Geodesic
Classification of periodic functions and the rate of convergence of their Fourier series
Izvestiya. Mathematics , Tome 28 (1987) no. 1, pp. 99-132.

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

The author proposes a classification for periodic functions that is based on grouping them according to the rate at which their Fourier coefficients tend to zero. The classes $L_\beta^\psi\mathfrak N$ thereby introduced coincide, for fixed values of the defining parameters, with the known classes $W^r$, $W^rH_\omega$, $W_\beta^r$, $W_\beta^rH_\omega$, and the like. Such an approach permits the classification of a wide spectrum of periodic functions, including infinitely differentiable, analytic, and entire functions. The asymptotic behavior of the deviations of the Fourier sums in these classes is studied. The assertions obtained in this direction contain results known earlier on approximation by Fourier sums of classes of differentiable functions. Bibliography: 18 titles. 
@article{IM2_1987_28_1_a5,
     author = {A. I. Stepanets},
     title = {Classification of periodic functions and the rate of convergence of their {Fourier} series},
     journal = {Izvestiya. Mathematics },
     pages = {99--132},
     publisher = {mathdoc},
     volume = {28},
     number = {1},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a5/}
}
TY  - JOUR
AU  - A. I. Stepanets
TI  - Classification of periodic functions and the rate of convergence of their Fourier series
JO  - Izvestiya. Mathematics 
PY  - 1987
SP  - 99
EP  - 132
VL  - 28
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a5/
LA  - en
ID  - IM2_1987_28_1_a5
ER  - 
%0 Journal Article
%A A. I. Stepanets
%T Classification of periodic functions and the rate of convergence of their Fourier series
%J Izvestiya. Mathematics 
%D 1987
%P 99-132
%V 28
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a5/
%G en
%F IM2_1987_28_1_a5
A. I. Stepanets. Classification of periodic functions and the rate of convergence of their Fourier series. Izvestiya. Mathematics , Tome 28 (1987) no. 1, pp. 99-132. http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a5/

[1] Kolmogoroff A., “Zur Grössenordmmg des Restglie des Fouriershen Reihen differenzierbarer Funktionen”, Ann. Math., 36:1 (1935), 521–526 | DOI | MR | Zbl

[2] Pinkevich V. T., “O poryadke ostatochnogo chlena ryada Fure funktsii, differentsiruemykh v smysle Veilya”, Izv. AN SSSR. Ser. matem., 4:6 (1940), 521–528 | MR | Zbl

[3] Nikolskii S. M., “Ob asimptoticheskom povedenii ostatka pri priblizhenii funktsii, udovletvoryayuschikh usloviyu Lipshitsa, summami Fure”, Izv. AN SSSR. Ser. matem., 4:6 (1940), 501–508 | MR | Zbl

[4] Nikolskii S. M., “O nekotorykh metodakh priblizheniya trigonometricheskimi summami”, Izv. AN SSSR. Ser. matem., 4:6 (1940), 509–520 | MR | Zbl

[5] Nikolskii S. M., “Asimptoticheskaya otsenka ostatka pri priblizhenii summami Fure”, Dokl. AN SSSR, 22:6 (1941), 386–389

[6] Nikolskii S. M., “Priblizhenie periodicheskikh funktsii trigonometricheskimi mnogochlenami”, Tr. Matem. instituta im. V. A. Steklova AN SSSR, 15, 1945, 1–76 | MR | Zbl

[7] Nikolskii S. M., “O lineinykh metodakh summirovaniya ryadov Fure”, Izv. AN SSSR. Ser. matem., 12:3 (1948), 259–278 | MR | Zbl

[8] Stepanets A. I., Ravnomernye priblizheniya trigonometricheskimi polinomami, Naukova dumka, Kiev, 1981, 340 pp. | MR

[9] Efimov A. V., “Priblizhenie, nepreryvnykh periodicheskikh funktsii summami Fure”, Izv. AN SSSR. Ser. matem., 24:2 (1960), 243–296 | MR | Zbl

[10] Efimov A. B., “Lineinye metody priblizheniya nekotorykh klassov nepreryvnykh periodicheskikh funktsii”, Tr. Matem. instituta im. V. A. Steklova AN SSSR, 62, 1961, 3–47 | MR

[11] Telyakovskii S. A., “O normakh trigonometricheskikh polinomov i priblizhenii differentsiruemykh funktsii lineinymi srednimi ikh ryadov Fure”, Tr. Matem. instituta im. V. A. Steklova AN SSSR, 62, 1961, 61–97

[12] Telyakovskii S. A., “Otsenka normy funktsii cherez ee koeffitsienty Fure, udobnaya v zadachakh teorii approksimatsii”, Tr. Matem. instituta im. V. A. Steklova AN SSSR, 109, 1971, 65–97

[13] Stepanets A. I., Klassy periodicheskikh funktsii i priblizhenie ikh elementov summami Fure, Preprint 83.10, Institut matematiki AN USSR, Kiev, 1983, 57 pp.

[14] Stepanets A. I., “Klassy periodicheskikh funktsii i priblizhenie ikh elementov lineinymi srednimi ikh ryadov Fure”, Mezhdunarodnaya konferentsiya po teorii priblizheniya funktsii (Tezisy dokladov), Kiev, 1983, 172

[15] Stepanets A. I, “Klassy periodicheskikh funktsii i priblizhenie ikh elementov trigonometricheskimi polinomami”, Mezhdunarodnyi kongress matematikov (Tezisy dokladov), Varshava, 1983, 30

[16] Nagy B., “Sur une classe génerale de procèdes de sommation pour les series de Fourier”, Hung. Acta Math., 1:3 (1948), 14–62 | MR

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

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