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The Capacity of Sets of Divergence of Certain Taylor Series on the Unit Circle

The Capacity of Sets of Divergence of Certain Taylor Series on the Unit Circle A simple and direct proof is given of a generalization of a classical result on the convergence of $$\sum _{k=0}^\infty a_k \text{ e }^{i k x}$$ ∑ k = 0 ∞ a k e ikx outside sets of x of an appropriate capacity zero, where $$f(z) = \sum _{k=0}^\infty a_k z^k$$ f ( z ) = ∑ k = 0 ∞ a k z k is analytic in the unit disc U and $$\sum _{k=0}^\infty k^\alpha |a_k|^2 < \infty $$ ∑ k = 0 ∞ k α | a k | 2 < ∞ with $$\alpha \in (0,1].$$ α ∈ ( 0 , 1 ] . We also discuss some convergence consequences of our results for weighted Besov spaces, for the classes of analytic functions in U for which $$ \sum _{k=1}^\infty k^\gamma |a_k|^p < \infty ,$$ ∑ k = 1 ∞ k γ | a k | p < ∞ , and for trigonometric series of the form $$\sum _{k=1}^\infty (\alpha _k \cos kx + \beta _k \sin kx)$$ ∑ k = 1 ∞ ( α k cos k x + β k sin k x ) with $$\sum _{k=1}^\infty k^\gamma (|\alpha _k|^p +|\beta _k|^p) < \infty $$ ∑ k = 1 ∞ k γ ( | α k | p + | β k | p ) < ∞ , where $$ \gamma>0, \, p>1.$$ γ > 0 , p > 1 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

The Capacity of Sets of Divergence of Certain Taylor Series on the Unit Circle

Computational Methods and Function Theory , Volume 19 (2) – May 9, 2019

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-019-00266-z
Publisher site
See Article on Publisher Site

Abstract

A simple and direct proof is given of a generalization of a classical result on the convergence of $$\sum _{k=0}^\infty a_k \text{ e }^{i k x}$$ ∑ k = 0 ∞ a k e ikx outside sets of x of an appropriate capacity zero, where $$f(z) = \sum _{k=0}^\infty a_k z^k$$ f ( z ) = ∑ k = 0 ∞ a k z k is analytic in the unit disc U and $$\sum _{k=0}^\infty k^\alpha |a_k|^2 < \infty $$ ∑ k = 0 ∞ k α | a k | 2 < ∞ with $$\alpha \in (0,1].$$ α ∈ ( 0 , 1 ] . We also discuss some convergence consequences of our results for weighted Besov spaces, for the classes of analytic functions in U for which $$ \sum _{k=1}^\infty k^\gamma |a_k|^p < \infty ,$$ ∑ k = 1 ∞ k γ | a k | p < ∞ , and for trigonometric series of the form $$\sum _{k=1}^\infty (\alpha _k \cos kx + \beta _k \sin kx)$$ ∑ k = 1 ∞ ( α k cos k x + β k sin k x ) with $$\sum _{k=1}^\infty k^\gamma (|\alpha _k|^p +|\beta _k|^p) < \infty $$ ∑ k = 1 ∞ k γ ( | α k | p + | β k | p ) < ∞ , where $$ \gamma>0, \, p>1.$$ γ > 0 , p > 1 .

Journal

Computational Methods and Function TheorySpringer Journals

Published: May 9, 2019

References