G. Akishev. Estimates for the best approximations of functions from the Nikol’skii–Besov class in the Lorentz space by trigonometric polynomials ... P. 5-27

We consider spaces of periodic functions of many variables, specifically, the Lorentz space $L_{p, \tau}(\mathbb{T}^{m})$ and the Nikol'skii-Besov space $S_{p, \tau, \theta}^{\bar{r}}B$, and study the best approximation of a function $f \in L_{p, \tau}(\mathbb{T}^{m})$ by trigonometric polynomials with the numbers of harmonics from a step hyperbolic cross. Sufficient conditions are established for a function $f \in L_{p, \tau_{1}}(\mathbb{T}^{m})$ to belong to a space $L_{q, \tau_{2}}(\mathbb{T}^{m})$ in the cases $1 <p <q <\infty$, $1 <\tau_{1}, \tau_{2} <\infty$ and $p = q$, $ 1 <\tau_{2} <\tau_{1} <\infty$. Estimates for the best approximations of functions from the Nikol'skii-Besov class $S_{p, \tau_{1}, \theta}^{\bar{r}}B$ in the norm of the space $L_{q, \tau_{2}}(\mathbb{T}^{m})$ are derived for different relations between the parameters $p$, $q$, $\tau_{1}$, $\tau_{2}$, and~$\theta$. For some relations between these parameters, it is shown that the estimates are exact.

Keywords: Lorentz space, Nikol'skii-Besov class, trigonometric polynomial, best approximation, hyperbolic cross

Received September 9, 2019

Revised May 20, 2020

Accepted May 25, 2020

Funding Agency: This work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Gabdolla Akishev, Dr. Phys.-Math. Sci., Prof., L.N.Gumilyov Eurasian National University, Nur–Sultan, 100008 Republic Kazakhstan; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: akishev_g@mail.ru

REFERENCES

1.   Stein E.M., Weiss G. Introduction to Fourier analysis on Euclidean spaces. Princeton: Princeton Univ. Press, 1971, 312 p. ISBN: 9781400883899 . Translated to Russian under the title Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh. Moscow: Mir Publ., 1974, 333 p.

2.   Kolyada V.I. Rearrangements of functions and embedding theorems. Russian Math. Surveys. 1989. Vol. 44, no. 5. P. 73–117.

3.   Edmunds D.E., Evans W.D. Hardy operators, function spaces and embedding. Berlin; Heidelberg: Springer-Verlag, 2004.

4.   Nikol’skii S.M. Approximation of functions of several variables and embedding theorems. Moscow: Nauka, 1977.

5.   Lizorkin P.I., Nikol’skii S.M. Spaces of functions of mixed smoothness from the decomposition point of view. Proc. Stekov Inst. Math., 1989. Vol. 187. P. 143–161.

6.   Amanov T.I. Prostranstva differentsiruemykh funktsii s dominiruyushchei smeshannoi proizvodnoi [Spaces of differentiable functions with dominant mixed derivative], Alma-Ata:Nauka, 1976. 224 p.

7.   Babenko K.I. Approximation by trigonometric polynomials in a certain class of periodic functions of several variables. Soviet Math. Dokl., 1960, vol. 1, pp. 672–675.

8.   Telyakovskii S.A. Some estimates for trigonometric series with quasiconvex coefficients. Mat. Sb., 1964, vol. 63, no. 3, pp.  426–444 (in Russian).

9.   Mityagin B.S. Approximation of functions in the spaces $L^p$ and $C$ on the torus. Mat. Sb. 1962, vol. 58, no. 4, pp. 397–414 (in Russian).

10.   Bugrov Ya.S. Approximation of function classes with the dominant mixed derivative. Mat. Sb. 1964, vol. 64, no. 3, pp. 410–418 (in Russian).

11.   Galeev E.M. Approximation of some classes of periodic functions of several variables by Fourier sums in the metric of $\tilde{L}_{p}$. Uspekhi Mat. Nauk., 1977, vol. 32, no. 4, pp. 251–252 (in Russian).

12.   Galeev E.M. Approximation by of Fourier sums of classes of functions with bounded derivatives. Mat. Zam., vol. 23, no. 2, pp. 197–212 (in Russian).

13.   Temlyakov V.N. Approximation of periodic functions of several variables with bounded mixed differences. Mat. Sb., 1980, vol. 113, no. 1, pp. 65–80.

14.   Temlyakov V.N. Approximation of functions with bounded mixed derivative. Tr. Mat. Inst. Steklov., 1986, vol. 178, pp. 3–112 (in Russian).

15.   Romanyuk A.S. Approximation of the Besov classes of periodic functions of several variables in the space $L_q$. Ukrain . Mat. Zh., 1991, vol. 43, pp. 1297–1306. doi: 10.1007/BF01061817 

16.   Romanyuk A.S. On estimates of approximation characteristics of the Besov classes of periodic functions of many variables. Ukrain . Mat. Zh., 1997, vol. 499, pp. 1409–1422. doi: 10.1007/BF02487348 

17.   Schmeisser H.-J., Sickel W. Spaces of functions of mixed smoothness and approximation from hyperbolic crosses. J. Approx. Th., 2004, vol. 128, no. 2, pp. 115–150. doi: 10.1016/j.jat.2004.04.007 

18.   Bekmaganbetov K.A., Toleugazy Y. On the order of the trigonometric diameter of the anisotropic Nikol‘skii-Besov class in the metric anisotropic Lorentz spaces. Anal. Math., 2019, vol. 45, no. 2, pp.  237–247. doi: 10.31489/2020M1/17-26 

19.   Tikhomirov V.M., Approximation theory. Current problems in mathematics. Fundamental directions, vol. 14, Moscow, 1987, pp. 103–260 (in Russian). [Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987, 41-02].

20.   Dinh Dung , Temlyakov V.N., Ullrich T. Hyperbolic cross approximation. Ser. Advanced Courses in Mathematics CRM Barcelona, Cham: Birkhauser, 2018, 222 p.

21.   Temlyakov V. Multivariate approximation. Cambridge: Cambridge University Press, 2018, 551 p.

22.   Kokilashvili V., Yildirir Y.E. On the approximation by trigonometric polynomials in weighted Lorentz spaces. Jour. Func. Spaces Applic., 2010, vol. 8, no. 1, pp.  67–86.

23.   Krein S.G., Petunin Yu.I., Semenov E.M. Interpolation of linear operators. Ser. Translat. Math. Monographs, vol. 54, Providence, R.I.: American Math. Soc., 1982, 375 p. Original Russian text published in Krein S.G., Petunin Yu.I., Semenov E.M. Interpolyatsiya lineinykh operatorov, Moscow: Nauka Publ., 1978, 400 p.

24.   Akishev G. On the orders M-terms approximations of classes of functions of the symmetrical space. Mat. Zh., 2014, vol. 14, no. 4, pp. 44–71 (in Russian).

25.   Yatsenko A.A. Iterative rearrangements of functions, and Lorentz spaces. Russian Mathematics (Izvestiya VUZ. Matematika), 1998, vol. 42, no. 4, pp. 71–75.

Cite this article as: G.Akishev. Estimates for the best approximations of functions from the Nikol’skii–Besov class in the Lorentz space by trigonometric polynomials, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 2, pp. 5–27.