Geodesic
Inequalities of Bernstein and Jackson--Nikol'skii Type
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 95-104.

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

We obtain Bernstein and Jackson–Nikol'skii inequalities for trigonometric polynomials with spectrum generated by the level surfaces of a function $\Lambda(t)$, and their sharpness is studied under a specific choice of $\Lambda(t)$. Estimates of the norms of derivatives of Dirichlet kernels with harmonics generated by the level surfaces of the function $\Lambda(t)$ are established in $L^p$.
Keywords: Bernstein-type inequality, Jackson–Nikol'skii inequality, Dirichlet kernel, trigonometric polynomial, spectrum of a polynomial, hyperbolic cross. 
@article{MZM_2006_80_1_a11,
     author = {M. B. Sikhov},
     title = {Inequalities of {Bernstein} and {Jackson--Nikol'skii} {Type}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {95--104},
     publisher = {mathdoc},
     volume = {80},
     number = {1},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a11/}
}
TY  - JOUR
AU  - M. B. Sikhov
TI  - Inequalities of Bernstein and Jackson--Nikol'skii Type
JO  - Matematičeskie zametki
PY  - 2006
SP  - 95
EP  - 104
VL  - 80
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a11/
LA  - ru
ID  - MZM_2006_80_1_a11
ER  - 
%0 Journal Article
%A M. B. Sikhov
%T Inequalities of Bernstein and Jackson--Nikol'skii Type
%J Matematičeskie zametki
%D 2006
%P 95-104
%V 80
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a11/
%G ru
%F MZM_2006_80_1_a11
M. B. Sikhov. Inequalities of Bernstein and Jackson--Nikol'skii Type. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 95-104. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a11/

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

[2] N. N. Pustovoitov, “Ortopoperechniki nekotorykh klassov periodicheskikh funktsii dvukh peremennykh s zadannoi mazhorantoi smeshannykh modulei nepreryvnosti”, Matem. zametki, 65:1 (1999), 107–117 | MR | Zbl

[3] E. M. Galeev, “Poryadkovye otsenki proizvodnykh periodicheskogo mnogomernogo $\alpha$-yadra Dirikhle v smeshannoi norme”, Matem. sb., 117(159):1 (1982), 32–43 | MR | Zbl

[4] V. N. Temlyakov, Priblizhenie funktsii s ogranichennoi smeshannoi proizvodnoi, Tr. MIAN SSSR, 178, 1986 | MR | Zbl

[5] B. S. Mityagin, “Priblizhenie funktsii v prostranstvakh $L^p$ i $C$ na tore”, Matem. sb., 58(100):4 (1962), 397–414 | Zbl

[6] N. S. Nikolskaya, “Priblizhenie differentsiruemykh funktsii mnogikh peremennykh summami Fure v metrike $L_p$”, Sib. matem. zh., 15:2 (1974), 395–412

[7] S. M. Nikolskii, “Neravenstva dlya tselykh funktsii konechnoi stepeni i ikh primenenie v teorii differentsiruemykh funktsii mnogikh peremennykh”, Tr. MIAN SSSR, 38, 1951, 244–278

[8] D. Jackson, “Certain problem of closest approximation”, Bull. Amer. Math. Soc., 39 (1933), 889–906 | DOI | Zbl

[9] E. S. Belinskii, “Dve ekstremalnye zadachi dlya trigonometricheskikh polinomov s zadannym chislom garmonik”, Matem. zametki, 49:1 (1991), 12–18 | MR | Zbl

[10] R. J. Nessel, G. Wilmes, “Nicolskii type inequalities for trigonometric polinomials and entire functions”, J. Austral. Math. Soc. Ser. A, 25 (1978), 7–18 | DOI | MR | Zbl

[11] V. A. Rodin, “Neravenstva dlya trigonometricheskikh polinomov s lakunami v prostranstvakh $L_p$”, Issledovaniya po teorii funktsii mnogikh peremennykh, Yaroslavl, 1990, 128–133 | Zbl

[12] E. S. Smailov, “O vliyanii geometricheskikh svoistv spektra mnogochlena na neravenstva raznykh metrik S. M. Nikolskogo”, Sib. matem. zh., 39:5 (1998), 1157–1163 | MR | Zbl

[13] M. B. Sikhov, “O neravenstvakh Dzheksona–Nikolskogo”, Vestnik KazGU. Ser. matem., mekh., inf., 2002, no. 2, 9–17 | MR