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 Mat. Sb. (N.S.), 1978, Volume 107(149), Number 2(10), Pages 245–258 (Mi msb2615)

Convergence of Fourier series almost everywhere and in the $L$-metric

Abstract: The following theorems are proved.
Theorem 1. There exists a constant $C>0$ such that for any function $f\in L(0,2\pi)$ there is a measurable function $F$ for which $|F|=|f|$, and
a) $\displaystyle\int_0^{2\pi}\sup_n|S_n(F)(x)| dx\leqslant C\int_0^{2\pi}|f(x)| dx$,
b) $\displaystyle\int_0^{2\pi}\sup_n|{\widetilde{S}}_n(F)(x)| dx\leqslant C\int_0^{2\pi}|f(x)| dx$,
c) $\displaystyle\int_0^{2\pi}|\widetilde{F}(x)| dx\leqslant C\int_0^{2\pi}|f(x)| dx$,
\noindent where $S_n(F)$ is a partial sum of the Fourier series of $F$, $\widetilde S_n(F)$ is a partial sum of the conjugate Fourier series, and $\widetilde F$ is the conjugate function to $F$.
\medskip Theorem 2. {\it For any function $f\in L(0,2\pi)$ and $\varepsilon>0$ there exists a measurable function $F$ such that $|F|=|f|$, $\mu\{x\in[0,2\pi):F(x)\ne f(x)\}<\varepsilon$ ($\mu$ is Lebesgue measure), and both the Fourier series of $F$ and its conjugate series converge almost everywhere and in the metric of $L$.}
Bibliography: 11 titles.

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English version:
Mathematics of the USSR-Sbornik, 1979, 35:4, 527–539

Bibliographic databases:

UDC: 517.51
MSC: Primary 42A20, 42A40; Secondary 42A04, 42A08

Citation: Sh. V. Kheladze, “Convergence of Fourier series almost everywhere and in the $L$-metric”, Mat. Sb. (N.S.), 107(149):2(10) (1978), 245–258; Math. USSR-Sb., 35:4 (1979), 527–539

Citation in format AMSBIB
\Bibitem{Khe78} \by Sh.~V.~Kheladze \paper Convergence of Fourier series almost everywhere and in the $L$-metric \jour Mat. Sb. (N.S.) \yr 1978 \vol 107(149) \issue 2(10) \pages 245--258 \mathnet{http://mi.mathnet.ru/msb2615} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=512010} \zmath{https://zbmath.org/?q=an:0404.42008} \transl \jour Math. USSR-Sb. \yr 1979 \vol 35 \issue 4 \pages 527--539 \crossref{https://doi.org/10.1070/SM1979v035n04ABEH001570} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1979JJ04900006} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. M. Olevskii, “Modifications of functions and Fourier series”, Russian Math. Surveys, 40:3 (1985), 181–224
2. M. G. Grigoryan, “On convergence of Fourier series in complete orthonormal systems in the $L^1$-metric and almost everywhere”, Math. USSR-Sb., 70:2 (1991), 445–466
3. M. G. Grigoryan, “On some properties of orthogonal systems”, Russian Acad. Sci. Izv. Math., 43:2 (1994), 261–289
4. S. Galstyan, R. I. Ovsepian, “Trigonometric series with rapidly decreasing coefficients”, Sb. Math., 187:11 (1996), 1577–1600
5. M. G. Grigoryan, “On the $L^p_\mu$-strong property of orthonormal systems”, Sb. Math., 194:10 (2003), 1503–1532
6. M. G. Grigoryan, A. A. Sargsyan, “Non-linear approximation of continuous functions by the Faber-Schauder system”, Sb. Math., 199:5 (2008), 629–653
7. Grigoryan, MG, “Unconditional C-strong property of Faber-Schauder system”, Journal of Mathematical Analysis and Applications, 352:2 (2009), 718
8. Grigoryan M.G., Sargsyan A.A., “On the coefficients of the expansion of elements from C[0,1] space by the Faber-Schauder system”, J Funct Spaces Appl, 9:2 (2011), 191–203
9. M. G. Grigoryan, “Modifications of functions, Fourier coefficients and nonlinear approximation”, Sb. Math., 203:3 (2012), 351–379
10. L. N. Galoyan, M. G. Grigoryan, A. Kh. Kobelyan, “Convergence of Fourier series in classical systems”, Sb. Math., 206:7 (2015), 941–979
11. M. G. Grigoryan, K. A. Navasardyan, “Universal functions in ‘correction’ problems guaranteeing the convergence of Fourier–Walsh series”, Izv. Math., 80:6 (2016), 1057–1083
12. M. G. Grigoryan, “Ob absolyutnoi skhodimosti ryadov Fure–Khaara v metrike $L^p(0,1)$, $0<p<1$”, Issledovaniya po lineinym operatoram i teorii funktsii. 46, Zap. nauchn. sem. POMI, 467, POMI, SPb., 2018, 34–54
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