On the bounded divergence by rearranged orthonormal systems
type: Item
instance of: scholarly article
title: On the bounded divergence by rearranged orthonormal systems
zbMATH Open document ID: 0828.42018
author: S. Sh. Galstyan
published in: Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences
publication date: 1996-01-14T00:00:00Z
review text: The following two theorems are proved: Theorem 1. Let \(\{\varphi_n: n= 1,2,\dots\}\) be either the trigonometric system or a basis in the space \(C[0, 1]\). Then there exist a continuous function \(f\) and a permutation \(\tau\) of the positive integers such that the rearranged expansion \[ \sum^\infty_{n= 1} a_{\tau(n)} \varphi_{\tau(n)}(t),\quad a_n:= \int^1_0 f(t) \varphi^*_n(t) dt,\tag{*} \] diverges a.e., where \(\{\varphi^*_n\}\) is the biorthogonal system of \(\{\varphi_n\}\), and the partial sums of the series (*) are uniformly bounded. Theorem 2. Let \(\{\varphi_n\}\) be a complete orthonormal system on \([0,1]\). Then there exist a continuous function \(f\) and a permutation \(\tau\) of the positive integers such that the rearranged Fourier series (*) of \(f\) diverges a.e. (this time \(\varphi^*_n\equiv \varphi_n\)), and the maximal function of the partial sums of the series (*) belongs to the space \(L^2[0, 1]\).
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42A20
zbMATH DE Number: 795392
zbMATH Keywords: trigonometric system
zbMATH Keywords: rearranged expansion
zbMATH Keywords: biorthogonal system
zbMATH Keywords: complete orthonormal system
zbMATH Keywords: rearranged Fourier series
zbMATH Keywords: maximal function
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
Recommended article: Q3755945
Recommended article: Q3343118
Recommended article: On bounded orthonormal systems of functions
Recommended article: On divergent Fourier series in orthogonal systems with bounded variations
Recommended article: Q4710496
Recommended article: On Lebesgue functions of uniformly bounded orthonormal systems
Recommended article: A complete orthonormal system of divergence
Recommended article: Q3361371
Recommended article: General orthonormal systems and absolute convergence
description: scientific article; zbMATH DE number 795392