hartogs domains
Motivated by practical applications, I present a novel and comprehensive framework for operator-valued positive definite kernels. This framework is applied to both operator theory and stochastic processes. The first application focuses on…
The usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants. From this representation one can deduce formulas for spacing probabilities in terms of Fredholm…
We utilize a connection between compositional kernels and branching processes via Mehler's formula to study deep neural networks. This new probabilistic insight provides us a novel perspective on the mathematical role of activation…
In this paper, we first answer Chen-Zhang's problem on $p$-Bergman metric proposed in \cite{CZ22}. Second, we prove the off-diagonal p-Bergman kernel function $K_p(z,w)$ is H\"older continuous of order (1-$\varepsilon$) about the second…
For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(\Omega)$ over some non-empty set $\Omega$, the Berezin range and the Berezin radius of $T$ are defined respectively, by $\text{Ber}(T) := \{\langle…
Explicit representations of the eigenvalues of the peridynamic operator have been recently derived in [5]. These representations are given in terms of generalized hypergeometric functions. Asymptotic analysis of the hypergeometric functions…
We construct the heat kernel on curvilinear polygonal domains in arbitrary surfaces for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems, including those of Zaremba type. We compute the short time asymptotic…
We present explicit formulae for Weber-Schafheitlin type integrals and give them an interpretation as the kernel of a physically relevant operator related to the hamiltonian of Aharanov and Bohm. In particular, we derive explicit formulae…
In our preprint q-alg/9703005 q-analogues of bounded symmetric domains were defined to be homogeneous spaces of the associated quantum groups. The investigation of a simplest among those domains, the quantum matrix ball, was started in…
We relate the regularity of the Bergman projection operator and the canonical solution operator to the Nebenh\"ulle of complete Hartogs domains.
We consider a kernel based harmonic analysis of "boundary," and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivated by) results and notions from classical…
Given only information in the form of similarity triplets "Object A is more similar to object B than to object C" about a data set, we propose two ways of defining a kernel function on the data set. While previous approaches construct a…
The first part I talk about the motivation for Lu Qi-Keng conjecture and the results about the presence or absence of zeroes of the Bergman kernel function of a bounded domain in ${\bf{C^n}}$. The second part I summarize the main results on…
We study how iterated and composed completely positive maps act on operator-valued kernels. Each kernel is realized inside a single Hilbert space where composition corresponds to applying bounded creation operators to feature vectors. This…
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…
Highly localized kernels constructed by orthogonal polynomials have been fundamental in recent development of approximation and computational analysis on the unit sphere, unit ball and several other regular domains. In this work we first…
We obtain the octonionic Bergman kernel for half space in the octonionic analysis setting by two different methods. As a consequence, we unify the kernel forms in both complex analysis and hyper-complex analysis.
Let $\Gamma\subset \mathrm{SU}((2,1),\mathbb{C})$ be a torsion-free cocompact subgroup. Let $\mathbb{B}^{2}$ denote the $2$-dimensional complex ball endowed with the hyperbolic metric $\mu_{\mathrm{hyp}}$, and let…
We establish an operator--theoretic correspondence between periodic Bernoulli kernels and Hermite polynomials, framed through the umbral calculus and a quantum analogy. Starting from the analytic master function $F^\ast$, the periodic…
We consider transcendental entire functions having doubly parabolic Baker domains, such that the Denjoy-Wolff point of the associated inner function is not a singularity. We describe in a very precise way the dynamics on the boundary from a…