On estimates of the order of the best $M$-term approximations of~functions of several variables in the anisotropic Lorentz -- Zygmund space
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 2, pp. 142-156.

Voir la notice de l'article provenant de la source Math-Net.Ru

The article considers the anisotropic Lorentz – Karamata space of periodic functions of several variables and the Nikol'skii – Besov class in this space. The order-sharp estimates are established for the best $M$-term trigonometric approximations of functions from the Nikol'skii-Besov class in the norm of another Lorentz – Zygmund space.
@article{ISU_2023_23_2_a0,
     author = {G. Akishev},
     title = {On estimates of the order of the best $M$-term approximations of~functions of several variables in the anisotropic {Lorentz} -- {Zygmund} space},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {142--156},
     publisher = {mathdoc},
     volume = {23},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2023_23_2_a0/}
}
TY  - JOUR
AU  - G. Akishev
TI  - On estimates of the order of the best $M$-term approximations of~functions of several variables in the anisotropic Lorentz -- Zygmund space
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2023
SP  - 142
EP  - 156
VL  - 23
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ISU_2023_23_2_a0/
LA  - ru
ID  - ISU_2023_23_2_a0
ER  - 
%0 Journal Article
%A G. Akishev
%T On estimates of the order of the best $M$-term approximations of~functions of several variables in the anisotropic Lorentz -- Zygmund space
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2023
%P 142-156
%V 23
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ISU_2023_23_2_a0/
%G ru
%F ISU_2023_23_2_a0
G. Akishev. On estimates of the order of the best $M$-term approximations of~functions of several variables in the anisotropic Lorentz -- Zygmund space. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 2, pp. 142-156. http://geodesic.mathdoc.fr/item/ISU_2023_23_2_a0/

[1] Bennett C., Sharpley R., Interpolation of Operators, Academic Press, Orlando, 1988, 469 pp. | MR | Zbl

[2] Stein E. M., Weiss G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1971, 312 pp. | MR | Zbl

[3] Blozinski A. P., “Multivariate rearrangements and Banach function spaces with mixed norms”, Transactions of the American Mathematical Society, 263:1 (1981), 149–167 | DOI | MR | Zbl

[4] Kolyada V. I., “On embedding theorems”, Nonlinear Analysis, Function spaces and Applic, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, 2007, 35–94 (data obrascheniya: 20.02.2022) http://dml.cz/dmlcz/702492 | Zbl

[5] Nikol'skii S. M., Approximation of Functions of Several Variables and Embedding Theorems, Nauka, M., 1977, 456 pp. (in Russian) | MR

[6] Amanov T. I., Spaces of Differentiable Functions with a Dominant Mixed Derivative, Nauka, Alma-Ata, 1976, 224 pp. (in Russian) | MR

[7] Lizorkin P. I., Nikol'skii S. M., “Spaces of functions of mixed smoothness from the decomposition point of view”, Proceedings of the Steklov Institute of Mathematics, 187 (1990), 163–184 | MR | Zbl

[8] Dũng D., Temlyakov V. N., Ullrich T., Hyperbolic Cross Approximation, Advanced Courses in Mathematics. CRM Barcelona, Springer, Basel–Berlin, 2018, 229 pp. | MR

[9] Belinskii E. S., “Approximation by a “floating” system of exponentials on the classes of smooth periodic functions with bounded mixed derivative”, Research on the Theory of Functions of Many Real Variables, ed. Brudnyi Yu. A., Yaroslavl State University Publ, Yaroslavl, 1988, 16–33 (in Russian) | MR

[10] Temlyakov V. N., “Approximations of functions with bounded mixed derivative”, Proceedings of the Steklov Institute of Mathematics, 178 (1989), 1–121 | MR

[11] Temlyakov V. N., “Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness”, Sbornik: Mathematics, 206:11 (2015), 1628–1656 | DOI | DOI | MR | Zbl

[12] Temlyakov V. N., “Constructive sparse trigonometric approximation for functions with small mixed smoothness”, Constructive Approximation, 45:3 (2017), 467–495 | DOI | MR | Zbl

[13] Romanyuk A. S., “Best $ M$-term trigonometric approximations of Besov classes of periodic functions of several variables”, Izvestiya: Mathematics, 67:2 (2003), 265–302 | DOI | DOI | MR | Zbl

[14] Bazarkhanov D. B., Temlyakov V. N., “Nonlinear tensor product approximation of functions”, Journal of Complexity, 31:6 (2015), 867–884 | DOI | MR | Zbl

[15] Bazarkhanov D. B., “Nonlinear trigonometric approximations of multivariate function classes”, Proceedings of the Steklov Institute of Mathematics, 293 (2016), 2–36 | DOI | DOI | MR | Zbl

[16] Akishev G. A., “On the exact estimations of the best $M$-terms approximation of the Besov class”, Siberian Electronic Mathematical Reports, 7 (2010), 255–274 (in Russian) | MR | Zbl

[17] Akishev G., “On the order of the $M$-term approximation classes in Lorentz spaces”, Matematical Journal. Almaty, 11:1 (2011), 5–29 (in Russian) | MR | Zbl

[18] Akishev G., On exact estimates of the order of approximation of functions of several variables in the anisotropic Lorentz – Zygmund space, 14 Jun 2021, 20 pp., arXiv: 2106.07188v2 [mathCA] | MR

[19] Akishev G., Estimates of the order of approximation of functions of several variables in the generalized Lorentz space, 31 May 2021, 18 pp., arXiv: 2105.14810v1 [mathCA] | MR

[20] Akishev G., “On estimates of the order of the best $M$-term approximations of functions of several variables in the anisotropic Lorentz – Karamata space”, Materials of the 21st International Saratov (Saratov, January 31 – February 4, 2022), Contemporary Problems of Function Theory and Their Applications, 21, Saratov State University Publ., Saratov, 2022, 13–16 (in Russian)