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2008/2009 Divergence in Measure of Rearranged Multiple Orthononal Fourier Series
Rostom Getsadze
Real Anal. Exchange 34(2): 501-520 (2008/2009).


Let $\{\varphi_n(x)$, $n=1,2,\dots\}$ be an arbitrary complete orthonormal system (ONS) on the interval $I:=[0,1)$ that consists of a.e. bounded functions. Then there exists a rearrangement $\{ \varphi_{\sigma_1(n)}$, $n=1,2, \dots\}$ of the system $\{\varphi_n(x)$, $n=1,2,\dots\}$ that has the following property: for arbitrary nonnegative, continuous and nondecreasing on $[0,\infty)$ function $\phi(u)$ such that $u\phi (u)$ is a convex function on $[0,\infty)$ and $\phi (u) = o(\ln u)$, $u \to \infty$, there exists a function $f \in L(I^2)$ such that $\int_{I^2} | f(x,y) |$ $\phi( | f(x,y) | )\;dx\; dy \infty$ and the sequence of the square partial sums of the Fourier series of $f$ with respect to the double system $\{ \varphi_{\sigma_1 (m)}(x)\varphi_{\sigma_1 (n)}(y)$, $m,n \in\N \}$ on $I^2$ is essentially unbounded in measure on $I^2$.


Download Citation

Rostom Getsadze. "Divergence in Measure of Rearranged Multiple Orthononal Fourier Series." Real Anal. Exchange 34 (2) 501 - 520, 2008/2009.


Published: 2008/2009
First available in Project Euclid: 29 October 2009

MathSciNet: MR2569201

Primary: 42B08
Secondary: 40B05

Keywords: divergence in measure , The double Haar system , unconditional convergence

Rights: Copyright © 2008 Michigan State University Press

Vol.34 • No. 2 • 2008/2009
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