Weighted Bergman-Dirichlet and Bargmann-Dirichlet spaces of order $m$: Explicit formulae for reproducing kernels and asymptotic

TL;DR

This paper introduces generalized weighted Bergman-Dirichlet and Bargmann-Dirichlet spaces, providing explicit formulas for their reproducing kernels and analyzing their asymptotic behavior as the domain radius grows.

Contribution

It offers a comprehensive description of new functional spaces and explicit formulas for their reproducing kernels, extending classical spaces in complex analysis.

Findings

Explicit formulas for reproducing kernels derived

Asymptotic behavior of kernels analyzed as R approaches infinity

Generalization of classical spaces to new functional spaces

Abstract

We introduce new functional spaces that generalize the weighted Bergman and Dirichlet spaces on the disk D(0,R) in the complex plane and the Bargmann-Fock spaces on the whole complex plane. We give a complete description of the considered spaces. Mainly, we are interested in giving explicit formulas for their reproducing kernel functions and their asymptotic behavior as R goes to infinity.