On weights which admit the reproducing kernel of Bergman type

Zbigniew Pasternak-Winiarski

doi:10.1155/s0161171292000012

Zbigniew Pasternak-Winiarski

Open Access

https://doi.org/10.1155/s0161171292000012

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Abstract

In this paper we consider (1) the weights of integration for which the reproducing kernel of the Bergman type can be defined, i.e., the admissible weights, and (2) the kernels defined by such weights. It is verified that the weighted Bergman kernel has the analogous properties as the classical one. We prove several sufficient conditions and necessary and sufficient conditions for a weight to be an admissible weight. We give also an example of a weight which is not of this class. As a positive example we consider the weight μ(z) = (Imz)2 defined on the unit disk in ℂ.

Highlights

In this paper we are concerned with the weights of integration, the so called admissible weights or a-weights for short for which the Bergman reproducing kernel can be defined
Reproducing kernels of Bergman type depending on weights of integration have been used to define the quantization of classical states in holomorphic models of quantum field theory (Odzijewicz [1])
Earlier they have appeared in studies of wave and Dirac equations (Jakobsen and Vergne [2]). Functions of this type have been considered in consequence of many mathematical problems (Burbea and Masani [3] and Mazur [4])

Summary

INTRODUCTION

In this paper we are concerned with the weights of integration, the so called admissible weights or a-weights for short for which the Bergman reproducing kernel can be defined. Reproducing kernels of Bergman type depending on weights of integration have been used to define the quantization of classical states in holomorphic models of quantum field theory (Odzijewicz [1]). Earlier they have appeared in studies of wave and Dirac equations (Jakobsen and Vergne [2]). We find a complete orthonormal system in the space L2H(D,p) of all holomorphic p-square integrable functions on D It makes possible the uniform approximation of the Bergman kernel Ka by polynomials on any compact subset of D x D, see Theorem 2.1, (i). Since the evaluation functionals axe all continuous we obtain ha for every w D

On the other hand for any k we have

REPRODUCING KERNEL OF BERGMAN TYPE

Similarly and