Geodesic
Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity
Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 651-665 
N. Yu. Antonov. Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity. Matematičeskie zametki, Tome 76 (2004) no. 5, pp. 651-665. http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a1/
@article{MZM_2004_76_5_a1,
     author = {N. Yu. Antonov},
     title = {Integrability of the {Majorants} of {Fourier} {Series} and {Divergence} of the {Fourier} {Series} of {Functions} with {Restrictions} on the {Integral} {Modulus} of {Continuity}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {651--665},
     year = {2004},
     volume = {76},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a1/}
}
TY  - JOUR
AU  - N. Yu. Antonov
TI  - Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity
JO  - Matematičeskie zametki
PY  - 2004
SP  - 651
EP  - 665
VL  - 76
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a1/
LA  - ru
ID  - MZM_2004_76_5_a1
ER  - 
%0 Journal Article
%A N. Yu. Antonov
%T Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity
%J Matematičeskie zametki
%D 2004
%P 651-665
%V 76
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_2004_76_5_a1/
%G ru
%F MZM_2004_76_5_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

We construct an example of a function from the class $H_1^{\omega^*}$ , where $\omega^*(t)=\sqrt{\log\log(t^{-1})/\log(t^{-1})}$, $0, whose trigonometric Fourier series is divergent almost everywhere. We obtain sharp integrability conditions for the majorants of the partial sums of trigonometric Fourier series in terms of whether the functions in question belong to the classes $H_1^\omega$.

[1] Korneichuk N. P., Ekstremalnye zadachi teorii priblizheniya, Nauka, M., 1976

[2] Zigmund A., Trigonometricheskie ryady, T. 2, Mir, M., 1965

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

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

[5] Prokhorenko V. I., “O raskhodyaschikhsya ryadakh Fure”, Matem. sb., 75 (117):2 (1968), 185–198 | MR | Zbl

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

[7] Chen Y. M., “On Kolmogoroff's divergent Fourier series”, Archiv der Mathematik, 14:2 (1963), 116–119 | DOI | Zbl

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

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

[10] Körner T. W., “Everywhere divergent Fourier series”, Colloq. Math., 45:1 (1981), 103–118 | MR | Zbl

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

[12] Ulyanov P. L., “O ryadakh po sisteme Khaara”, Matem. sb., 63 (105):3 (1964), 356–391 | MR | Zbl

[13] Ulyanov P. L., “Vlozhenie nekotorykh klassov funktsii $H_p^\omega$”, Izv. AN SSSR. Ser. matem., 32:3 (1968), 649–686 | MR | Zbl

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