Real Analysis Exchange

Divergence in Measure of Rearranged Multiple Orthononal Fourier Series

Rostom Getsadze

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Abstract

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 $I^2$.

Article information

Source
Real Anal. Exchange Volume 34, Number 2 (2008), 501-520.

Dates
First available in Project Euclid: 29 October 2009

Permanent link to this document
http://projecteuclid.org/euclid.rae/1256835201

Mathematical Reviews number (MathSciNet)
MR2569201

Subjects
Primary: 42B08: Summability
Secondary: 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)

Keywords
The double Haar system unconditional convergence divergence in measure

Citation

Getsadze, Rostom. Divergence in Measure of Rearranged Multiple Orthononal Fourier Series. Real Anal. Exchange 34 (2008), no. 2, 501--520. http://projecteuclid.org/euclid.rae/1256835201.


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References

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