Yanchenko S. Extremal problems of approximation theory of classes of smooth functions of one and many variables

Українська версія

Thesis for the degree of Doctor of Science (DSc)

State registration number

0524U000281

Thesis Registration Form

0524U000281.pdf

Applicant for

Specialization

  • 01.01.01 - Математичний аналіз

17-09-2024

Specialized Academic Board

Д26.206.01

Institute of Mathematics of the National Academy of Sciences of Ukraine

Essay

The thesis is devoted to solving a wide range of extremal problems of the theory of functions related to the approximation of functional classes (Sobolev classes, Nikol'skii–Besov classes, and also their generalizations) by various methods and finding optimal ones among them in one sense or another. The direction of research about the approximation of classes of functions, that are endowed with some differential properties, which are described in terms of smoothness modules or a certain differentiation operation, has been gaining popularity and actively developing since the 1930s years of the 20th century and this is caused, in our opinion, by two circumstances. On the one hand, establishing estimates of the approximation characteristics of functional classes in unexplored situations introducing requires the creation of new methods and approaches, which plays an important role for the development of the theory of approximation itself, and on the other hand, they have practical applications in some related fields of science and technology. In the thesis considers the problems of finding estimates of: exact upper bounds of the values of the best orthogonal trigonometric approximations of the functions from the mentioned classes, the best approximations of the functions using entire functions of the exponential type, with the support of their Fourier transform in sets of the finite Lebesgue measure (step hyperbolic cross, d-dimensional “parallelepipeds”), approximations of functions from the corresponding classes by their step-hyperbolic Fourier sums, M-dimensional Kolmogorov widths, entropy numbers, etc. The vast majority of the results of the thesis are presented in the form of exact-order estimates of the mentioned characteristics of linear on nonlinear approximation. The deepened interest in problems of nonlinear approximation (of the best orthogonal trigonometric approximations, of the best M-term trigonometric approximations, etc.) is due, first of all, to the fact that in many cases nonlinear approximation methods turned out to be more effective in comparison with linear methods. This direction of research is connected with the works of such famous mathematicians: E.S. Belinskiy, R. DeVora, W. Sickel, D. Dung, R.S. Ismagilov, T. Kuhn, S.B. Kashin, V.E. Mayorov, A.S. Romanyuk, A.S. Serdyuk, O.I. Stepanets, V.M. Temlyakov, H. Tribel, T. Ullrich, Wang Heping, Sun Yongsheng. In particular, due to the modification and improvement of discretization and decomposition methods, a number of new important scientific results were obtained for the first time: 1) For periodic functions from the Nikol'skii–Besov classes with dominating mixed smoothness, d≥2, in the metric of the space of quasi-continuous functions, we have found exect-order estimates of the M-dimensional Kolmogorov widths and entropy numbers. 2) We obtained exact-order estimates of the best orthogonal trigonometric approximations, orthotransverses, and approximation characteristics close to them of the Nikol'skii-Besov classes of periodic functions of one and many variables with dominating mixed smoothness in the Lebesgue subspaces B1,1(Td) and B∞,1(Td)). In some cases, we studied the behavior of the corresponding approximation characteristics of the Sobolev classes. 3) We investigated the approximation of functions from the classes with dominanting mixed smoothness, defined in R^d, and their generalizations, by entire functions of the exponential type with the supports of their Fourier transform in a step hyperbolic cross with and approximation by entire functions of a special forms. We found the exact-order estimates of these quantities. Also we showed that there are situations when estimates of approximation by entire functions of a special form have better order than the corresponding estimates for approximation by entire functions of the exponential type with the support of their Fourier transform in the step hyperbolic cross.

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Research papers

Romanyuk A. S., Yanchenko S.Ya. Estimates for the entropy numbers of the Nikol’skii–Besov classes of functions with mixed smoothness in the space of quasi-continuous functions. Math. Nachr. 2023, 296 (6), 2575–2587, https://doi.org/10.1002/mana.202100202.

Romanyuk A. S., Yanchenko S.Ya. Approximation of the classes of periodic functions of one and many variables from the Nikol’skii–Besov and Sobolev spaces. Ukrainian Math. J. 2022, 74 (6), 967 – 980, https://doi.org/10.1007/s11253-022-02110-5; translation of Ukrain. Mat. Zh. 2022, 74 (6), 844 – 855, https://doi.org/10.37863/umzh.v74i6.7141.

Romanyuk A. S., Yanchenko S.Ya. Kolmogorov widths of the Nikol’skii–Besov classes of periodic functions of many variables in the space of quasicontinuous functions. Ukrainian Math. J. 2022, 74 (2), 251 – 265, https://doi.org/10.1007/s11253-022-02061-x; translation of Ukrain. Mat. Zh. 2022, 74 (2), 220 – 232, https://doi.org/10.37863/umzh.v74i2.6932.

