zbMATH — the first resource for mathematics

The ortho-diameters of Nikol’skii and Besov classes in the Lorentz spaces. (English. Russian original) Zbl 1189.46019
Russ. Math. 53, No. 2, 21-29 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 2, 25-33 (2009).
V. N. Temlyakov has introduced the notion of the ortho-diameter of functional classes and he has obtained estimates for ortho-diameters of Sobolev and Nikol’skii classes in Lebesgue spaces [Sov. Phys. Dokl. 267, 314–317 (1982; Zbl 0524.41013); “Approximation of functions with bounded mixed derivative” (Trudy Matematicheskogo Instituta im. V. A. Steklova 178; Moskva: “Nauka”) (1986; Zbl 0625.41028)]. This research has been developed in [D. Zung, Math. USSR, Sb. 59, 247–267 (1988); translation from Mat. Sb., Nov. Ser. 131(173), No. 2(10), 251–271 (1986; Zbl 0634.42005)], E. M. Galeev [Math. Notes 43, No. 2, 110–118 (1988); translation from Mat. Zametki 43, No. 2, 197–211 (1988; Zbl 0659.42008)] and N. N. Pustovoitov [Izv. Math. 64, No. 1, 121–141 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 1, 123–144 (2000; Zbl 1007.42004)]. The aim of the paper under review is to estimate the ortho-diameters of Nikol’skij and Besov classes in norms of anisotropic Lorentz spaces.

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
41A30 Approximation by other special function classes
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
Full Text: DOI
[1] A. P. Blozinski, ”Multivariate Rearrangements and Banach Function Spaces with Mixed Norms,” Trans. Amer. Math. Soc. 263, 149–167 (1981). · Zbl 0462.46020 · doi:10.1090/S0002-9947-1981-0590417-X
[2] E. D. Nursultanov, ”About Coefficients of Multiple Fourier Series from L p-Spaces,” Izvestiya Ross. Akad. Nauk, Ser. Matem. 64(1), 95–122 (2000). · doi:10.4213/im275
[3] S. M. Nikol’skii, Approximation of Multivariable Functions and Inclusion Theorems (Nauka, Moscow, 1977) [in Russian].
[4] P. I. Lizorkin and S. M. Nikol’skii, ”Functions Spaces of Mixed Smoothness from a Decomposition Point of. View,” Trudy MIAN 187, 143–161 (1989).
[5] T. I. Amanov, Spaces of Differentiable Functions with Dominating Mixed Derivative (Nauka, Alma-Ata, 1976) [in Russian].
[6] V. N. Temlyakov, ”Ortho-Diameters of Some Classes of Functions of Several Variables,” Sov. Phys. Dokl. 267, 314–317 (1982).
[7] V. N. Temlyakov, ”Approximation of Functions with Bounded Mixed Derivative,” Trudy MIAN SSSR 178 (1986). · Zbl 0625.41028
[8] Din’ Zung, ”Approximation of Multivariable Functions by Trigonometric Polynomials on a Torus,” Matem. Sborn. 131(2), 251–271 (1986).
[9] E. M. Galeev, ”Orders of the Orthoprojection Widths of Classes of Periodic Functions of One and of Several Variables,” Matem. Zametki 43(2), 197–211 (1988). · Zbl 0659.42008
[10] N. N. Pustovoitov, ”Ortho-Diameters of Some Classes of Periodic Two-Variable Functions with a Given Majorant of Mixed Modules of Continuity,” Izv. Ross. Aka. Nauk, Ser. Matem. 64, 124–144 (2000).
[11] G. A. Akishev, ”Inequalities of Different Metrics for Polynomials and Their Applications,” in Proceedings of the 10th Inter-University Conf., Almaty, October 7–9, 2004 (Almaty, 2005), pp. 53–60.
[12] G. A. Akishev, ”Orders of Approximation of Besov Classes by Trigonometric Polynomials,” Vestnik KarGU, Ser. Matem., No. 3, 9–16 (2004).
[13] G. A. Akishev, ”Orders of Approximation of Functional Classes in the Marcinkiewicz Space,” Matem. Zhurnal (Almaty) 4, 10–19 (2004). · Zbl 1133.41315
[14] V. A. Rodin, ”The Jackson and Nikol’skii Inequalities for Trigonometric Polynomials in a Symmetric Space,” in Proceedings of the 7th Winter School, Drogobych, 1974, pp. 133–139.
[15] L. A. Sherstneva, ”The Nikol’skii Inequalities for Trigonometric Polynomials in the Lorentz Spaces,” Vestn. MGU, Ser. Matem., Mekhan., No. 4, 75–79 (1984). · Zbl 0563.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.