Hardy and Paley inequalities for fully-odd Vilenkin systems
type: Item
instance of: scholarly article
title: Hardy and Paley inequalities for fully-odd Vilenkin systems
zbMATH Open document ID: 0928.42010
DOI: 10.1023/A:1006534802172
published in: Acta Mathematica Hungarica
publication date: 1999-10-31T00:00:00Z
review text: Choose a nonatomic probability space \(\Omega\). Let \((\psi_n)^\infty_{n=-\infty}\) for be a Vilenkin system on \(\Omega\). Denote the probability measure on \(\Omega\) by \(d\omega\). Given a function \(f\) in \(L^1(d\omega)\) and an integer \(n\), let \[ \widehat f (n)=\int_\Omega f(\omega)\overline{\psi_n(\omega)}d\omega. \] We also regard \(\widehat f\) as a function on the set \(\Gamma=\{\psi_n\}^\infty_{n=-\infty}\) and use the notation \(\widehat f(\psi_n)\) rather than \(\widehat f(n)\). One of the main results is: Theorem 1. Let \(f\in L^1(d\omega)\) and \(\widehat f(\psi_n)=0\) for all \(n<0\). Then \[ \sum_{n>0}\frac{| \widehat f(\psi_n)| }{n}<\infty. \] Moreover, if \((\lambda_j)\) is a Paley sequence, then \[ \sum^\infty_{j=1} | \widehat f(\psi_{\lambda_j})| ^2<\infty. \]
Mathematics Subject Classification ID: 42C10
zbMATH DE Number: 1354962
zbMATH Keywords: Vilenkin systems
zbMATH Keywords: Hardy inequality
zbMATH Keywords: Paley inequality
zbMATH Keywords: Paley sequence
author: John J. F. Fournier
MaRDI profile type: Publication
full work available at URL: a:1006534802172
OpenAlex ID: W1730499902
description: scientific article; zbMATH DE number 1354962