Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc

TL;DR

This paper develops a reproducing kernel for weighted Bergman spaces on the symmetrized polydisc by embedding these spaces into the polydisc's Bergman spaces, enabling explicit kernel computation.

Contribution

It introduces an isometric embedding and derives an explicit kernel formula for weighted Bergman spaces on the symmetrized polydisc, including Szeg"{o} and Bergman kernels.

Findings

Explicit orthonormal basis for the subspace

Computed kernel functions for the symmetrized polydisc

Unified approach to Szeg"{o} and Bergman kernels

Abstract

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szeg\"{o} and the Bergman kernel on the symmetrized polydisc.