Geodesic
Order estimates of smallest norms, with respect to the choice of $N$ harmonics, of derivatives of the Dirichlet and Favard kernels
Sbornik. Mathematics, Tome 72 (1992) no. 2, pp. 567-578 
È. M. Galeev. Order estimates of smallest norms, with respect to the choice of $N$ harmonics, of derivatives of the Dirichlet and Favard kernels. Sbornik. Mathematics, Tome 72 (1992) no. 2, pp. 567-578. http://geodesic.mathdoc.fr/item/SM_1992_72_2_a16/
@article{SM_1992_72_2_a16,
     author = {\`E. M. Galeev},
     title = {Order estimates of smallest norms, with respect to the choice of $N$ harmonics, of derivatives of the {Dirichlet} and {Favard} kernels},
     journal = {Sbornik. Mathematics},
     pages = {567--578},
     year = {1992},
     volume = {72},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1992_72_2_a16/}
}
TY  - JOUR
AU  - È. M. Galeev
TI  - Order estimates of smallest norms, with respect to the choice of $N$ harmonics, of derivatives of the Dirichlet and Favard kernels
JO  - Sbornik. Mathematics
PY  - 1992
SP  - 567
EP  - 578
VL  - 72
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1992_72_2_a16/
LA  - en
ID  - SM_1992_72_2_a16
ER  - 
%0 Journal Article
%A È. M. Galeev
%T Order estimates of smallest norms, with respect to the choice of $N$ harmonics, of derivatives of the Dirichlet and Favard kernels
%J Sbornik. Mathematics
%D 1992
%P 567-578
%V 72
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1992_72_2_a16/
%G en
%F SM_1992_72_2_a16

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

The Dirichlet kernel is defined for periodic functions of several variables; it consists of $N$ harmonics and has minimal order of the norm with respect to the choice of harmonics of the mixed Weyl derivative in the space $\tilde L_q$. A similar problem on the minimal order of the norm is solved for the Favard kernel. Both problems generalize to the case of several derivatives.

[1] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[2] Galeev E. M., “Poryadkovye otsenki proizvodnykh periodicheskogo mnogomernogo $\alpha$-yadra Dirikhle v smeshannoi norme”, Matem. sb., 117(159) (1982), 32–43 | MR | Zbl

[3] Temlyakov V. N., “Priblizhenie periodicheskikh funktsii neskolkikh peremennykh s ogranichennoi smeshannoi proizvodnoi”, DAN SSSR, 248:3 (1979), 527–531 | MR | Zbl

[4] Temlyakov V. N., “Priblizhenie periodicheskikh funktsii neskolkikh peremennykh s ogranichennoi smeshannoi proizvodnoi”, Tr. MIAN, 156, 1980, 233–260 | MR | Zbl

[5] Temlyakov V. N., “Priblizhenie periodicheskikh funktsii neskolkikh peremennykh s ogranichennoi smeshannoi raznostyu”, Matem. sb., 113(155) (1980), 65–80 | MR | Zbl

[6] Bugrov Ya. S., “Priblizhenie klassa funktsii s dominiruyuschei smeshannoi proizvodnoi”, Matem. sb., 64(106):3 (1964), 410–418 | MR | Zbl

[7] Din Zung, “O priblizhenii periodicheskikh funktsii mnogikh peremennykh”, UMN, 38:6 (1983), 111–112 | MR | Zbl

[8] Din Zung, “Priblizhenie klassov gladkikh funktsii mnogikh peremennykh”, Tr. seminara im. I. G. Petrovskogo, 1984, no. 10, 207–226 | Zbl

[9] Maiorov V. E., “Trigonometricheskie poperechniki sobolevskikh klassov”, Teoriya funktsii i priblizhenii, Tr. 2-i Saratovskoi zimnei shkoly. Mezhvuz. nauch. sb. (24 yanvarya–5 fevralya 1984 g.), 3, Izd-vo Sarat. un-ta, Saratov, 1986, 20–22 | MR

[10] Temlyakov V. N., “Priblizhenie periodicheskikh funktsii neskolkikh peremennykh trigonometricheskimi polinomami i poperechniki nekotorykh klassov funktsii”, Izv. AN SSSR. Ser. matem., 49:5 (1985), 986–1030 | MR

[11] Benedec A., Panzone R., “The spaces $L^p$ with mixed norm”, Duke Math. J., 28:3 (1961), 301–324 | DOI | MR

[12] Galeev E. M., “Priblizhenie nekotorykh klassov periodicheskikh funktsii mnogikh peremennykh summami Fure v metrike $\tilde{L}_p$”, UMN, 32:4 (1977), 251–252 | MR | Zbl

[13] Galeev E. M., “Poperechniki po Kolmogorovu nekotorykh klassov periodicheskikh funktsii mnogikh peremennykh”, Konstruktivnaya teoriya funktsii, Tr. Mezhdunarodnoi konf. po konstruktivnoi teorii funktsii (NRB), Sofiya, 1984, 27–32 | Zbl

[14] Galeev E. M., “Approximation of periodic functions of one and several variables”, Constructive theory of functions'87, Sofia, 1988, 138–144 | MR | Zbl