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Sbornik: Mathematics, 1996, Volume 187, Issue 3, Pages 365–384
DOI: https://doi.org/10.1070/SM1996v187n03ABEH000116
(Mi sm116)
 

This article is cited in 6 scientific papers (total in 6 papers)

Hilbert transform and exponential integral estimates of rectangular sums of double Fourier series

G. A. Karagulian

Institute of Mathematics, National Academy of Sciences of Armenia
References:
Abstract: A new integral estimate for rectangular partial sums of double Fourier series is obtained. The main result of the paper is the following.
Theorem. {\it For any $f\in L\log L(\mathbf T^2)$ and $\delta>0$ there exists a set $E_{\delta,f}\in\mathbf T^2$, $|E_{\delta,f}|>(2\pi)^2-\delta$ such that}
\begin{align*} &1)\quad \int_{E_{\delta,f}}\exp\biggl[\frac{c_1\delta|S_{N,M}(x,y,f)|}{\|f\|_{L\log L(\mathbf T^2)}}\biggr]^{1/2}\,dx\,dy\leqslant C_2, \qquad N,M=1,2,\dots, \\ &2)\quad \lim_{N,M\to\infty}\int_{E_{\delta,f}}\bigl[\exp(|S_{N,M}(x,y,f)-f(x,y)|)^{1/2}-1\bigr]\,dx\,dy=0. \end{align*}

This theorem yields estimates almost everywhere for rectangular sums of double Fourier series and convergence in $L^p$ on sets of large measure.
Received: 10.01.1995
Bibliographic databases:
UDC: 517.51
MSC: Primary 44A15; Secondary 40B05, 40A05
Language: English
Original paper language: Russian
Citation: G. A. Karagulian, “Hilbert transform and exponential integral estimates of rectangular sums of double Fourier series”, Sb. Math., 187:3 (1996), 365–384
Citation in format AMSBIB
\Bibitem{Kar96}
\by G.~A.~Karagulian
\paper Hilbert transform and exponential integral estimates of rectangular sums of double Fourier series
\jour Sb. Math.
\yr 1996
\vol 187
\issue 3
\pages 365--384
\mathnet{http://mi.mathnet.ru//eng/sm116}
\crossref{https://doi.org/10.1070/SM1996v187n03ABEH000116}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1400345}
\zmath{https://zbmath.org/?q=an:0870.42003}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1996VE21900003}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0030305808}
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  • https://doi.org/10.1070/SM1996v187n03ABEH000116
  • https://www.mathnet.ru/eng/sm/v187/i3/p55
  • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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