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

2022/04/08 by Anton Freund, Ulrich Kohlenbach, Freund, Anton +1
Mathematics · #03F10 #47H10 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2204.04100

openalex publication_date 2022/04/08 · openalex created_date 2022/04/15 · openalex updated_date 2026/07/28

Abstract

We analyze a proof of Bruck to obtain an explicit rate of asymptotic regularity for Cesàro means in uniformly convex Banach spaces. Our rate will only depend on a norm bound and a modulus η 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 η(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.

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