Geodesic
Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables
Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 429-442.

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We obtain order-sharp estimates of best approximations to the classes $B^r_{p,\theta}$ of periodic functions of several variables in the space $L_q$, $1\le p,q\le\infty$, by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes $B^{r_1}_{1,\theta}$ in the space $L_1$.
Keywords: class $B^r_{p,\theta}$ of periodic functions, trigonometric polynomial, hyperbolic cross, Fourier hyperbolic sum
Mots-clés : Bernoulli kernel, Valée-Poussin kernel, Fejér kernel. 
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A. S. Romanyuk. Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables. Matematičeskie zametki, Tome 87 (2010) no. 3, pp. 429-442. http://geodesic.mathdoc.fr/item/MZM_2010_87_3_a10/

[1] A. S. Romanyuk, “Priblizhenie klassov Besova periodicheskikh funktsii mnogikh peremennykh v prostranstve $L_q$”, Ukr. matem. zhurn., 43:10 (1991), 1398–1408 | MR | Zbl

[2] A. S. Romanyuk, “Priblizhenie klassov $B_{p,\theta}^r$ periodicheskikh funktsii mnogikh peremennykh lineinymi metodami i nailuchshie priblizheniya”, Matem. sb., 195:2 (2004), 91–116 | MR | Zbl

[3] A. S. Romanyuk, “Nailuchshie priblizheniya i poperechniki klassov periodicheskikh funktsii mnogikh peremennykh”, Matem. sb., 199:2 (2008), 93–114 | MR | Zbl

[4] O. V. Besov, “O nekotorom semeistve funktsionalnykh prostranstv. Teoremy vlozheniya i prodolzheniya”, Dokl. AN SSSR, 126:6 (1959), 1163–1165 | MR | Zbl

[5] S. M. Nikolskii, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR | Zbl

[6] P. I. Lizorkin, S. M. Nikolskii, “Prostranstva funktsii smeshannoi gladkosti s dekompozitsionnoi tochki zreniya”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam. Chast 13, Sbornik rabot, Tr. MIAN SSSR, 187, Nauka, M., 1989, 143–161 | MR | Zbl

[7] V. N. Temlyakov, “Priblizhenie funktsii s ogranichennoi smeshannoi proizvodnoi”, Tr. MIAN SSSR, 178, 1986, 3–113 | MR | Zbl

[8] S. M. Nikolskii, “Neravenstva dlya tselykh funktsii konechnoi stepeni i ikh primenenie v teorii differentsiruemykh funktsii mnogikh peremennykh”, Sbornik statei. Posvyaschaetsya akademiku Ivanu Matveevichu Vinogradovu k ego 60-letiyu, Tr. MIAN SSSR, 38, Izd-vo AN SSSR, M., 1951, 244–278 | MR | Zbl

[9] V. N. Temlyakov, Approximation of Periodic Functions, Comput. Math. Anal. Ser., Nova Science Publ., Commack, NY, 1993 | MR | Zbl

[10] V. N. Temlyakov, “Greedy algorithm and $m$-term trigonometric approximation”, Constr. Approx, 14:4 (1998), 569–587 | DOI | MR | Zbl