A convergence theorem for harmonic measures with applications to Taylor series

Gardiner, Stephen; Manolaki, Myrto

doi:10.1090/proc/12764

A convergence theorem for harmonic measures with applications to Taylor series
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by Stephen J. Gardiner and Myrto Manolaki PDF

Proc. Amer. Math. Soc. 144 (2016), 1109-1117 Request permission

Abstract:

Let $f$ be a holomorphic function on the unit disc, and let $(S_{n_{k}})$ 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.

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