The asymptotic behaviour of the Cesàro operator
Andrew K. J. Pritchard, David Seifert
TL;DR
The paper analyzes the Cesàro operator $T$ on the convergent sequence space $c$ and, by leveraging the Katznelson--Tzafriri theorem and Laguerre polynomial techniques, characterizes when the orbits $(T^n x)$ converge and at what rate. It proves that norm convergence is equivalent to the simple boundary condition $x_0=\lim_{k\to\infty} x_k$ and identifies the limit as $P x$, the projection onto constant sequences, with a precise norm decay $\|T^n(I-T)\|\to0$. A quantitative rate of convergence $O(n^{-1/2})$ is obtained under a concrete series condition on $x$, shown to be optimal in general and achievable more rapidly on dense subspaces; the approach extends to Cesàro operators on spaces of continuous functions, yielding parallel convergence and rate results for $C[0,1]$ and $C_\infty[0,\infty)$. Together, the results provide a unified, functional-analytic view of Cesàro orbit asymptotics beyond the discrete setting.
Abstract
We study the asymptotic behaviour of orbits $(T^nx)_{n\ge0}$ of the classical Cesàro 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 Cesàro operator defined on spaces of continuous functions, establishing new and improved results in this setting, too.