A quantified Tauberian theorem for sequences

TL;DR

This paper presents a quantified version of Ingham's Tauberian theorem for bounded vector sequences, providing decay rate estimates based on boundary function smoothness, and offers a new proof of the Katznelson-Tzafriri theorem.

Contribution

It extends Tauberian theorems to vector sequences with explicit decay estimates and introduces a novel proof of the quantified Katznelson-Tzafriri theorem.

Findings

Provides decay rate estimates for vector sequences based on boundary function smoothness

Develops a new proof technique for the quantified Katznelson-Tzafriri theorem

Enhances understanding of the relationship between boundary behavior and sequence decay

Abstract

The main result of this paper is a quantified version of Ingham's Tauberian theorem for bounded vector-valued sequences rather than functions. It gives an estimate on the rate of decay of such a sequence in terms of the behaviour of a certain boundary function, with the quality of the estimate depending on the degree of smoothness this boundary function is assumed to possess. The result is then used to give a new proof of the quantified Katznelson-Tzafriri theorem recently obtained in [21].