Summability

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 α > 1 / 2, and the rate of convergence is of the order O ( n − 1 / 2 ). The inequality α > 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.