Geodesic
Everywhere Divergent
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 50-59.

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For a function $f\in L^1({\mathbb T})$, we investigate the sequence $(C,1)$ of mean values $\Phi(|S_k(x,f)-f(x)|)$, where $\Phi (t)\colon [0,+\infty)\to [0,+\nobreak \infty)$, $\Phi (0)=\nobreak 0$, is a continuous increasing function. We prove that if $\Phi $ increases faster than exponentially, then these means can diverge everywhere. Divergence almost everywhere of such means was established earlier.
Keywords: Fourier series, means of Fourier series, the space $L^1({\mathbf T})$. 
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G. A. Karagulian. Everywhere Divergent. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 50-59. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a6/

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

[2] A. Zigmund, Trigonometricheskie ryady, t. 1, Mir, M., 1965 | MR

[3] N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[4] J. Marcinkiewicz, “Sur la sommabilitee forte des series de Fourier”, J. London Math. Soc., 14 (1939), 162–168 | DOI | MR | Zbl

[5] A. Zygmund, “On the convergence and summability of power series on the circle of convergence”, Proc. London Math. Soc., 47 (1941), 326–350 | DOI | MR

[6] A. Zigmund, Trigonometricheskie ryady, t. 2, Mir, M., 1965 | MR

[7] K. I. Oskolkov, “O silnoi summiruemosti ryadov Fure”, Tr. MIAN, 172, 1985, 280–290 | MR | Zbl

[8] V. A. Rodin, “The space BMO and strong means of Fourier series”, Ann. Math., 16 (1990), 291–302 | DOI | MR | Zbl

[9] L. D. Gogoladze, “O silnoi summiruemosti pochti vsyudu”, Matem. sb., 135 (1988), 158–169 | MR

[10] F. John, L. Nirenberg, “On functions of bounded mean oscilation”, Comm. Pure Appl. Math., 14 (1961), 415–426 | DOI | MR | Zbl

[11] Dzh. Garnet, Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR

[12] G. A. Karagulyan, “O raskhodimosti silnykh $\Phi$-srednikh ryadov Fure”, Izv. AN Armenii. Ser. “Matematika”, 26:2 (1991), 159–162 | MR | Zbl

[13] G. A. Karagulyan, O skhodimosti i summiruemosti trigonometricheskikh i obschikh ortogonalnykh ryadov Fure, Diss. ... d.f.m.n., M., 2002

[14] V. Totik, “Notes on Fourier series strong approximations”, J. of App. Theory, 43 (1985), 105–111 | DOI | MR | Zbl

[15] V. Totik, “On the strong approximation of Fourier series”, Acta Math. Acad. Sci Hung., 1–2 (1980), 157–172 | MR

[16] G. A. Karagulian, “On the convergence to infinity of Fourier series along dense sub-sequence of numbers”, Analysis Math., 18:4 (1992), 249–259 | DOI | MR | Zbl

[17] S. V. Konyagin, “O vsyudu raskhodyaschegosya trigonometricheskogo ryada Fure”, Matem. sb., 191:1 (2000), 97–120 | MR | Zbl

[18] S. V. Bochkarev, “O probleme gladkosti funktsii, ryady Fure–Uolsha kotorykh raskhodyatsya pochti vsyudu”, Dokl. RAN, 371:4 (2000), 370–373

[19] V. Totik, “On the divergence of Fourier series,”, Publicationes Mathematicae (Debrecen), 29:3–4 (1982), 251–264 | MR | Zbl

[20] Y. Katznelson, “Sur les ensembles de divergence des series trigonometriques”, Studia Math., 26 (1966), 301–304 | MR | Zbl