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

Abstract

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