Ergodic theorems and approximation theorems with rates
type: Item
instance of: scholarly article
title: Ergodic theorems and approximation theorems with rates
zbMATH Open document ID: 0971.47005
author: Sen-Yen Shaw
published in: Taiwanese Journal of Mathematics
publication date: 2001-10-18T00:00:00Z
review text: The paper devoted to the study of some strong ergodic theorems, uniform ergodic theorems and ergodic theorems with rates. It is diveded into five sections. The first section is an introduction. In the second section it is given a review of some results in connections with \(A\)-ergodic nets, which were firstly introduced in [\textit{S.-Y. Shaw}, J. Funct. Anal. 87, No. 2, 428-441 (1989; Zbl 0704.47006)]. In this section for \(A\)-ergodic nets strong ergodic theorems, uniform ergodic theorems and ergodic theorems with rates are formulated. In the third section it is defined an approximation process and \(A\)-regularized approximation process on a Banach space \(X\). For these processes optimal convergence, non-optimal convergence and sharpness theorems are proved. In the forth section it is introduced, more exectly generalized \(r\)-times integrated solution family for the Volterra equation: \[ u(t)=f(t)+\int^{t}_{0}a(t-s)Au(s) ds, \quad t\geq 0, \] for \(f\in C([0;\infty);X)\), where \(A\) is a closed linear operator on \(X\) and \(a\in L^{1}_{\text{loc}}(\mathbb{R}^+)\) is a positive kernel. The notion of an \(r\)-times integrated solution family is an extention of the concept of \(n\)-times integrated semigroups and \(n\)-times integrated cosine functions. As applications of the general results to be formulated in sections 2 and 3, for \(r\)-times integrated solution family it is proved uniform ergodic theorem, strong ergodic theorem with rates. In section 5 for \(r\)-times integrated solution family local ergodic theorems with rates are proved. There are also established approximationsproperties of theorems given in section 4.
Mathematics Subject Classification ID: 47A35
Mathematics Subject Classification ID: 47A58
Mathematics Subject Classification ID: 47D62
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47D09
zbMATH DE Number: 1547610
zbMATH Keywords: \(A\)-ergodic net
zbMATH Keywords: \(A\)-regularized approximation process
zbMATH Keywords: Dunford-Pettis property
zbMATH Keywords: \(r\)-times integrated solution family
zbMATH Keywords: Grothendieck space
zbMATH Keywords: strong ergodic theorems
zbMATH Keywords: uniform ergodic theorems
zbMATH Keywords: ergodic theorems with rates
zbMATH Keywords: optimal convergence
zbMATH Keywords: sharpness theorems
zbMATH Keywords: Volterra equation
reviewed by: Farruh Mukhamedov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500407254
OpenAlex ID: W4235221436
DOI: 10.11650/TWJM/1500407254
description: scientific article; zbMATH DE number 1547610