The asymptotic behaviour of the Ces\`aro operator

TL;DR

This paper investigates the long-term behavior of the Cesàro operator on convergent sequences and continuous functions, providing new proofs, optimality results, and rates of convergence using operator theory techniques.

Contribution

It introduces new non-probabilistic proofs and optimality results for the asymptotic behavior of the Cesàro operator on sequences and function spaces.

Findings

Characterization of sequences with convergent Cesàro orbits

Provision of convergence rates for certain sequences

Extension of results to spaces of continuous functions

Abstract

We study the asymptotic behaviour of orbits $(T^nx)_{n\ge0}$ of the classical Ces\`aro operator $T$ for sequences $x$ in the Banach space $c$ of convergent sequences. We give new non-probabilistic proofs, based on the Katznelson-Tzafriri theorem and one of its quantified variants, of results which characterise the set of sequences $x\in c$ that lead to convergent orbits and, for sequences satisfying a simple additional condition, we provide a rate of convergence. These results are then shown, again by operator-theoretic techniques, to be optimal in different ways. Finally, we study the asymptotic behaviour of the Ces\`aro operator defined on spaces of continuous functions, establishing new and improved results in this setting, too.