Estimates of $M$–term approximations of functions of several variables in the Lorentz space by a constructive method

G. Akishev

Eurasian mathematical journal, Tome 15 (2024) no. 2, pp. 8-32.

Résumé

In the paper, the Lorentz space $L_{q,r}(\mathbb{T}^m)$ of periodic functions of several variables, the Nikol'skii–Besov class $S_{q,\tau,\theta}^{\overline{r}}$ and the associated class $W_{q,r}^{a,b,\overline{r}}$ for $1$, $\tau\infty$, $1\leqslant\theta\leqslant\infty$ are considered. Estimates are established for the best $M$-term trigonometric approximations of functions of the classes $W_{q,\tau_1}^{a,b,\overline{r}}$ and $S_{q,\tau_1,\theta}^{\overline{r}}B$ in the norm of the space $L_{p,\tau_2}(\mathbb{T}^m)$ for different relations between the parameters $q$, $\tau_1$, $p$, $\tau_2$, $a$, $\theta$. The proofs of the theorems are based on the constructive method developed by V.N. Temlyakov.

Bibliographie

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

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

[3] 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

[4] G. Akishev, “Estimates for best approximations of functions from the logarthmic smoothness class in the Lorentz space”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 23, no. 3, 2018, 3–21 | DOI | MR

[5] G. Akishev, “Estimation of the best approximations of the functions Nikol'skii Besov class in the space of Lorentz by trigonometric polynomials”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 26, no. 2, 2020, 5–27 | DOI | MR

[6] G. Akishev, “On the best $M$-term approximations of the functions in the space of Lorentz”, International Voronezh Winter Mathematical School “Modern methods of the theory of functions and related problems” (Voronezh (Russia) January 28–February 2, 2021), 29–31

[7] G. Akishev, “On estimates of the best $n$-term approximations of functions of the Nikol'skii-Besov class in the Lorentz space”, International Conference on Algebra, Analysis and Geometry (Kazan, August 22-28, 2021), Kazan, 2021, 165–167 | MR

[8] T. I. Amanov, Spaces of differentiable functions with dominant mixed derivative, Nauka, Alma-Ata, 1976 (in Russian) | MR

[9] S. Artamonov, K. Runovski, H. J. Schmeisser, “Methods of trigonometric approximation and generalized smoothness, II”, Eurasian Math. J., 13:4 (2022), 18–43 | MR | Zbl

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

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

[12] E. S. Belinsky, “Approximation by a 'floating' system of exponents on the classes of smooth periodic functions”, Sb. Math., 132:1 (1987), 20–27 | MR

[13] E. S. Belinsky, “Approximation of periodic functions by a 'floating' system of exponents and trigonometric diametrs”, Research on the theory of functions of many real variables, Yaroslavl' State University, 1984, 10–24 (in Russian) | MR

[14] E. S. Belinsky, “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, Yaroslavl' State University, 1988, 16–33 (in Russian) | MR

[15] V. I. Burenkov, A. Senouci, “Equivalent semi-norms for Nikol'skii-Besov spaces”, Eurasian Math. J., 14:4 (2023), 15–22 | MR

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

[17] Dinh Dung, “On asymptotic order of $n$-term approximations and non-linear $n$-widths”, Vietnam Journal Math., 27:4 (1999), 363–367 | MR | Zbl

[18] Dinh Düng, V.N. Temlyakov, T. Ullrich, Hyperbolic cross approximation, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser, Basel; Springer, Berlin, 2018, 222 pp. | MR | Zbl

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

[20] M. Hansen, W. Sickel, “Best $M$-term approximation and Sobolev-Besov spaces of dominating mixed smoothness the case of compact embeddings”, Constr. Approx., 36:1 (2012), 1–51 | DOI | MR | Zbl

[21] R. S. Ismagilov, “Widths of sets in linear normed spaces and the approximation of functions by trigonometric polynomials”, Uspehi mathem. nauk, 29:3 (1974), 161–178 | MR | Zbl

[22] B. S. Kashin, “Approximation properties of complete orthnormal systems”, Trudy Mat. Inst. Steklov, 172 (1985), 187–201 | MR

[23] P. I. Lizorkin, S. M. Nikol'skii, “Spaces of functions of mixed smoothness from the decomposition point of view”, Proc. Stekov Inst. Math., 187 (1989), 143–161 | MR

[24] V. E. Maiorov, “Trigonometric diametrs of the Sobolev classes $W_p^r$ in the space $L_q$”, Math. Notes, 40:2 (1986), 161–173 | DOI | MR | Zbl

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

[26] S. M. Nikol'skii, Approximation of functions of several variables and embedding theorems, Nauka, M., 1977 | MR

[27] A. S. Romanyuk, “On the best $M$-term trigonometric approximations for the Besov classes of periodic functions of many variables”, Izv. Ros. Akad. Nauk, Ser. Mat., 67:2 (2003), 61–100 | DOI | MR | Zbl

[28] A. S. Romanyuk, “Best trigonometric approximations for some classes of periodic functions of several variables in uninorm metric”, Math. Notes, 82:1 (2007), 216–228 | DOI | MR | Zbl

[29] A. L. Shidlich, Approximation of certain classes of functions of several variables by greedy approximates in the integral metrics, 12 Feb 2013, 16 pp., arXiv: 1302.2790v1 [mathCA] | Zbl

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

[31] S. A. Stasyuk, “Best $M$-term trigonometric approximation of periodic function of several variables from Nikol'skii–Besov classes for small smoothness”, Jour. Approx. Theory, 177 (2014), 1–16 | DOI | MR | Zbl

[32] S. A. Stasyuk, “Approximating characteristics of the analogs of Besov classes with logarithmic smoothness”, Ukr. Math. Jour., 66:4 (2014), 553–560 | DOI | MR | Zbl

[33] S. B. Stechkin, “On the absolute convergence of orthogonal series”, Doklad Akadem. Nauk SSSR, 102:2 (1955), 37–40 | MR | Zbl

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

[35] V. N. Temlyakov, “Approximation of functions with bounded mixed derivative”, Tr. Mat. Inst. Steklov, 178 (1986), 3–112 | MR

[36] V. N. Temlyakov, Greedy approximation, Cambridge Univer. Press, Cambridge, 2011 | MR | Zbl

[37] V. N. Temlyakov, “Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness”, Sb. Math., 206:11 (2015), 131–160 | DOI | MR | Zbl

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

[39] V. N. Temlyakov, Multivariate approximation, Cambridge University Press, Cambridge, 2018 | MR | Zbl

[40] N. T. Tleukhanova, A. Bakhyt, “On trigonometric Fourier series multipliers in $\lambda_{p,q}$”, Eurasian Math. J., 12:1 (2021), 103–106 | DOI | MR | Zbl

[41] Wang Heping, Sun Yongsheng, “Representation and $M$-term approximation for anisotropic classes”, Theory of Approximation of Functions and Applications, eds. C. M. Nikol'skii et al, 2003, 250–268