Geodesic
On everywhere divergence of trigonometric Fourier series
Sbornik. Mathematics, Tome 191 (2000) no. 1, pp. 97-120 Cet article a éte moissonné depuis la source Math-Net.Ru

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The following theorem is established. Theorem. {\it Let a function $\varphi\colon[0,+\infty)\to[0,+\infty)$ and a sequence $\{\psi(m)\}$ satisfy the following condition: the function $\varphi(u)/u$ is non-decreasing on $(0,+\infty)$, $\psi(m)\geqslant 1$ $(m=1,2,\dots)$ and $\varphi(m)\psi(m)=o(m\sqrt{\ln m}/\sqrt{\ln\ln m}\,)$ as $m\to\infty$. Then there is a function $f\in L[-\pi,\pi]$ such that $$ \int _{-\pi}^\pi\varphi(|f(x)|)\,dx<\infty $$ and $\limsup_{m\to\infty}S_m(f,x)/\psi(m)=\infty$ for all $x\in[-\pi,\pi]$ here $S_m(f)$ is the $m$-th partial sum of the trigonometric Fourier series of $f$}. 
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     author = {S. V. Konyagin},
     title = {On everywhere divergence of trigonometric {Fourier} series},
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     year = {2000},
     volume = {191},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_1_a3/}
}
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S. V. Konyagin. On everywhere divergence of trigonometric Fourier series. Sbornik. Mathematics, Tome 191 (2000) no. 1, pp. 97-120. http://geodesic.mathdoc.fr/item/SM_2000_191_1_a3/

[1] Carleson L., “On convergence and growth of partial sums of Fourier series”, Acta Math., 116 (1966), 133–157 | DOI | MR

[2] Antonov N. Yu., “Convergence of Fourier series”, East J. Approx., 2 (1996), 187–196 | MR | Zbl

[3] Sjölin P., “An inequality of Paley and convergence a.e. of Walsh–Fourier series”, Ark. Mat., 7 (1969), 551–570 | DOI | MR | Zbl

[4] Kolmogoroff A. N., “Une série de Fourier–Lebesque divergente presque partout”, Fund. Math., 4 (1923), 324–328 | Zbl

[5] Kolmogoroff A. N., “Une série de Fourier–Lebesque divergente partout”, C. R. Acad. Sci. Paris, 183 (1926), 1327–1329

[6] Bari N. K., Trigonometricheskie ryady, GIFML, M., 1961 | MR

[7] Körner T. W., “Everywhere divergent Fourier series”, Colloq. Math., 45 (1981), 103–118 | MR | Zbl

[8] Hardy G. H., “On the summability of Fourier's series”, Proc. London Math. Soc., 12 (1913), 365–372 | DOI | Zbl

[9] Chen Y. M., “An almost everywhere divergent Fourier series of the class $L(\log^+\log^+L)^{1-\varepsilon}$”, J. London Math. Soc., 44 (1969), 643–654 | DOI | MR | Zbl

[10] Oskolkov K. I., “A class of I. M. Vinogradov's series and its applications in harmonic analysis”, Progress in approximation theory. An international perspective, Proceedings of the international conference on approximation theory (Tampa, South Florida, USA, March 19–22, 1990), eds. A. A. Gonchar et al., Springer-Verlag, New York, 1992, 353–402 | MR | Zbl

[11] Oskolkov K. I., “Podposledovatelnosti summ Fure integriruemykh funktsii”, Tr. MIAN, 167, Nauka, M., 1985, 239–260 | MR | Zbl

[12] Tandori K., “Ein Divergenzsatz für Fourierreihen”, Acta Sci. Math., 30 (1969), 43–48 | MR | Zbl

[13] Totik V., “On the divergence of Fourier series”, Publ. Math. Debrecen, 29 (1982), 251–264 | MR | Zbl

[14] Stein E. M., “On limits of sequences of operators”, Ann. Math., 74 (1961), 140–170 | DOI | MR | Zbl

[15] Kakhan Zh.-P., Sluchainye funktsionalnye ryady, Mir, M., 1973 | MR | Zbl

[16] Konyagin S. V., Ruzsa I. Z., Schlag W., “On uniformly distributed dilates of finite integer sequences”, J. Number Theory (to appear) | MR | Zbl