Real Analysis Exchange

Divergence in Measure of Rearranged Multiple Orthononal Fourier Series

Rostom Getsadze

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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

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

First available in Project Euclid: 29 October 2009

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Mathematical Reviews number (MathSciNet)

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

The double Haar system unconditional convergence divergence in measure


Getsadze, Rostom. Divergence in Measure of Rearranged Multiple Orthononal Fourier Series. Real Anal. Exchange 34 (2008), no. 2, 501--520.

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  • A. M. Garsia, Topics in Almost Everywhere Convergence, Lectures in Advanced Mathematics, 4, Markham Publishing Co., Chicago, 1970.
  • G. A. Karagulyan, Divergence of double Fourier series in complete orthonormal systems, (Russian) Izv. Akad. Nauk Armyan. SSR Ser. Mat., 24(2) (1989), 147–159, 200; translation in Soviet J. Contemp. Math. Anal., 24(2) (1989), no. 2, 44–56.
  • B. S. Kashin, A. A. Saakyan, Orthogonal Series, Translated from the Russian by Ralph P. Boas, Translation edited by Ben Silver, Translations of Mathematical Monographs, 75, American Mathematical Society, Providence, RI, 1989.
  • A. M. Olevskiĭ, Fourier Series with Respect to General Orthogonal Systems, Translated from the Russian by B. P. Marshall and H. J. Christoffers, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, 86, Springer-Verlag, New York-Heidelberg, 1975.
  • A. M. Olevskiĭ, Divergent series for complete systems in $L\sp{2}$, (Russian) Dokl. Akad. Nauk SSSR, 138 (1961), 545–548. English translation: Soviet Math. Dokl., 2(6) (1961), 669–672.
  • A. M. Olevskiĭ, Divergent Fourier series, Izv. Akad. Nauk SSSR Ser. Mat., 27 (1963), 343–366.
  • G. E. Tkebuchava, Divergence of multiple Fourier series with respect to bases, (English translation) Soviet Math. Dokl., 40(2) (1990), 346–348.