Geodesic
Extension of Functions from Isotropic Nikol'skii–Besov Spaces and Their Approximation together with Derivatives
Matematičeskie zametki, Tome 108 (2020) no. 5, pp. 714-724 Cet article a éte moissonné depuis la source Math-Net.Ru

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Isotropic Nikol'skii–Besov spaces with norms whose definition, instead of the modulus of continuity of certain order of partial derivatives of functions of fixed order, involves the "$L_p$-averaged" modulus of continuity of functions of the corresponding order are studied. We construct continuous linear mappings of such spaces of functions given on bounded domains of $(1,\dots,1)$-type (in the wide sense) to the usual isotropic Nikol'skii–Besov spaces on $\mathbb{R}^d$, which are extension operators of these functions; this implies the coincidence of these spaces on the domains mentioned above. It is established that any bounded domain in $\mathbb{R}^d$ with Lipschitz boundary is a domain of $(1,\dots,1)$-type (in the wide sense). We also establish the weak asymptotics of approximation characteristics related to the problem of the recovery of functions together with their derivatives from the values of these functions at a given number of points, to the Stechkin problem for the differentiation operator, and to the problem of describing the asymptotics of widths for isotropic classes of Nikol'skii and Besov in these domains.
Mots-clés : isotropic Nikol'skii–Besov spaces
Keywords: extension of functions, equivalent norms, recovery of functions, approximation of an operator, width. 
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S. N. Kudryavtsev. Extension of Functions from Isotropic Nikol'skii–Besov Spaces and Their Approximation together with Derivatives. Matematičeskie zametki, Tome 108 (2020) no. 5, pp. 714-724. http://geodesic.mathdoc.fr/item/MZM_2020_108_5_a6/

[1] O. V. Besov, “Prodolzhenie funktsii za predely oblasti s sokhraneniem differentsialno-raznostnykh svoistv v $L_p$”, Matem. sb., 66 (108):1 (1965), 80–96 | MR | Zbl

[2] Yu. A. Brudnyi, “Teorema prodolzheniya dlya odnogo semeistva funktsionalnykh prostranstv”, Issledovaniya po lineinym operatoram i teorii funktsii. VI, Zap. nauchn. sem. LOMI, 56, Izd-vo «Nauka», Leningrad. otd., L., 1976, 170–173 | MR | Zbl

[3] V. I. Burenkov, “O prodolzhenii funktsii s sokhraneniem polunormy”, Dokl. AN SSSR, 228:4 (1976), 779–782 | MR | Zbl

[4] V. N. Konovalov, “Prodolzhenie funktsii mnogikh peremennykh s sokhraneniem differentsialno-raznostnykh svoistv”, Ukr. matem. zhurn., 36:3 (1984), 304–308 | Zbl

[5] S. N. Kudryavtsev, “Nailuchshaya tochnost vosstanovleniya funktsii konechnoi gladkosti po ikh znacheniyam v zadannom chisle tochek”, Izv. RAN. Ser. matem., 62:1 (1998), 21–58 | DOI | MR | Zbl

[6] S. N. Kudryavtsev, “Zadacha Stechkina dlya operatora chastnogo differentsirovaniya na klassakh funktsii konechnoi gladkosti”, Matem. zametki, 67:1 (2000), 77–86 | DOI | MR | Zbl

[7] S. N. Kudryavtsev, “Poperechniki klassov funktsii konechnoi gladkosti v prostranstvakh Soboleva”, Matem. zametki, 77:4 (2005), 535–539 | DOI | MR | Zbl

[8] S. N. Kudryavtsev, “Prodolzhenie funktsii iz neizotropnykh prostranstv Nikolskogo–Besova i priblizhenie ikh proizvodnykh”, Izv. RAN. Ser. matem., 82:5 (2018), 78–130 | DOI | Zbl

[9] S. N. Kudryavtsev, Extending Functions From Isotropic Nikolskii–Besov Spaces and Their Approximating With Derivatives, 2019, arXiv: 1805.10625v2

[10] S. M. Nikolskii, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR | Zbl

[11] S. N. Kudryavtsev, Extending Functions From Nonisotropic Nikolskii–Besov Spaces and Approximating Their Derivatives (With Supplement), 2018, arXiv: 1703.09734v3

[12] V. M. Tikhomirov, Nekotorye voprosy teorii priblizhenii, Izd-vo Mosk. un-ta, M., 1976 | MR