Geodesic
On estimates for orders of best $M$-term approximations
Ufa mathematical journal, Tome 15 (2023) no. 1, pp. 1-20 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper we consider a well-known class of weakly varying functions and by these functions we define an anisotropic Lorentz-Karamata space of $2\pi$-periodic functions of many variables. Particular cases of these spaces are anisotropic Lorentz-Zygmund and Lorentz spaces. In the anisotropic Lorentz-Karamata space we define an analogue of Nikolskii-Besov space. The main aim of the paper is to find sharp orders of best $M$-term trigonometric approximation of functions from Nikolskii-Besov space by the norm of another anisotropic Lorentz-Karamata space. In the paper we establish order sharp two-sided estimates of best $M$-term trigonometric approximations for the functions from the Nikolskii-Besov space in the anisotropic Lorentz-Karamata space in various metrics. In order to prove an upper bound for $M$-term approximations, we employ an idea of the greedy algorithms proposed by V.N. Temlyakov and we modify it for the anisotropic Lorentz-Karamata space.
Keywords: $M$–term approximation.
Mots-clés : Lorentz-Karamata space, Nikolskii-Besov space 
@article{UFA_2023_15_1_a0,
     author = {G. A. Akishev},
     title = {On estimates for orders of best $M$-term approximations},
     journal = {Ufa mathematical journal},
     pages = {1--20},
     year = {2023},
     volume = {15},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2023_15_1_a0/}
}
TY  - JOUR
AU  - G. A. Akishev
TI  - On estimates for orders of best $M$-term approximations
JO  - Ufa mathematical journal
PY  - 2023
SP  - 1
EP  - 20
VL  - 15
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/UFA_2023_15_1_a0/
LA  - en
ID  - UFA_2023_15_1_a0
ER  - 
%0 Journal Article
%A G. A. Akishev
%T On estimates for orders of best $M$-term approximations
%J Ufa mathematical journal
%D 2023
%P 1-20
%V 15
%N 1
%U http://geodesic.mathdoc.fr/item/UFA_2023_15_1_a0/
%G en
%F UFA_2023_15_1_a0
G. A. Akishev. On estimates for orders of best $M$-term approximations. Ufa mathematical journal, Tome 15 (2023) no. 1, pp. 1-20. http://geodesic.mathdoc.fr/item/UFA_2023_15_1_a0/

[1] S.G. Krejn, Yu.I. Petunin, E.M. Semenov, Interpolation of linear operators, Amer. Math. Soc., Providence, R.I., 1982 | MR | MR | Zbl

[2] A.P. Blozinski, “Multivariate rearragements and Banach function spaces with mixed norms”, Trans. Amer. Math. Soc., 263 (1981), 146–167 | DOI | MR

[3] A.A. Yatsenko, “Iterative rearrangements of functions, and Lorentz spaces”, Russ. Math., 42:5 (1998), 71–75 | MR | Zbl

[4] V.I. Kolyada, “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 | MR | Zbl

[5] N.K. Bari, S.B. Stechkin, “Best approximations and differential properties of two conjugate functions”, Trudy Mosk. Matem. Obshch., 5 (1956), 483–522 | Zbl

[6] D.E. Edmunds, W.D. Evans, Hardy operators, function spaces and embedding, Springer-Verlag, Berlin–Heidelberg, 2004 | MR

[7] E. Seneta, Regularly varying functions, Springer, Berlin, 1976 | MR | Zbl

[8] E.M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971 | MR | Zbl

[9] S.M. Nikol'skij, Approximation of functions of several variables and imbedding theorems, Springer-Verlag, Berlin, 1975 | MR | MR | Zbl

[10] P.I. Lizorkin, S.M. Nikol'skii, “Functional spaces of mixed smoothness from decompositional point of view”, Proc. Steklov Inst. Math., 187 (1990), 163–184 | MR | Zbl

[11] T.I. Amanov, Spaces of differentiable functions with a dominating mixed derivative, Nauka, Alma-Ata, 1976 | MR

[12] R.S. Ismagilov, “Diameters of sets in normed linear spaces and the approximation of functions by trigonometric polynomials”, Russ. Math. Surv., 29:3 (1974), 169–186 | DOI | MR | Zbl

