Ces\`aro summability of Taylor series in weighted Dirichlet spaces

TL;DR

This paper proves that Taylor series in weighted Dirichlet spaces with superharmonic weights are summable to the original function under Cesàro means for orders greater than 1/2, with this threshold being optimal.

Contribution

It establishes the Cesàro summability of Taylor series in weighted Dirichlet spaces with superharmonic weights for orders above 1/2, highlighting a sharp contrast with classical results.

Findings

Cesàro summability holds for lpha > 1/2

The threshold lpha = 1/2 is sharp

Contrast with classical disk algebra case

Abstract

We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,\alpha)$-summable to the function in the norm of the space, provided that $\alpha>1/2$. We further show that the constant $1/2$ is sharp, in marked contrast with the classical case of the disk algebra.