Romanyuk A. S., Yanchenko S.Ya. Estimates of approximating characteristics and the properties of the operators of best approximation for the classes of periodic functions in the space 𝐵1,1. Ukrainian Math. J. 2022, 73 (8), 1278 – 1298, https://doi.org/10.1007/s11253-022-01990-x; translation of Ukrain. Mat. Zh. 2021, 73 (8), 1102 – 1119, https://doi.org/10.37863/ umzh.v73i8.6755.

Yanchenko S.Ya., Radchenko O.Ya. Approximation of the Nikol’skii-Besov functional classes by entire functions of a special form. Carpathian Math. Publ. 2021, 13 (3), 851 – 861, https://doi.org/10.15330/cmp.13.3.851-861.

Yanchenko S.Ya., Radchenko O.Ya. Approximating characteristics of the Nikol’skii–Besov classes 𝑆𝑟1,θ𝐵(R𝑑). Ukrainian Math. J. 2020, 71 (10), 1608 – 1626, https://doi.org/10.1007/s11253-020-01734-9; translation of Ukrain. Mat. Zh. 2019, 71 (10), 1405 – 1421.

Yanchenko S.Ya. Approximation of the Nikol’skii–Besov functional classes by entire functions of a special form. Carpathian Math. Publ. 2020, 12 (1), 148 – 156, https://doi.org/10.15330/cmp.12.1.148-156.

Yanchenko S.Ya. Best approximation of the functions from anisotropic Nikol’skii–Besov classes defined in R𝑑. Ukrainian Math. J. 2018, 70(4), 661 – 670; translation of Ukrain. Mat. Zh. 2018, 70 (4), 574 – 582, https://doi.org/10.1007/s11253-018-1523-y.

Yanchenko S.Ya., Stasyuk S. A. Approximative characteristics of functions from the classes 𝑆Ω𝑝,θ𝐵 with a given majorant of mixed moduli of continuity. J. Math. Sci. (N. Y.) 2018, 235 (1), 103 – 115, https://doi.org/10.1007/s10958-018-4062-z; translated of Ukr. Mat. Visn. 2018, 15 (1), 132 – 148.

Yanchenko S.Ya. Order estimates of approximation characteristics of functions from the anisotropic Nikol’skii–Besov classes. J. of Math. Sci. (N. Y.) 2018, 234 (1), 98 – 105, https://doi.org/10.1007/s10958-018-3984-9; translated of Ukr. Mat. Visn. 2017, 14 (4), 595 – 604.

Yanchenko S.Ya. Order estimates for the approximative characteristics of functions from the classes 𝑆Ω𝑝,θ𝐵(R𝑑) with a given majorant of generalized mixed modules of smoothness in the uniform metric. Ukrainian Math. J. 2017, 68 (12), 1975 – 985, https://doi.org/10.1007/s11253-017-1342-6; translation of Ukrain. Mat. Zh. 2016, 68 (12), 1705 – 1714.

Stasyuk S. А., Yachenko S. Ya. Approximation of functions from Nikolskii–Besov type classes of generalized mixed smoothness. Anal. Math. 2015, 41 (4), 311 – 334, https://doi.org/10.1007/s10476-015-0305-0.

Yanchenko S.Ya. Approximation of functions from the isotropic Nikol’skii–Besov classes in the uniform and integral metrics. Ukrainian Math. J. 2016, 67 (10), 1599 – 1610, https://doi.org/10.1007/s11253-016-1175-8; translation of Ukrain. Mat. Zh. 2015, 67 (10), 1423 – 1433.

Янченко С. Я. Порядковi оцiнки апроксимативних характеристик функцiй з узагальнених класiв мiшаної гладкостi типу Нiкольського–Бєсова. Теорiя наближення функцiй та сумiжнi питання: Зб. праць Iн-ту математики НАН України 2014, 11 (3), 330 – 343.

Yanchenko S.Ya. Approximation of functions from the classes 𝑆𝑟𝑝,θ𝐵 in the uniform metric. Ukrainian Math. J. 2013, 65 (5), 771 – 779, https://doi.org/10.1007/s11253-013-0813-7; translation of Ukrain. Mat. Zh. 2013, 65 (5), 698 – 705.

Миронюк В. В., Янченко С. Я. Наближення функцiй з узагальнених класiв Нiкольського–Бєсова цiлими функцiями у просторах Лебега. Мат. Студiї 2013, 39 (2), 190 – 202.

Янченко С. Я. Оцiнки апроксимативних характеристик класiв функцiй 𝑆𝑟𝑝,θ𝐵(R𝑑) у рiвномiрнiй метрицi. Теорiя наближення функцiй та сумiжнi питання: Зб. праць Iн-ту математики НАН України 2013, 10 (1), 328 – 340.

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