The article is based on experience of implementation of computer algebra system Bergman and on analysis of other symbolic computation systems. It is noted that many symbolic computation systems meet similar interface problems. Necessity of strict separation of calculation engine and interface shell, functions of these components, and requirements to them are motivated. The described approach wi...

Canonical divisors in Bergman spaces can be found as solutions of extremal problems. We derive a formula for certain extremal functions in the weighted Bergman spaces Aα for α > −1 and 1 ≤ p <∞. This leads to a study of the zeros of a specific family of hypergeometric functions.

In this paper we will discuss local coordinates canonically corresponding to a Kähler metric. We will also discuss the C ∞ convergence of Bergman metrics following Tian's result on C 2 convergence of Bergman metrics. At the end we present an interesting characterization of ample line bundle that could be useful in arithmetic geometry.

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 .

For an arbitrary unimodular Lie group G, we construct strongly continuous unitary representations in the Bergman space of a strongly pseudoconvex neighborhood of G in the complexification of its underlying manifold. In particular, the Bergman spaces of these manifolds are infinite-dimensional.

We obtain a conceptually new differential geometric proof of P.F. Klembeck’s result (cf. [9]) that the holomorphic sectional curvature kg(z) of the Bergman metric of a strictly pseudoconvex domain Ω ⊂ C approaches −4/(n + 1) (the constant sectional curvature of the Bergman metric of the unit ball) as z → ∂Ω.

We treat the complex harmonic function on the Np–ball which is defined by the Np–norm related to the Lie norm. As a subspace, we treat Hardy spaces and consider the Bergman kernel on those spaces. Then, we try to construct the Bergman kernel in a concrete form in 2–dimensional Euclidean space. Introduction. In [2], [4], [6] and [7], we studied holomorphic functions and analytic functionals on t...

Let D be a bounded domain in C". The invariant distance in D is given by I I v I \ is l \\ '/2\ l/2 Kp(z,i)K„(w,w) It is shown that one half of the length of a piecewise C1 curve y: [a, b] -* D with respect to the Bergman metric is equal to the length of y measured by pD, which implies that the associated inner distance p*D coincides (up to the factor {) with the Bergman-distance. Also it was p...