Open Access
2013/2014 Essential Divergence in Measure of Multiple Orthogonal Fourier Series
Rostom Getsadze
Real Anal. Exchange 39(1): 91-100 (2013/2014).


In the present paper we prove the following theorem: \\ Let \(\{\vf _{m,n}(x,y)\}_{m,n=1}^{\infty} \) be an arbitrary uniformly bounded double orthonormal system on \(I^2:=[0,1]^2\) such that for some increasing sequence of positive integers \(\{N_n\}_{n=1}^\infty \) the Lebesgue functions \(L_{N_n,N_n}(x,y)\) of the system are bounded below a. e. by \( \ln^{1+\epsilon} N_n \), where \(\epsilon \) is a positive constant. Then there exists a function \(g \in L(I^2)\) such that the double Fourier series of \(g\) with respect to the system \(\{\vf _{m,n}(x,y)\}_{m,n=1}^{\infty} \) essentially diverges in measure by squares on \(I^2\). The condition is critical in the logarithmic scale in the class of all such systems %\footnote{ 2000 Mathematics Subject Classification : Primary 42B08; Secondary 40B05. % Key words and phrases: %Essential divergence in measure, orthogonal Fourier series, Lebesgue functions}.


Download Citation

Rostom Getsadze. "Essential Divergence in Measure of Multiple Orthogonal Fourier Series." Real Anal. Exchange 39 (1) 91 - 100, 2013/2014.


Published: 2013/2014
First available in Project Euclid: 1 July 2014

zbMATH: 1296.13021
MathSciNet: MR1006530

Primary: 26B10 , 42B08
Secondary: 40B05

Keywords: convergence , Essential , Fourier , measure

Rights: Copyright © 2013 Michigan State University Press

Vol.39 • No. 1 • 2013/2014
Back to Top