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 known as the Schwarz function. Simplified proofs of several other well known facts about quadrature domains will fall out along the way. Finally, Bergman re...

In this note, we point out that a large family of n × n matrix valued kernel functions defined on the unit disc D ⊆ C, which were constructed recently in [9], behave like the familiar Bergman kernel function on D in several different ways. We show that a number of questions involving the multiplication operator on the corresponding Hilbert space of holomorphic functions on D can be answered usi...

We prove that the Bergman kernel function associated to a finitely connected domain Ω 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 function whose existence is deduced by means of the field of meromorphic functions on the double of Ω. Because many other functions of conformal mapping and poten...

We compute the explicit formula of the Bergman kernel for a nonhomogeneous domain {(z1, z2) ∈ C2 : |z1|1 + |z2|2 < 1} for any positive integers q1 and q2. We also prove that among the domains Dp := {(z1, z2) ∈ C2 : |z1|1 + |z2|2 < 1} in C2 with p = (p1, p2) ∈ N2, the Bergman kernel is represented in terms of closed forms if and only if p = (p1, 1), (1, p2), or p = (2, 2).

Throughout this paper by using the frame theory we give a short proof for atomic decomposition for weighted Bergman space. In fact we show that the weighted Bergman space L 2 a (dA α) admit an atomic decomposition i.e every analytic function in this space can be presented as a linear combination of " atoms " defined using the normalized reproducing kernel of this space .

We study a positive reproducing kernel for holomorphic functions on complex domains. This kernel, which induces what has now come to be known as the Berezin transform, is manufactured from the Bergman kernel using an idea of L. K. Hua. The kernel has important analytic and geometric properties which we develop in some detail.

We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line bundles.