LAUSR

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∞akeikx outside sets of x of an appropriate capacity zero, where $$f(z) = \sum _{k=0}^\infty a_k z^k$$f(z)=∑k=0∞akzk is analytic in the unit disc U and $$\sum _{k=0}^\infty k^\alpha |a_k|^2 < \infty $$∑k=0∞kα|ak|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∞kγ|ak|p<∞, and for trigonometric series of the form $$\sum _{k=1}^\infty (\alpha _k \cos kx + \beta _k \sin kx)$$∑k=1∞(αkcoskx+βksinkx) with $$\sum _{k=1}^\infty k^\gamma (|\alpha _k|^p +|\beta _k|^p) < \infty $$∑k=1∞kγ(|αk|p+|βk|p)<∞, where $$ \gamma>0, \, p>1.$$γ>0,p>1.