R.E. Bruck, proof mining and a rate of asymptotic regularity for ergodic averages in Banach spaces

TL;DR

This paper provides an explicit rate of asymptotic regularity for Cesàro means in uniformly convex Banach spaces, using proof mining techniques to connect geometric properties with convergence rates.

Contribution

It explicitly determines the rate of asymptotic regularity based on the modulus of convexity and Rademacher type, advancing the proof mining approach in functional analysis.

Findings

Explicit rate depends on norm bound and modulus of convexity

Determined Rademacher type and constant from the modulus

Connects geometric properties to convergence in Banach spaces

Abstract

We analyze a proof of Bruck to obtain an explicit rate of asymptotic regularity for Ces\`aro means in uniformly convex Banach spaces. Our rate will only depend on a norm bound and a modulus $\eta$ of uniform convexity. One ingredient for the proof by Bruck is a result of Pisier, which shows that every uniformly convex (in fact every uniformly nonsquare) Banach space has some Rademacher type $q>1$ with a suitable constant $C_q$. We explicitly determine $q$ and $C_q$, which only depend on the single value $\eta(1)$ of our modulus. Beyond these specific results, we summarize how work of Bruck has inspired developments in the proof mining program, which applies tools from logic to obtain results in various areas of mathematics.