Almost everywhere convergence of weighted averages
type: Item
instance of: scholarly article
title: Almost everywhere convergence of weighted averages
zbMATH Open document ID: 0736.28008
DOI: 10.1007/BF01444724
author: Roger Jones
author: Alexandra Ionescu Tulcea
author: J. Rosenblatt
published in: Mathematische Annalen
publication date: 1992-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164965
review text: Given a sequence \((\mu_ n)\) of probability measures on \(Z\), and an invertible measure-preserving transformation \(\tau\) of a probability space \((X,\beta,m)\), the averages \(\mu_ nf(x)=\sum^{\infty}_{k=- \infty}\mu_ n(k)f(\tau^ kx)\) are bounded operators on \(L^ p(m)\), \(1\leq p\leq\infty\). Sufficient conditions are given in terms of the Fourier transforms \((\hat\mu_ n(\gamma))\) on \(T=\{\gamma:|\gamma|=1\}\) for the a.e. convergence of \(\mu_ nf(x)\) for all \(f\in L^ p(m)\), \(1<p\leq\infty\). In particular, if \(\lim_{n\to\infty}\sum^{\infty}_{k=-\infty}|\mu_ n(k)-\mu_ n(k-1)|=0\), then there exists a subsequence \((\mu_{n_ m})\) such that for all systems \((X,\beta,m,\tau)\) and \(f\in L^ p(m), 1<p\leq\infty\), \(\lim_{n\to\infty}\mu_{n_ m}f(x)\) exists a.e. If \(\mu_ n=\mu^ n\) for some fixed strictly aperiodic finitely supported measure \(\mu\), then \((n_ m)\) can be any subsequence with \(n_{m+1}>n^{\beta}_ m\) for some \(\beta>1\). Related theorems, examples, and generalizations to group actions are also described.
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 4215991
zbMATH Keywords: probability measures
zbMATH Keywords: invertible measure-preserving transformation
zbMATH Keywords: probability space
zbMATH Keywords: averages
zbMATH Keywords: bounded operators
zbMATH Keywords: Fourier transforms
zbMATH Keywords: group actions
MaRDI profile type: Publication
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OpenAlex ID: W2040580319
description: scientific article; zbMATH DE number 4215991