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The capacity of sets of divergence of Certain Taylor Series on the unit circle
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Date
2019-05-09
Authors
Twomey, J. Brian
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Springer
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Abstract
A simple and direct proof is given of a generalization of a classical result on the convergence of ∑∞k=0ak e ikx outside sets of x of an appropriate capacity zero, where f(z)=∑∞k=0akzk is analytic in the unit disc U and ∑∞k=0kα|ak|2<∞ with 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 ∑∞k=1kγ|ak|p<∞, and for trigonometric series of the form ∑∞k=1(αkcoskx+βksinkx) with ∑∞k=1kγ(|αk|p+|βk|p)<∞ , where γ>0,p>1.
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Keywords
Convergence of Taylor series , Trigonometric series , Capacity , Exceptional sets , Dirichlet-type spaces , Analytic Besov spaces
Citation
Twomey, J. B. (2019) 'The Capacity of Sets of Divergence of Certain Taylor Series on the Unit Circle', Computational Methods and Function Theory, 19(2), pp. 227-236. doi: 10.1007/s40315-019-00266-z
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© Springer-Verlag GmbH Germany, part of Springer Nature 2019. This is a post-peer-review, pre-copyedit version of an article published in Computational Methods and Function Theory. The final authenticated version is available online at: http://dx.doi.org/10.1007/s40315-019-00266-z