Geodesic
Almost Everywhere Divergent Subsequences of Fourier Sums of Functions from $\varphi(L)\cap H_1^\omega$
Matematičeskie zametki, Tome 85 (2009) no. 4, pp. 502-515 Cet article a éte moissonné depuis la source Math-Net.Ru

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For a gap sequence of natural numbers $\{n_k\}^\infty_{k=1}$, for a nondecreasing function $\varphi\colon[0,+\infty)\to[0,+\infty)$ such that $\varphi(u)=o(u\ln\ln u)$ as $u\to\infty$, and a modulus of continuity satisfying the condition $(\ln k)^{-1}=O(\omega(n_k^{-1}))$, we present an example of a function $F\in\varphi(L)\cap H_1^\omega$ with an almost everywhere divergent subsequence $\{S_{n_k}(F,x)\}$ of the sequence of partial sums of the trigonometric Fourier series of the function $F$.
Mots-clés : Fourier sum
Keywords: gap sequence, trigonometric Fourier series, modulus of continuity, Dirichlet kernel, Lebesgue measurability, Jensen's inequality. 
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N. Yu. Antonov. Almost Everywhere Divergent Subsequences of Fourier Sums of Functions from $\varphi(L)\cap H_1^\omega$. Matematičeskie zametki, Tome 85 (2009) no. 4, pp. 502-515. http://geodesic.mathdoc.fr/item/MZM_2009_85_4_a1/

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

[2] A. Kolmogoroff, “Une série de Fourier–Lebesgue divergente partout”, C. R. Acad. Sci. Paris, 183 (1926), 1327–1328 | Zbl

[3] R. P. Gosselin, “On the divergence of Fourier series”, Proc. Amer. Math. Soc., 9:2 (1958), 278–282 | DOI | MR | Zbl

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

[5] S. V. Konyagin, “O raskhodimosti vsyudu podposledovatelnostei chastnykh summ trigonometricheskikh ryadov Fure”, Tr. In-ta matem. i mekh. UrO RAN, 11:2 (2005), 112–119 | MR | Zbl

[6] S. V. Konyagin, “O raskhodimosti vsyudu trigonometricheskikh ryadov Fure”, Matem. sb., 191:1 (2000), 103–126 | MR | Zbl

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

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

[9] K. I. Oskolkov, “Podposledovatelnosti summ Fure integriruemykh funktsii”, Sovremennye problemy matematiki. Matematicheskii analiz, algebra, topologiya, Tr. MIAN, 167, Nauka, M., 1985, 239–260 | MR | Zbl

[10] N. Yu. Antonov, “O raskhodimosti podposledovatelnostei summ Fure funktsii s ogranicheniyami na integralnyi modul nepreryvnosti”, Izv. TulGU. Ser. Matem. Mekh. Inform., 12:1 (2006), 12–26

[11] N. Yu. Antonov, “Integriruemost mazhorant summ Fure i raskhodimost ryadov Fure funktsii s ogranicheniyami na integralnyi modul nepreryvnosti”, Matem. zametki, 76:5 (2004), 651–665 | MR | Zbl

[12] Sh. V. Kheladze, “O raskhodimosti vsyudu ryadov Fure funktsii iz klassa $L\varphi(L)$”, Tr. Tbilissk. matem. in-ta, 89 (1988), 51–59 | MR | Zbl

[13] P. L. Ulyanov, “O ryadakh po sisteme Khaara”, Matem. sb., 63:3 (1964), 356–391 | MR | Zbl

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