We obtain exact-order estimates for the best bilinear approximations of Nikol’skii–Besov classes in the functional spaces L q (π 2d ).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
O. V. Besov, “Investigation of one family of functional spaces in connection with imbedding and continuity theorems,” Tr. Mat. Inst. Akad. Nauk SSSR, 60, 42–61 (1961).
S. M. Nikol’skii, “Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of many variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 38, 244–278 (1951).
P. I. Lizorkin, “Generalized Hölder spaces \( B_{{p,\theta}}^{(r) } \) and their relation to Sobolev spaces \( L_p^{(r) } \),” Sib. Mat. Zh., 9, No. 5, 1127–1152 (1968).
E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. I,” Math. Ann., 63, 433–476 (1907).
M. Sh. Birman and M. Z. Solomyak, “Estimates for singular numbers of integral operators,” Usp. Mat. Nauk, 32, No. 1, 17–84 (1977).
N. V. Miroshin and V. V. Khromov, “On one problem of the best approximation of functions of many variables,” Mat. Zametki, 32, No. 5, 721–727 (1982).
R. S. Ismagilov, “Widths of sets in linear normed spaces and approximation of functions by trigonometric polynomials,” Usp. Mat. Nauk, 29, No. 3, 161–178 (1974).
C. A.Micchelli and A. Pinkus, “Some problems in the approximation of functions of two variables and n-widths of integral operators,” J. Approxim. Theory, 24, 51–77 (1978).
E. M. Cheney, “The best approximation of multivariate functions by combinations of univariate ones,” J. Approxim. Theory, Ser. IV, 1–26 (1983).
V. N. Temlyakov, “Approximation of periodic functions of many variables by combinations of functions depending on a smaller number of variables,” Tr. Mat. Inst. Akad. Nauk SSSR, 173, 243–252 (1986).
V. N. Temlyakov, “Estimates for the best approximations of periodic functions,” Tr. Mat. Inst. Akad. Nauk SSSR, 181, 250–267 (1988).
V. N. Temlyakov, “Bilinear approximation and applications,” Tr. Mat. Inst. Akad. Nauk SSSR, 187, 191–215 (1989).
V. N. Temlyakov, “Bilinear approximation and related problems,” Tr. Mat. Inst. Akad. Nauk SSSR, 194, 229–248 (1992).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivative,” Tr. Mat. Inst. Akad. Nauk SSSR, 178, 1–112 (1986).
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 and V. S. Romanyuk, “Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables,” Ukr. Mat. Zh., 62, No. 4, 536–551 (2010); English translation: Ukr. Math. J., 62, No. 4, 612–629 (2010).
M.-B. A. Babaev, “Approximation of Sobolev classes of functions by sums of products of functions of a smaller number of variables,” Mat. Zametki, 48, No. 6, 10–21 (1990).
M.-B. A. Babaev, “On the order of approximation of the Sobolev class \( W_q^r \) by bilinear forms,” Mat. Zametki, 182, No. 1, 122–129 (1991).
B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow (1984).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 685–697, May, 2012.
Rights and permissions
About this article
Cite this article
Romanyuk, A.S., Romanyuk, V.S. Best bilinear approximations of functions from Nikol’skii–Besov classes. Ukr Math J 64, 781–796 (2012). https://doi.org/10.1007/s11253-012-0678-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0678-1