[13] E.S. Belinsky, “Approximation of periodic functions with a ‘floating’ system of exponentials and trigonometric widths”, Studies in theory of functions of many real variabels, Yaroslavl State Univ., Yaroslavl, 1984, 10–24

[14] É.S. Belinskii, “Approximation by a “floating” system of exponentials on classes of smooth periodic functions”, Math. USSR-Sb., 60:1 (1988), 19–27 | DOI | MR

[15] É.S. Belinskii, “Approximation by a ‘floating’ system of exponentials on classes of periodic functions with a bounded mixed derivative”, Studies in theory of functions of many real variabels, Yaroslavl State Univ., Yaroslavl, 1988, 16–33

[16] Y. Makovoz, “On trigonometric $n$–widths and their generalization”, J. Approx. Theory., 41:4 (1984), 361–366 | DOI | MR | Zbl

[17] V.E. Maiorov, “Trigonometric diameters of the Sobolev classes $W^{r}_{p}$ in the space $L_p$”, Math. Notes, 40:2 (1986), 590–597 | DOI | MR | Zbl

[18] R.A. DeVore, “Nonlinear approximation”, Acta Numerica, 7 (1998), 51–150 | DOI | MR | Zbl

[19] V.N. Temlyakov, “Approximations of functions with bounded mixed derivative”, Proc. Steklov Inst. Math., 178 (1989), 1–121 | MR | Zbl

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

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

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

[23] M. Hansen, W. Sickel, “Best $m$–term approximation and Lizorkin–Triebel spaces”, J. Approx. Theory, 163 (2011), 923–954 | DOI | MR | Zbl

[24] S.A. Stasyuk, “Best $m$–term trigonometric approximation for the classes $B_{p, \theta}^{r}$ of functions of low smoothness”, Ukr. Math. Jour., 62:1 (2010), 114–122 | DOI | MR | Zbl

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

[26] D.B. Bazarkhanov, “Nonlinear trigonometric approximations of multivariate function classes”, Proc. Steklov Inst. Math., 293 (2016), 2–36 | DOI | DOI | MR | Zbl

[27] Dũng Dinh, V.N. Temlyakov, T. Ullrich, Hyperbolic Cross Approximation, Advanced Courses in Mathematics. CRM Barcelona, Birkhüser/Springer, Basel/Berlin, 2018 | MR | Zbl

[28] G. Akishev, “On the orders of an $M$-term approximation of classes of periodic functions”, Matem. Zhurn., 6:4 (2006), 5–14 | MR

[29] G. Akishev, “On estimates for the best $M$-term approximation of the Besov class”, Matem. Zhurn., 7:1 (2007), 5–14 | MR | Zbl

[30] G. Akishev, “On the exact estimations of the best $M$-terms approximation of the Besov class”, Sibir. Elektr. Matem. Izv., 7 (2010), 255–274 | Zbl

[31] G. Akishev, “On the order of the $M$-th approach of the classes approximation in the Lorentz space”, Matem. Zhurn., 11:1 (2011), 5–29 | MR | Zbl

[32] G. Akishev, “Trigonometric widths of the Nikol'skii-Besov classes in the Lebesgue space with mixed norm”, Ukr. Math. J., 66:6 (2014), 723–732 | DOI | MR | Zbl

[33] G. Akishev, “On $M$–term approximations of the Nikol'skii-Besov class”, Hacet. J. Math. Stat., 45:2 (2016), 297–310 | MR | Zbl

[34] G. Akishev, “Estimations of the best $M$–term approximations of functions in the Lorentz space with constructive methods”, Bull. Karaganda Univer. Math. ser., 3 (2017), 13–26 | DOI | MR

[35] G. Akishev, Estimates of the order of approximation of functions of several variables in the generalized Lorentz space, 2021, arXiv: 2105.14810 | MR

[36] G. Akishev, On exact estimates of the order of approximation of functions of several variables in the anisotropic Lorentz-Zygmund space, 2021, arXiv: 2106.07188 | MR

[37] G. Akishev, On estimates of the order of approximation of functions of several variables in the anisotropic Lorentz - Karamata space, 2021, arXiv: 2106.12761 | MR