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A convergence theorem for harmonic measures with applications to Taylor series


Authors: Stephen J. Gardiner and Myrto Manolaki
Journal: Proc. Amer. Math. Soc. 144 (2016), 1109-1117
MSC (2010): Primary 30B30, 30C85, 30K05, 31A15, 31B20
DOI: https://doi.org/10.1090/proc/12764
Published electronically: September 11, 2015
MathSciNet review: 3447664
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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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Additional Information

Stephen J. Gardiner
Affiliation: School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
Email: stephen.gardiner@ucd.ie

Myrto Manolaki
Affiliation: Department of Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada
Email: arhimidis8@yahoo.gr

DOI: https://doi.org/10.1090/proc/12764
Received by editor(s): August 25, 2014
Received by editor(s) in revised form: January 26, 2015, and February 12, 2015
Published electronically: September 11, 2015
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2015 American Mathematical Society