Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Nikol’skii–Besov-type classes of periodic functions of one and many variables in the space B∞,1.
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A. S. Romanyuk, “Entropy numbers and widths for the classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Ukr. Mat. Zh., 68, No. 10, 1403–1417 (2016); English translation: Ukr. Math. J., 68, No. 10, 1620–1636 (2017).
A. S. Romanyuk and V. S. Romanyuk, “Approximating characteristics of the classes of periodic multivariate functions in the space B∞,1,” Ukr. Mat. Zh., 71, No. 2, 271–282 (2019); English translation: Ukr. Math. J., 71, No. 2, 308–321 (2019).
A. S. Romanyuk and V. S. Romanyuk, “Estimation of some approximating characteristics of the classes of periodic functions of one and many variables,” Ukr. Mat. Zh., 71, No. 8, 1102–1115 (2019); English translation: Ukr. Math. J., 71, No. 8, 1257–1272 (2020).
A. S. Romanyuk and V. S. Romanyuk, “Approximating characteristics and properties of operators of best approximation of classes of functions from the Sobolev and Nikol’skii–Besov classes,” Ukr. Mat. Visn., 17, No. 3, 372–395 (2020).
A. S. Romanyuk and S. Ya. Yanchenko, “Estimates of approximating characteristics and the properties of operators of best approximation for the classes of periodic functions in the space B1,1,” Ukr. Mat. Zh., 73, No. 8, 1102–1119 (2021); English translation: Ukr. Math. J., 73, No. 8, 1278–1298 (2022).
M. V. Hembars’kyi and S. B. Hembars’ka, “Widths of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of many variables in the space B1,1,” Ukr. Mat. Visn., 15, No. 1, 43–57 (2018).
M. V. Hembars’kyi, S. B. Hembars’ka, and K. V. Solich, “Best approximations and widths of the classes of periodic functions of one and many variables in the space B∞,1,” Mat. Stud., 51, No. 1, 74–85 (2019).
O. V. Fedunyk-Yaremchuk, M. V. Hembarskyi, and S. B. Hembarska, “Approximative characteristics of the Nikol’skii–Besov-type classes of periodic functions in the space B∞,1,” Carpathian Math. Publ., 12, No. 2, 376–391 (2020).
D. Ding, V. N. Temlyakov, and T. Ullrich, Hyperbolic Cross Approximation, Birkh¨auser (2018).
S. N. Bernstein, Collected Works, Vol. 2. Constructive Theory of Functions (1931–1953) [in Russian], Akad. Nauk SSSR, Moscow (1954).
S. B. Stechkin, “On the order of the best approximation of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 15, 219–242 (1951).
N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two adjoint functions,” Tr. Mosk. Mat. Obshch., 5, 483–522 (1956).
Y. Sun and H. Wang, “Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness,” Tr. Mat. Inst. Ros. Akad. Nauk, 219, 356–377 (1997).
T. I. Amanov, “Representation and embedding theorems for the function spaces \( {S}_{p,\theta}^{(r)} \) B(ℝn) and \( {S}_{p,\theta}^{(r)} \) (0 ≤ xj ≤ 2τ; j = 1, . . . ,n),” Tr. Mat. Inst. Akad. Nauk SSSR, 77, 5–34 (1965).
P. I. Lizorkin and S. M. Nikol’skii, “Spaces of functions of mixed smoothness from the decomposition point of view,” Tr. Mat. Ins. Akad. Nauk SSSR, 187, 143–161 (1989).
S. M. Nikol’skii, “Functions with predominant mixed derivative satisfying the multiple Hölder condition,” Sib. Mat. Zh., 4, No. 6, 1342–1364 (1963).
N. N. Pustovoitov, “Representation and approximation of periodic functions of many variables with given mixed modulus of continuity,” Anal. Math., 20, 35–48 (1994).
É. S. Belinskii, “Approximation by a ‘floating’ system of exponents on the classes of periodic functions with bounded mixed derivative,” in: Investigations into the Theory of Functions of Many Real Variables [in Russian], Yaroslavl. Univ., Yaroslavl (1988), pp. 16–33.
A. S. Romanyuk, “Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials,” Ukr. Mat. Zh., 48, No. 1, 80–89 (1996); English translation: Ukr. Math. J., 48, No. 1, 90–100 (1996).
A. S. Romanyuk, “Approximation of classes of periodic functions of many variables,” Mat. Zametki, 71, No. 1, 109–121 (2002).
A. S. Romanyuk, “Bilinear and trigonometric approximations of the Besov classes \( {B}_{p,\theta}^r \) of periodic functions of many variables,” Izv. Ros. Akad. Nauk, Ser. Mat., 70, No. 2, 69–98 (2006).
A. S. Romanyuk, “Best trigonometric approximations of the classes of periodic functions of many variables in uniform metric,” Mat. Zametki, 82, No. 2, 247–261 (2007).
A. S. Romanyuk, Approximate Characteristics of the Classes of Periodic Functions of Many Variables [in Russian], Proc. of the Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv (2012).
A. F. Konohrai and S. A. Stasyuk, “Best orthogonal trigonometric approximations of the classes \( {B}_{p,\theta}^{\Omega} \) of periodic functions of many variables,” in: Proc. of the Institute of Mathematics, National Academy of Sciences of Ukraine [in Ukrainian], Kyiv, 4, No. 1 (2007), pp. 151–171.
V. N. Temlyakov, Approximation of Periodic Functions, Nova Science Publ., New York (1993).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 6, pp. 772–783, June, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i6.7070.
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Hembars’ka, S.B., Zaderei, P.V. Best Orthogonal Trigonometric Approximations of the Nikol’skii–Besov-Type Classes of Periodic Functions in the Space B∞,1. Ukr Math J 74, 883–895 (2022). https://doi.org/10.1007/s11253-022-02115-0
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DOI: https://doi.org/10.1007/s11253-022-02115-0