A representation for the reproducing kernel of a weighted Bergman space

TL;DR

This paper provides a canonical representation of the reproducing kernel for weighted Bergman spaces with specific weight functions, enabling explicit formulas for zero divisors related to finite zero sets.

Contribution

It introduces a new representation of the reproducing kernel for weighted Bergman spaces with Blaschke product weights, connecting kernel values and derivatives at the origin.

Findings

Explicit formula for the reproducing kernel in weighted Bergman spaces.

Representation of the contractive zero divisor for finite zero sets.

Connection between kernel values and derivatives at the origin.

Abstract

For a weight function in the unit disk which is the modulus of a finite product of powers of Blaschke factors, we give a canonical representation for the reproducing kernel of the corresponding weighted Bergman space in terms of the values of the kernel and its derivatives at the origin. This yields a formula for the contractive zero divisor of a Bergman space corresponding to a finite zero set.