Multiple Rademacher series in symmetric spaces
type: Item
instance of: scholarly article
title: Multiple Rademacher series in symmetric spaces
zbMATH Open document ID: 0944.42019
DOI: 10.1007/BF02465195
publication date: 2000-09-18T00:00:00Z
review text: Let \(r_1,r_2,\dots\) denote the Rademacher system on the interval \([0,1]\). The author announces several results concerning representation of functions as series of the form \[ f(x)= \sum_{i\neq j} a_{i,j} r_k(x) r_j(x). \] It seems that in rearrangement invariant spaces \(X\) (e.g., \(L^p\) spaces, Orlicz spaces and Marcinkiewicz spaces), the norm of such an \(f\) is equivalent to the \(\ell^2\) norm of the coefficients \(\{a_{i,j}\}\) if and only if \(X\) contains the closure of \(L^\infty\) in the Orlicz space \(L_M\), where \(M(x)= e^x- 1\). On the other hand, when \(1\leq p<2\), there are no rearrangement invariant spaces in which the norm of \(f\) is equivalent to the \(\ell^p\) norm of the coefficients \(\{a_{i,j}\}\).
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1407629
zbMATH Keywords: Rademacher system
zbMATH Keywords: rearrangement invariant spaces
zbMATH Keywords: Orlicz spaces
author: Sergey Astashkin
reviewed by: William R. Wade
MaRDI profile type: Publication
cites work: Q5762102
cites work: Q3997990
cites work: Q3202684
cites work: Q3955929
cites work: Rademacher series in symmetric spaces
cites work: Q3272715
cites work: Q4196887
cites work: On Orlicz sequence spaces
cites work: Q4383236
full work available at URL: https://doi.org/10.1007/bf02465195
OpenAlex ID: W2030009783
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1407629