Geodesic
Embedding theorems and best approximations
Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 213-224 
È. A. Storozhenko. Embedding theorems and best approximations. Sbornik. Mathematics, Tome 26 (1975) no. 2, pp. 213-224. http://geodesic.mathdoc.fr/item/SM_1975_26_2_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

We establish necessary and sufficient conditions, in terms of best approximations, for a function in $L^p(0,2\pi)$ ($0) to belong to $L^q(0,2\pi)$ ($q). The proofs depend on the properties of equimeasurable functions, which were applied by Ul'yanov in the theory of the embedding of certain classes $H_p^\omega$ for $p\geqslant1$ (RZhMat., 1969, 2B109). We also obtain some relationships among moduli of continuity in different metrics, which let us generalize results of Hardy and Littlewood (Math. Z., 28, No 4 (1928), 612–634) to the case $0 and prove converses for nonincreasing functions. Bibliography: 11 titles.

[1] P. L. Ulyanov, “Vlozhenie nekotorykh klassov funktsii $H_p^{\omega}$”, Izv. AN SSSR, seriya matem., 32 (1968), 649–686

[2] P. L. Ulyanov, “Teoremy vlozheniya i sootnosheniya mezhdu nailuchshimi priblizheniyami (modulyami nepreryvnosti) v raznykh metrikakh”, Matem. sb., 81(123) (1970), 104–131

[3] E. A. Storozhenko, “Neobkhodimye i dostatochnye usloviya dlya vlozheniya nekotorykh klassov funktsii”, Izv. AN SSSR, seriya matem., 37 (1973), 386–398 | Zbl

[4] K. I. Oskolkov, S. A. Telyakovskii, “K otsenkam P. L. Ulyanova dlya integralnykh modulei nepreryvnosti”, Izv. AN ArmSSR, Matematika, 6:5 (1971), 406–411 | MR

[5] M. P. Korniichuk, “Pro spividnoshennya mizh modulyami neperervnosti funktsii ta ikh perestavlen”, DAN URSR, 1973, no. 9, 794–796

[6] Adriano Garsia, “Combinatorial inequalities and smoothness of functions”, Notices Amer. Math. Soc., A-115:151 (1974)

[7] E. A. Storozhenko, “O nekotorykh teoremakh vlozheniya”, Matem. zametki, 17 (1975)

[8] A. F. Timan, Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, Moskva, 1960

[9] S. B. Stechkin, “O poryadke nailuchshikh priblizhenii nepreryvnykh funktsii”, Izv. AN SSSR, seriya matem., 15 (1951), 219–242 | Zbl

[10] G. Khardi, D. Littlvud, G. Polia, Neravenstva, IL, Moskva, 1948

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