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

TL;DR

This paper proves that Taylor series in weighted Dirichlet spaces are Nörlund summable under certain conditions on the summation sequence, with sharp bounds on the parameter and convergence rate.

Contribution

It establishes new Nörlund summability results for Taylor series in weighted Dirichlet spaces, including sharp conditions on the summation sequence.

Findings

Taylor series are Nörlund summable for all α > 1/2

Convergence rate is O(n^{-1/2})

Conditions on the summation sequence's growth are necessary and sufficient

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.