# Convergence speed of weighted Bergman kernels towards extremal functions

## TL;DR

This paper studies how weighted Bergman kernels, constructed via Bernstein-Markov inequalities on high tensor powers of line bundles, uniformly converge to extremal functions, providing explicit convergence speed estimates.

## Contribution

It introduces a method to construct inner products leading to weighted Bergman kernels that converge uniformly to extremal functions with quantifiable speed.

## Key findings

- Established uniform convergence of weighted Bergman kernels to extremal functions.
- Derived explicit bounds on the convergence speed.
- Applied Bernstein-Markov inequalities to high tensor powers of line bundles.

## Abstract

We construct inner products by the Bernstein-Markov inequality on spaces of holomorphic sections of high powers of a line bundle. The corresponding weighted Bergman kernel functions converge to an extremal function. We obtain a uniform convergence speed.