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Tauberian theorems for sequences and the Katznelson--Tzafriri theorem

Andrew K. J. Pritchard, David Seifert

TL;DR

The paper provides a streamlined, quantitative treatment of Tauberian theorems for Banach-space valued sequences and their link to the Katznelson--Tzafriri phenomenon. By replacing a fixed cutoff with a flexible family of differentiable functions, it derives improved decay rates and extends the framework from a single boundary singularity to multiple singularities on the unit circle. It yields explicit decay bounds for sequence terms via boundary-data regularity, and translates these into quantified KT-type results for power-bounded and Ritt/E-Ritt operators. The multi-singularity extension further yields generalized KT bounds and decay results for $E$-Ritt operators, broadening applications in discrete ergodic theory and operator theory on Banach spaces.

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.

Tauberian theorems for sequences and the Katznelson--Tzafriri theorem

TL;DR

The paper provides a streamlined, quantitative treatment of Tauberian theorems for Banach-space valued sequences and their link to the Katznelson--Tzafriri phenomenon. By replacing a fixed cutoff with a flexible family of differentiable functions, it derives improved decay rates and extends the framework from a single boundary singularity to multiple singularities on the unit circle. It yields explicit decay bounds for sequence terms via boundary-data regularity, and translates these into quantified KT-type results for power-bounded and Ritt/E-Ritt operators. The multi-singularity extension further yields generalized KT bounds and decay results for -Ritt operators, broadening applications in discrete ergodic theory and operator theory on Banach spaces.

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.

Paper Structure

This paper contains 3 sections, 10 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Tauberian Theorems with Single Boundary Singularity
  3. Tauberian Theorems with Several Boundary Singularities

Key Result

Theorem 2.2

Let $X$ be a complex Banach space and suppose that $x \in \ell^\infty(\mathbb{Z}_+;X)$ satisfies If there exists a boundary function $F_x \in L^1_{\mathrm{loc}}(\mathbb{T} \setminus \{1\}; X)$ for $G_x$, then $x \in c_0(\mathbb{Z}_+;X)$. Furthermore, if $m\colon (0,\pi] \to (0,\infty)$ is a continuous, non-increasing function, then the following hold:

Theorems & Definitions (17)

  • Definition 2.1
  • Theorem 2.2: Sei15_Tauberian
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • ...and 7 more