On N\"orlund summability of Taylor series in weighted Dirichlet spaces

Abstract

In this note we show that the Taylor series of a function in a weighted Dirichlet space is (generalized) N\"orlund summable, provided that the sequence determining the N\"orlund operator is non-decreasing and has finite upper growth rate. In particular the Taylor series is N\"orlund summable for all $\alpha>1/2$, and the rate of convergence is of the order $O(n^{-1/2})$. The inequality $\alpha>1/2$ is sharp. On the other hand if the Taylor series is N\"orlund summable and the partial sums of the determining sequence enjoy a certain growth condition then the determining sequence has finite lower growth rate. An analogue result is derived for a non-increasing sequence that is uniformly bounded away from zero.