Geodesic
Convergence of Fourier series almost everywhere and in the $L$-metric
Sbornik. Mathematics, Tome 35 (1979) no. 4, pp. 527-539 
Sh. V. Kheladze. Convergence of Fourier series almost everywhere and in the $L$-metric. Sbornik. Mathematics, Tome 35 (1979) no. 4, pp. 527-539. http://geodesic.mathdoc.fr/item/SM_1979_35_4_a5/
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     title = {Convergence of {Fourier} series almost everywhere and in the $L$-metric},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] A. Zigmund, Trigonometricheskie ryady, t. 1, izd-vo “Mir”, Moskva, 1965 | MR

[2] N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, Moskva, 1961 | MR

[3] O. D. Tsereteli, “K voprosu ob integriruemosti sopryazhennykh funktsii”, Matem. zametki, 4:4 (1968), 461–465 | Zbl

[4] O. D. Tsereteli, “O nekotorykh zadachakh, voznikayuschikh v teorii ryadov Fure”, Seminar in-ta prikl. matem. Tbil. un-ta, annotatsii dokladov, 6, 1972, 33–36

[5] O. D. Tsereteli, “O skhodimosti pochti vsyudu ryadov Fure”, Soobsch. AN GruzSSR, 57:1 (1970), 21–24 | Zbl

[6] L. Carlesson, “On convergence and growth of partial sums of Fourier series”, Acta Math., 116 (1966), 135–157 | DOI | MR

[7] H. A. Luzin, Integral i trigonometricheskii ryad, Gostekhizdat, Moskva–Leningrad, 1951

[8] A. N. Kolmogorov, “Sur les fonctions harmoniques conjuguées et les séries de Fourier”, Fundam. Math., 7 (1925), 23–28

[9] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Gostekhizdat, Moskva, 1957 | MR

[10] N. A. Hunt, “On the convergence of Fourier series”, Orthogonal expantions and their continuous analogues, SIU press, Carbondal, Illinois, 1968, 235–255 | MR

[11] P. Sjölin, “An inequelity of Paley and convergence a. e. of Walsh–Fourier series”, Arkiv Math., 7 (1969), 551–570 | DOI | MR | Zbl