Related papers: The Bergman kernel: explicit formulas, deflation, …
Herein, the theory of Bergman kernel is developed to the weighted case. A general form of weighted Bergman reproducing kernel is obtained, by which we can calculate concrete Bergman kernel functions for specific weights and domains.
We prove an explicit formula for the Bergman kernel of polarized abelian varieties. As applications, we show that if two positive line bundles represent the same first Chern class and have identical Bergman kernel functions for some tensor…
We prove a number of quadratic transformations of elliptic Selberg integrals (conjectured in an earlier paper of the author), as well as studying in depth the "interpolation kernel", an analytic continuation of the author's elliptic…
We present an exact, closed-form expression for the Newtonian potential of the characteristic function associated with two overlapping discs in the plane. This setting naturally arises when discretising nonlocal interaction terms present in…
We consider the weighted Bergman space $A^2_\psi(\Bn)$ of all holomorphic functions on $\Bn$ square integrable with respect to a particular exponential weight measure $e^{-{\psi}} dV$ on $\Bn$, where \begin{align*}…
In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on $\ell^2$--valued Bergman-type spaces. This paper generalizes many well--known results about classical function spaces to their…
We compute explicitly the Bergman kernels of all two dimensional monomial polyhedra, a class of domains including the Hartogs triangle and some of its generalizations. The kernel is computed from the representation of such domains as…
We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any…
We calculate the exact value and find the polynomial of the best weighted polynomial approximation of kernels of the form $\frac {A+Bt}{(t^2+\lambda^2)^{s+1}}$, where $A$ and $B$ are fixed complex numbers, $\lambda>0$, $s\in {\mathbb N}$,…
We consider polynomial Bergman kernels with respect to exponentially varying weights $e^{-n \mathscr Q(z)}$ depending on a potential $\mathscr Q:\mathbb C^d\to\mathbb R$. We use these kernels to construct determinantal point processes on…
In this paper we study two multicritical correlation kernels and prove that they converge to the Pearcey kernel in a certain double scaling limit. The first kernel appears in a model of non-intersecting Brownian motions at a tacnode. The…
We obtain ceratin estimates for the reproducing kernels of large weighted Bergman spaces. Applications of these estimates to boundedness of the Bergman projection on $L^p(\D,\omega ^{p/2})$, complex interpolation and duality of weighted…
We present a technique for computing explicit, concrete formulas for the weighted Bergman kernel on a planar domain with weight the modulus squared of a meromorphic function in the case that the meromorphic function has a finite number of…
In this paper, we are concerned with Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$ on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of…
The goal of this note is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in…
Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their…
We~identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in~the sense of Shimura, on~bounded symmetric domains. This also yields a description of the analogous kernels for spaces of…
We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer…
\begin{abstract} We obtain sharp $L^p\rightarrow L^q$ hypercontractive inequalities for the weighted Bergman spaces on the unit disk $\mathbb{D}$ with the usual weights \\ $\frac{\alpha-1}{\pi}(1-|z|^2)^{\alpha-2},\alpha>1$ for $q\geq 2,$…
For appropriate domains $\Omega_{1}, \Omega_{2}$ we consider mappings $\Phi_{\mathbf A}:\Omega_{1}\to\Omega_{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $\mathcal A^{2}(\Omega_{1})$ into finitely many…