Estimates of \(M\)-term approximations of functions of several variables in the Lorentz space by a constructive method. (English) Zbl 07896573
Summary: 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 \(1<q, \tau<\infty, 1\leqslant\theta\leqslant\infty\) 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.
MSC:
| 41A10 | Approximation by polynomials |
| 41A25 | Rate of convergence, degree of approximation |
| 42A05 | Trigonometric polynomials, inequalities, extremal problems |
References:
| [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. · Zbl 1329.41035 |
| [2] | G. Akishev, On the order of the M -term approximation classes in Lorentz spaces. Matematical Journal. Almaty. 11 (2011), no. 1, 5-29. · Zbl 1488.42005 |
| [3] | G. Akishev Estimations of the best M -term approximationsof functions in the Lorentz space with constructive methods. Bull. Karaganda Univer. Math. Ser. 3 (2017), 13-26. |
| [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 (2018), no. 3, 3-21. |
| [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 (2020), no. 2, 5-27 . |
| [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). Pro-ceedings. Kazan, (2021), 165-167. |
| [8] | T.I. Amanov, Spaces of differentiable functions with dominant mixed derivative. Nauka, Alma-Ata, 1976 (in Russian). |
| [9] | S. Artamonov, K. Runovski, H.-J. Schmeisser, Methods of trigonometric approximation and generalized smooth-ness. II, Eurasian Math. J., 13 (2022), no. 4, 18-43. · Zbl 1513.46058 |
| [10] | D.B. Bazarkhanov, V.N. Temlyakov, Nonlinear tensor product approximation of functions. J. Complexity, 31 (2015), no. 6, 867-884 · Zbl 1332.41026 |
| [11] | D.B. Bazarkhanov, Nonlinear trigonometric approximation of multivariate function classes. Proceedings Steklov Inst. Math., 293 (2016), 2-36. · Zbl 1351.42002 |
| [12] | E.S. Belinsky, Approximation by a ’floating’ system of exponents on the classes of smooth periodic functions. Sb. Math., 132 (1987), no. 1, 20 -27. · Zbl 0642.42003 |
| [13] | E.S. Belinsky, Approximation of periodic functions by a ’floating’ system of exponents and trigonometric diametrs. Issledovanija po teorii funkcij mnogih veshhestvennyh peremennyh, Jaroslavl’ Research on the theory of functions of many real variables, Yaroslavl’ State University, (1984), 10-24 (in Russian). · Zbl 0595.42002 |
| [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). · Zbl 0749.42009 |
| [15] | V.I. Burenkov, A. Senouci, Equivalent semi-norms for Nikol’skii-Besov spaces, Eurasian Math. J., 14 (2023), no. 4, 15-22 · Zbl 07823364 |
| [16] | R.A. De Vore, Nonlinear approximation. Acta Numerica, 7 (1998), 51-150 . · Zbl 0931.65007 |
| [17] | Dinh Dung, On asymptotic order of n -term approximations and non-linear n-widths. Vietnam Journal Math. 27 (1999), no. 4, 363-367. · Zbl 0968.41013 |
| [18] | Dinh Dũng, V.N. Temlyakov, T. Ullrich, Hyperbolic cross approximation. Advanced Courses in Mathematics. CRM Barcelona. (Birkhäuser/Springer, Basel/Berlin (2018)), 222 pp. |
| [19] | M. Hansen, W. Sickel, Best m-term approximation and Lizorkin -Triebel spaces. Jour. Approx. Theory. 163 (2011), 923-954. · Zbl 1229.41028 |
| [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 (2012), no. 1, 1-51. · Zbl 1251.41005 |
| [21] | R.S. Ismagilov, Widths of sets in linear normed spaces and the approximation of functions by trigonometric polynomials. Uspehi mathem. nauk. 29 (1974), no. 3, 161-178. · Zbl 0303.41039 |
| [22] | B.S. Kashin, Approximation properties of complete orthnormal systems. Trudy Mat. Inst. Steklov. 172 (1985), 187-201. · Zbl 0581.42018 |
| [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. · Zbl 0707.46025 |
| [24] | V.E. Maiorov, Trigonometric diametrs of the Sobolev classes W r p in the space L q . Math. Notes, 40 (1986), no. 2, 161-173. · Zbl 0623.41024 |
| [25] | Y. Makovoz, On trigonometric n-widths and their generalization. J. Approx.Theory, 41 (1984), no. 4, 361-366. · Zbl 0543.41027 |
| [26] | S.M. Nikol’skii, Approximation of functions of several variables and embedding theorems. Nauka, Moscow, 1977. · Zbl 0496.46020 |
| [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 (2003), no. 2, 61-100. · Zbl 1066.42001 |
| [28] | A.S. Romanyuk, Best trigonometric approximations for some classes of periodic functions of several variables in uninorm metric. Math. Notes , 82 (2007), no. 1, 216-228. · Zbl 1219.41006 |
| [29] | A.L. Shidlich, Approximation of certain classes of functions of several variables by greedy approximates in the integral metrics. ArXiv:1302.2790v1 [mathCA] 12 Feb, (2013), 1-16. |
| [30] | S.A. Stasyuk, Best m-term trigonometric approximation for the classes B r p,θ of functions of low smoothness. |
| [31] | Ukr. Math. Jour. 62 (2010), no. 1, 114-122. |
| [32] | 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. · Zbl 1480.42003 |
| [33] | S.A. Stasyuk, Approximating characteristics of the analogs of Besov classes with logarithmic smoothness. Ukr. Math. Jour. 66 (2014), no. 4, 553-560. · Zbl 1320.42006 |
| [34] | S.B. Stechkin, On the absolute convergence of orthogonal series, Doklad. Akadem. Nauk SSSR. 102 (1955), no. 2, 37-40. · Zbl 0068.05104 |
| [35] | E.M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press, Princeton, 1971. · Zbl 0232.42007 |
| [36] | V.N. Temlyakov, Approximation of functions with bounded mixed derivative. Tr. Mat. Inst. Steklov. 178 (1986), 3-112. · Zbl 0625.41028 |
| [37] | V.N. Temlyakov, Greedy approximation. Cambridge Univer. Press, Cambridge , 2011. · Zbl 1279.41001 |
| [38] | V.N. Temlyakov Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness. Sb. Math. 206 (2015), no. 11 131-160. · Zbl 1362.41009 |
| [39] | V.N. Temlyakov, Constructive sparse trigonometric approximation for functions with small mixed smoothness. Constr. Approx. 45 (2017), no. 3, 467-495. · Zbl 1373.42009 |
| [40] | V.N. Temlyakov, Multivariate approximation. Cambridge University Press, Cambridge, 2018. · Zbl 1428.41001 |
| [41] | N.T. Tleukhanova, A. Bakhyt On trigonometric Fourier series multipliers in λ p,q . Eurasian Math. J., 12 (2021), no. 1, 103-106. · Zbl 1474.42049 |
| [42] | Wang Heping, Sun Yongsheng, Representation and m-term approximation for anisotropic classes, C.M. Nikol’skii, et al. (Eds.), Theory of Approximation of Functions and Applications, 2003, 250-268. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
