Tauberian theorems for sequences and the Katznelson--Tzafriri theorem

TL;DR

This paper offers an alternative, slightly strengthened proof of a Tauberian theorem for vector-valued sequences, connecting decay rates with boundary function properties, and applies it to Katznelson--Tzafriri and Ritt operator results.

Contribution

It provides a new proof and an improved version of a Tauberian theorem, enabling more precise quantifications in operator theory.

Findings

Enhanced proof of a vector-valued Tauberian theorem

Quantified versions of Katznelson--Tzafriri theorem derived

Applications to Ritt operators demonstrated

Abstract

In this note, we present an alternative proof of a quantified Tauberian theorem for vector-valued sequences first proved in \cite{Sei15_Tauberian}. The theorem relates the decay rate of a bounded sequence with properties of a certain boundary function. We present a slightly strengthened version of this result, and illustrate how it can be used to obtain quantified versions of the Katznelson--Tzafriri theorem as well as results on Ritt operators.