We establish an exact-order estimate for the deviation of partial Fourier sums of periodic functions of many variables from the classes \( B_{{p,\theta}}^{\Omega } \) in the space L p with p = 1, ∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. L. Gol’dman, “Imbedding theorems for the anisotropic Nikol’skii–Besov spaces with moduli of continuity of the general form,” Tr. Mat. Inst. Akad. Nauk SSSR, 170, 84–106 (1984).
G. A. Kalyabin, “Imbedding theorems for generalized Besov and Liouville spaces,” Dokl. Akad. Nauk SSSR, 232, No. 6, 1245–1248 (1977).
Liu Yongping and Xu Cuiqiao, “The infinite-dimensional widths and optimal recovery of generalized Besov classes,” J. Complexity, 18, 815–832 (2002).
Xu Guiqiao, “The n-widths for generalized periodic Besov classes,” Acta Math. Sci., Ser. B, Engl. Ed., 25B, No. 4, 663–671 (2005).
S. A. Stasyuk, “Approximation of the classes \( B_{{1,\theta}}^{\omega } \) of periodic functions in the space L1 by Fourier sums,“ in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 7, No. 1 (2010), pp. 338–344.
S. A. Stasyuk, “Approximation of the classes \( B_{{p,\theta}}^{\omega } \) of periodic functions of many variables by polynomials with spectra in cubic domains,” Mat. Stud., 35, No. 1, 66–73 (2011).
S. P. Voitenko, “Best M-term trigonometric approximations of the classes \( B_{{p,\theta}}^{\Omega } \) of periodic functions of many variables,” Ukr. Mat. Zh., 61, No. 9, 1189–1199 (2009); English translation: Ukr. Math. J., 61, No. 9, 1401–1416 (2009).
S. P. Voitenko, “Best orthogonal trigonometric approximations of the classes \( B_{{p,\theta}}^{\Omega } \) of periodic functions of many variables,” Ukr. Mat. Zh., 61, No. 11, 1473–1484 (2009); English translation: Ukr. Math. J., 61, No. 11, 1728–1742 (2009).
K. V. Solich, “Bilinear approximations of the classes \( B_{{p,\theta}}^{\Omega } \) of periodic functions of many variables,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], 7, No. 1 (2010), pp. 325–337.
S. N. Bernstein, Constructive Theory of Functions (1931–1953). Collected Works [in Russian], Vol. 2, Izd. Akad. Nauk SSSR, Moscow (1954).
N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugate functions,” Tr. Mosk. Mat. Obshch., 5, 483–522 (1956).
S. M. Nikol’skii, “Inequalities for the entire functions of finite power and their application to the theory of differentiable functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).
O. V. Besov, “On a family of functional spaces. Theorems of imbedding and extension,” Dokl. Akad. Nauk SSSR, 126, No. 6, 1163–1165 (1959).
V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).
V. N. Temlyakov, Approximation of Periodic Functions, Nova Science, New York (1993).
G. M. Fikhtengol’ts, A Course in Differential and Integral Calculus [in Russian], Vol. 3, Nauka, Moscow (1976).
A. S. Romanyuk, “Approximation of the classes \( B_{{p,\theta}}^r \) of periodic functions of one and many variables,” Mat. Zametki, 87, No. 3, 429–442 (2010).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 9, pp. 1204–1213, September, 2012.
Rights and permissions
About this article
Cite this article
Myronyuk, V.V. Approximation of the classes \( {\boldsymbol{B}}_{{ {\boldsymbol{p}},{\boldsymbol{\theta}}}}^{\Omega } \) of periodic functions of many variables by Fourier sums in the space L p with p = 1, ∞. Ukr Math J 64, 1370–1381 (2013). https://doi.org/10.1007/s11253-013-0722-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0722-9