Geodesic
On Equivalent Normings in the Spaces $H_p^r$ on Compact Homogeneous Spaces
Matematičeskie zametki, Tome 68 (2000) no. 6, pp. 898-909 
S. S. Platonov. On Equivalent Normings in the Spaces $H_p^r$ on Compact Homogeneous Spaces. Matematičeskie zametki, Tome 68 (2000) no. 6, pp. 898-909. http://geodesic.mathdoc.fr/item/MZM_2000_68_6_a10/
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     title = {On {Equivalent} {Normings} in the {Spaces} $H_p^r$ on {Compact} {Homogeneous} {Spaces}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $M$ be an arbitrary compact Riemannian symmetric space of rank 1. The function spaces $H_p^r$ of Nikol'skii type were introduced earlier by means of averaged differences along geodesics. In the present paper we give an equivalent description of these spaces and norms in them by using the Laplace–Beltrami operator. The results obtained generalize the results of Nikol'skii and Lizorkin on the spaces $H_p^r$ over the sphere.

[1] Nikolskii S. M., Lizorkin P. I., “Approksimatsiya funktsii na sfere”, Izv. AN SSSR. Ser. matem., 51:3 (1987), 635–651 | Zbl

[2] Nikolskii S. M., Lizorkin P. I., “Ob ekvivalentnykh normirovkakh gëlderovykh prostranstv na sfere”, Issledovaniya po teorii priblizheniya funktsii, Ufa, 1987, 7–25 | Zbl

[3] Besse A., Mnogoobraziya s zamknutymi geodezicheskimi, Mir, M., 1981

[4] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964 | Zbl

[5] Tikhomirov V. M., “Teoriya priblizhenii”, Itogi nauki i tekhniki. Sovrem. probl. matem. Fund. napravleniya, 14, VINITI, M., 1987, 103==260 | Zbl

[6] Ragozin D. L., “Polynomial approximation on compact manifolds and homogeneous spaces”, Trans. Amer. Math. Soc., 150 (1971), 41–53 | DOI | MR

[7] Kamzolov A. I., “Ob interpolyatsionnoi formule Rissa i neravenstve Bernshteina dlya funktsii na odnorodnykh prostranstvakh”, Matem. zametki, 15:6 (1974), 967–978 | MR | Zbl

[8] Kamzolov A. I., “O neravenstve Bora–Favara dlya funktsii na kompaktnykh simmetricheskikh prostranstvakh ranga 1”, Matem. zametki, 33:2 (1983), 187–193 | MR | Zbl

[9] Ivanov V. A., “O neravenstvakh Bernshteina–Nikolskogo i Favara na kompaktnykh odnorodnykh prostranstvakh ranga 1”, UMN, 38:3 (1983), 179–180 | MR | Zbl

[10] Platonov S. S., “Priblizheniya funktsii na kompaktnykh simmetricheskikh prostranstvakh ranga 1”, Matem. sb., 188:5 (1997), 113–130 | MR | Zbl

[11] Khelgason S., Gruppy i geometricheskii analiz, Mir, M., 1987

[12] Bavinck H., “A special classes of Jacobi series and some applications”, J. Math. Anal. Appl., 37 (1972), 767–797 | DOI | MR | Zbl