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 Mat. Sb., 2004, Volume 195, Number 2, Pages 91–116 (Mi msb801)

Approximability of the classes $B_{p,\theta}^r$ of periodic functions of several variables by linear methods and best approximations

A. S. Romanyuk

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: Several questions of the approximability by linear methods of the Besov classes $B_{1,\theta}^r$ and $B_{p,\theta}^r$ of periodic functions of several variables, $1\leqslant p<\infty$, are considered alongside their best approximations in the spaces $L_1$ and $L_\infty$, respectively. Taken for approximation aggregates are trigonometric polynomials with spectrum in the step hyperbolic cross. Sharp (in order) estimates of the deviations of step hyperbolic Fourier sums on the classes $B_{p,\theta}^r$, $1\leqslant p<\infty$, in the $L_\infty$ space are also obtained.

DOI: https://doi.org/10.4213/sm801

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English version:
Sbornik: Mathematics, 2004, 195:2, 237–261

Bibliographic databases:

UDC: 517.5
MSC: 41A35, 46E35

Citation: A. S. Romanyuk, “Approximability of the classes $B_{p,\theta}^r$ of periodic functions of several variables by linear methods and best approximations”, Mat. Sb., 195:2 (2004), 91–116; Sb. Math., 195:2 (2004), 237–261

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb/v195/i2/p91

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. S. Romanyuk, “Best Trigonometric Approximations for Some Classes of Periodic Functions of Several Variables in the Uniform Metric”, Math. Notes, 82:2 (2007), 216–228
2. A. S. Romanyuk, “Best approximations and widths of classes of periodic functions of several variables”, Sb. Math., 199:2 (2008), 253–275
3. S. A. Stasyuk, “Best Approximations of Periodic Functions of Several Variables from the Classes $B^\Omega_{p,\theta}$”, Math. Notes, 87:1 (2010), 102–114
4. A. S. Romanyuk, “Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables”, Math. Notes, 87:3 (2010), 403–415
5. Sickel W., Ullrich T., “Spline interpolation on sparse grids”, Appl Anal, 90:3–4 (2011), 337–383
6. Pomahiok A.C., “Diameters and best approximation of the classes B-p(r) of periodic functions of several variables”, Anal Math, 37:3 (2011), 181–213
7. N. N. Pustovoitov, “Approximation of periodic functions in the classes $H_q^\Omega$ by linear methods”, Sb. Math., 203:1 (2012), 88–110
8. S. A. Stasyuk, “Nailuchshee priblizhenie periodicheskikh funktsii neskolkikh peremennykh iz klassov $MB^\omega_{p,\theta}$ v ravnomernoi metrike”, Tr. IMM UrO RAN, 18, no. 4, 2012, 258–266
9. Konograi A.F., “Approximation of Classes Bp, Theta (Omega) of Periodic Functions in Several Variables by Linear Methods”, Anal. Math., 39:3 (2013), 217–233
10. Yanchenko S.Ya., “Approximation of Functions From the Classes S (R) (P, Theta) B in the Uniform Metric”, Ukr. Math. J., 65:5 (2013), 771–779
11. Romanyuk A.S., “Estimation of the Entropy Numbers and Kolmogorov Widths for the Nikol'skii–Besov Classes of Periodic Functions of Many Variables”, Ukr. Math. J., 67:11 (2016), 1739–1757
12. Yanchenko S.Ya., “Order Estimates For the Approximating Characteristics of Functions From the Classes With a Given Majorant of Mixed Modules of Continuity in the Uniform Metric”, Ukr. Math. J., 68:12 (2017), 1975–1985
13. Romanyuk A.S., “Trigonometric and Linear Widths For the Classes of Periodic Multivariate Functions”, Ukr. Math. J., 69:5 (2017), 782–795
14. Romanyuk A.S., “Entropy Numbers and Widths For the Nikol'Skii-Besov Classes of Functions of Many Variables in the Space l-Infinity”, Anal. Math., 45:1 (2019), 133–151
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