A note on weighted poly-Bergman spaces

TL;DR

This paper introduces weighted poly-Bergman spaces in the unit disc, providing explicit orthonormal bases and reproducing kernels, utilizing disc polynomials as key tools for their characterization.

Contribution

It presents the first explicit construction of weighted poly-Bergman spaces and their bases, extending the classical theory with concrete formulas and kernel expressions.

Findings

Explicit orthonormal bases for weighted poly-Bergman spaces

Closed-form expressions for reproducing kernels

Use of disc polynomials as a fundamental tool

Abstract

We introduce \`a la Vasilevski the weighted poly-Bergman spaces in the unit disc and provide concrete orthonormal basis and give close expression of their reproducing kernel. The main tool in the description if these spaces is the so-called disc polynomials that form an orthogonal basis of the whole weighted Hilbert space.