##### Work number - M 29 AWARDED

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Presented Institute of Mathematics of NAS of Ukraine. __

__Authors__:

Pozharska Kateryna Vitaliivna – PhD, junior researcher,

Stepaniuk Tetiana Anatoliivna – PhD, researcher,

Yanchenko Serhii Yakovych – PhD, doctoral candidate, senior researcher.

The work is devoted to the development of new methods and to the improvement of known methods of investigation of a number of approximation characteristics of linear and nonlinear approximation of functional classes. Namely, in the work the authors established estimates of corresponding quantities on the Nilol’skii-Besov classes of periodic and nonperiodic functions of one and several variables, on the generalizations of these classes, and also on the classes of 2π-periodic (ψ,β)- differentiable functions. The work presents the scientific results, which were obtained by authors during 2013-2019 in the department of theory of functions of the Institute of Mathematics of NAS of Ukraine. Given scientific work has theoretical character.

In the paper the authors were able to significantly supplement the known results by new estimates for the best approximations, approximations by Fourier sums, best orthogonal approximations, best M-term trigonometric approximations, best bilinear approximations, entropy numbers, worst-case integration error and the energies on the unit sphere.

In the framework of this work the authors have used and improved the world known, effective methods of solving the problems of approximation theory, combining with classical methods of approximation theory, which are used for solving certain problems, and also methods of the theory of functional spaces, measure theory, etc. The authors developed the methods, which were proposed in the works of O.I. Stepanets, A.S. Romanyuk, V.N. Temlyakov (USA), S. Yongsheng, W. Heping (China), I. H. Sloan (Australia), K. Hesse (Germany), J. S. Brauchart (Austria).

The results of this work and used methods can be applied to problems of estimation of singular numbers of integral operators, Kolmogorov widths of corresponding functional classes, and also can be applied to solving of other problems of approximation theory.

**Number of publications**: 33 articles in Ukrainian and international proffessional editions and 41 abstracts of scientific conferences. 23 articles are published in international journals, including 21 articles in journals, which have impact-factor and print articles in English.