A class of sequences with a strong average property
type: Item
instance of: scholarly article
title: A class of sequences with a strong average property
zbMATH Open document ID: 0585.60016
DOI: 10.1016/0001-8708(85)90090-8
author: V. Losert
published in: Advances in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: A sequence \((m_ k)\) of probability measures on \({\mathbb{N}}\) is called ergodic if lim \(<f,m_ k>=<f,\lambda >\) holds for each continuous f on the Bohr compactification b\({\mathbb{Z}}\) of \({\mathbb{Z}}\). A bounded sequence (g(n)) is called p-sequence, if lim \(<g,m_ k>=p\) for each ergodic sequence \((m_ k)\) on \({\mathbb{N}}\). It is shown that (g(n)) is a p-sequence for some \(p\in {\mathbb{R}}\) if and only if there exist, for each \(\epsilon >0\), measures \(\mu_ 1\), \(\mu_ 2\) on the torus \(T={\mathbb{R}}/{\mathbb{Z}}\) with \[ {\hat \mu}_ 1\leq g\leq {\hat \mu}_ 2\quad on\quad {\mathbb{N}}\quad and\quad \lim N^{-1}\sum^{N}_{1}({\hat \mu}_ 2(n)-{\hat \mu}_ 1(n))<\epsilon. \] (\({\hat \mu}\) denotes the Fourier transform).
reviewed by: Ulrich Krengel
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 43A55
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 3938131
zbMATH Keywords: Bohr compactification
zbMATH Keywords: ergodic sequence
zbMATH Keywords: p-sequence
MaRDI profile type: Publication
full work available at URL: 0001-8708(85)90090-8
OpenAlex ID: W2057982473
cites work: The Weighted Pointwise Ergodic Theorem and the Individual Ergodic Theorem Along Subsequences
cites work: Uniformity in weak convergence
cites work: On Ergodic Sequences of Measures
cites work: The individual ergodic theorem for p-sequences
cites work: p-Sets for random walks
cites work: Ergodic theorems for operator sequences
cites work: Q5634392
cites work: Q3881310
cites work: The method of Grothendieck-Ramirez and weak topologies in C(T)
cites work: Q4767350
cites work: On the existence of uniformly distributed sequences in compact topological spaces. II
cites work: Uniform Distribution on Locally Compact Groups
cites work: Q3293614
cites work: Q4087471
description: scientific article; zbMATH DE number 3938131