We establish the exact-order estimates for the best bilinear approximations of the Nikol’skii–Besov classes \( {B}_{\mathrm{p},\theta}^{\mathrm{r}} \) of periodic functions of several variables. We also determine the orders of singular numbers for the integral operators with kernels from the classes \( {B}_{\mathrm{p},\theta}^{\mathrm{r}} \).
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V. N. Temlyakov, “Estimates for the best bilinear 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. Ros. Akad. Nauk, 194, 229–248 (1992).
A. S. Romanyuk, “Bilinear and trigonometric approximations of the Besov classes \( {B}_{\mathrm{p},\theta}^{\mathrm{r}} \) 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 the 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).
A. S. Romanyuk and V. S. Romanyuk, “Best bilinear approximations of functions from Nikol’skii–Besov classes,” Ukr. Mat. Zh., 64, No. 5, 685–697 (2012); English translation: Ukr. Math. J., 64, No. 5, 781–796 (2012).
A. S. Romanyuk and V. S. Romanyuk, “Best bilinear approximations for the classes of functions of many variables,” Ukr. Mat. Zh., 65, No. 12, 1681–1699 (2013); English translation: Ukr. Math. J., 65, No. 12, 1862–1882 (2014).
A. S. Romanyuk, “Best trigonometric and bilinear approximations for the classes of functions of many variables,” Mat. Zametki, 94, No. 3, 401–415 (2013).
S. M. Nikol’skii, Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).
O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1975).
T. I. Amanov, Spaces of Differentiable Functions with Predominant Mixed Derivative [in Russian], Nauka, Alma-Ata (1976).
A. S. Romanyuk, Approximate Characteristics of the Classes of Periodic Functions of Many Variables [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2012).
E. Schmidt, “Zur Theorie der linearen und nichlincaren Integralgleichungen. I,” Math. Ann., 63, 433–476 (1907).
M. Sh. Birman and M. Z. Solomyak, “On the estimates for singular numbers of integral operators. II,” Vestn. Leningrad. Univ., 3, No. 13, 21–28 (1967).
M. Sh. Birman and M. Z. Solomyak, “Estimation of the singular numbers of integral operators,” Usp. Mat. Nauk, 32, No. 1, 17–84 (1977).
F. Smithies, “The eigenvalues and singular values of integral equations,” Proc. London Math. Soc. (2), 43, 255–279 (1937).
J. A. Cochran, “Summability of the singular values of L 2 kernels. Analogies with Fourier series,” Enseign. Math., 22, No. 1-2, 141–157 (1976).
A. Pich, “Eigenvalues of integral operators,” Dokl. Akad. Nauk SSSR, 247, No. 6, 1324–1327 (1979).
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. 68, No. 9, pp. 1240–1250, September, 2016.
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Romanyuk, A.S., Romanyuk, V.S. Estimation of the Best Linear Approximations for the Classes \( {B}_{\mathrm{p},\theta}^{\mathrm{r}} \) and Singular Numbers of the Integral Operators. Ukr Math J 68, 1424–1436 (2017). https://doi.org/10.1007/s11253-017-1304-z
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DOI: https://doi.org/10.1007/s11253-017-1304-z