International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 1, Pages 1-14

On weights which admit the reproducing kernel of Bergman type

Zbigniew Pasternak-Winiarski

Institute of Mathematics, Technical University of Warsaw, Pl. Jedności Robotniczej 1, Warsaw 00-661, Poland

Received 7 February 1990

Copyright © 1992 Zbigniew Pasternak-Winiarski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 .