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 ∑∞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.