A convergence theorem for harmonic measures with applications to Taylor series

TL;DR

This paper proves a convergence theorem for harmonic measures and applies it to show that subsequences of Taylor polynomials of holomorphic functions converge nontangentially to the function at boundary points, providing new insights into boundary behavior.

Contribution

It introduces a new convergence theorem for harmonic measures and applies it to analyze boundary limits of Taylor polynomial subsequences of holomorphic functions.

Findings

Subsequences of Taylor polynomials converge nontangentially to the function at boundary points.

The convergence theorem for harmonic measures is of independent interest.

Results give new understanding of boundary behavior of universal Taylor series.

Abstract

Let $f$ be a holomorphic function on the unit disc, and $(S_{n_{k}})$ be a subsequence of its Taylor polynomials about $0$. It is shown that the nontangential limit of $f$ and lim$_{k\rightarrow \infty }S_{n_{k}}$ agree at almost all points of the unit circle where they simultaneously exist. This result yields new information about the boundary behaviour of universal Taylor series. The key to its proof lies in a convergence theorem for harmonic measures that is of independent interest.