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Spurious limit functions of Taylor series

Spurious limit functions of Taylor series It is known that, generically in the space $$H({\mathbb {D}})$$ H ( D ) of functions holomorphic in the unit disc $${\mathbb {D}}$$ D , the sequences $$(S_n f)$$ ( S n f ) of partial sums of Taylor series behave extremely erratically on the unit circle $${\mathbb {T}}$$ T . According to a result of Gardiner and Manolaki, the situation changes in a significant way if $$f \in H({\mathbb {D}})$$ f ∈ H ( D ) has nontangential limits on subsets of $${\mathbb {T}}$$ T of positive arc length measure. In this case each convergent subsequence tends to the nontangential limit function almost everywhere. We consider the question to which extent in spaces of holomorphic functions where nontangential limits are guaranteed, “spurious” limit functions, that is, limit functions different than the nontangential limit may appear on small subsets of $${\mathbb {T}}$$ T . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Spurious limit functions of Taylor series

Analysis and Mathematical Physics , Volume 9 (2) – May 23, 2019

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-019-00322-w
Publisher site
See Article on Publisher Site

Abstract

It is known that, generically in the space $$H({\mathbb {D}})$$ H ( D ) of functions holomorphic in the unit disc $${\mathbb {D}}$$ D , the sequences $$(S_n f)$$ ( S n f ) of partial sums of Taylor series behave extremely erratically on the unit circle $${\mathbb {T}}$$ T . According to a result of Gardiner and Manolaki, the situation changes in a significant way if $$f \in H({\mathbb {D}})$$ f ∈ H ( D ) has nontangential limits on subsets of $${\mathbb {T}}$$ T of positive arc length measure. In this case each convergent subsequence tends to the nontangential limit function almost everywhere. We consider the question to which extent in spaces of holomorphic functions where nontangential limits are guaranteed, “spurious” limit functions, that is, limit functions different than the nontangential limit may appear on small subsets of $${\mathbb {T}}$$ T .

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: May 23, 2019

References