Geodesic
On uniform approximation of functions by Fourier sums
Sbornik. Mathematics, Tome 42 (1982) no. 4, pp. 515-538 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper studies traditional problems on uniform approximation of a continuous $2\pi$-periodic function $f$ by its $n$th Fourier sums $S_n(f)$. To this end the deviation $\|f-S_n(f)\|_{C_{2\pi}}$ is estimated in terms of some new functional characteristics. As an application of the estimates a number of known results (due to Lebesgue, Salem, Stechkin, Ul'yanov, Oskolkov, and others) are obtained. Bibliography: 18 titles. 
@article{SM_1982_42_4_a4,
     author = {E. A. Sevast'yanov},
     title = {On uniform approximation of functions by {Fourier} sums},
     journal = {Sbornik. Mathematics},
     pages = {515--538},
     year = {1982},
     volume = {42},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1982_42_4_a4/}
}
TY  - JOUR
AU  - E. A. Sevast'yanov
TI  - On uniform approximation of functions by Fourier sums
JO  - Sbornik. Mathematics
PY  - 1982
SP  - 515
EP  - 538
VL  - 42
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1982_42_4_a4/
LA  - en
ID  - SM_1982_42_4_a4
ER  - 
%0 Journal Article
%A E. A. Sevast'yanov
%T On uniform approximation of functions by Fourier sums
%J Sbornik. Mathematics
%D 1982
%P 515-538
%V 42
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1982_42_4_a4/
%G en
%F SM_1982_42_4_a4
E. A. Sevast'yanov. On uniform approximation of functions by Fourier sums. Sbornik. Mathematics, Tome 42 (1982) no. 4, pp. 515-538. http://geodesic.mathdoc.fr/item/SM_1982_42_4_a4/

[1] Lebesgue H., “Sur la représentation trigonométrique approchée des fonctions satisfaisant á une condition de Lipschitz”, Bull. Soc. Math. France, 38 (1910), 184–210 | MR | Zbl

[2] Sevastyanov E. A., “Kusochno monotonnaya approksimatsiya i $\Phi$-variatsii”, Anal. Math., 1:2 (1975), 141–164 | DOI | MR

[3] Lagrange R., “Sur oscillations d'ordre superieur d'une fonction numérique”, Ann. scient. Ecole norm. supér., 82:2 (1965), 101–130 | MR | Zbl

[4] Dolzhenko E. P., Sevastyanov E. A., “O priblizheniyakh funktsii v khausdorfovoi metrike posredstvom kusochno monotonnykh (v chastnosti, ratsionalnykh) funktsii”, Matem. sb., 101 (143) (1976), 508–541 | Zbl

[5] Hardy G. H., Littlewood J. E., “A convergence criterion for Fourier series”, Math. Z., 28:4 (1928), 612–634 | DOI | MR | Zbl

[6] Salem R., “New theorems on the convergence of Fourier series”, Indag. Math., 16 (1954), 550–555 | MR

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

[8] Dzyadyk V. S., Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977 | MR | Zbl

[9] Geronimus Ya. L., “O nekotorykh teoremakh vlozheniya”, Izv. VUZov, Matematika, 1965, no. 6, 53–62 | MR | Zbl

[10] Andrienko V. A., “Vlozhenie nekotorykh klassov funktsii”, Izv. AN SSSR, ser. matem., 31 (1957), 1311–1326 | MR

[11] Zigmund A., Trigonometricheskie ryady, T. I, Mir, M., 1965 | MR

[12] Salem R., “Essais sur les séries trigonométriques”, Actual. Sci. et Industr., No 862, Paris, 1940 | MR

[13] Oskolkov K. I., “Obobschennaya variatsiya, indikatrisa Banakha i ravnomernaya skhodimost ryadov Fure”, Matem. zametki, 12:3 (1972), 313–324 | MR | Zbl

[14] Chanturiya Z. A., “O ravnomernoi skhodimosti ryadov Fure”, Matem. sb., 100 (142) (1976), 534–554 | Zbl

[15] Sevastyanov E. A., “Kusochno monotonnaya i ratsionalnaya approksimatsii i ravnomernaya skhodimost ryadov Fure”, Anal. Math., 1:4 (1975), 283–295 | DOI | MR

[16] Stechkin S. B., “O priblizhenii nepreryvnykh funktsii summami Fure”, UMN, VII:4(50) (1952), 139–141

[17] Akhobadze T. I., “Nekotorye klassy funktsii i trigonometricheskie ryady Fure”, Soobsch. AN Gruz. SSR, 86:1 (1977), 49–52 | MR | Zbl

[18] Ulyanov P. L., “Ob absolyutnoi i ravnomernoi skhodimosti ryadov Fure”, Matem. sb., 72(114) (1967), 193–224