Gabdolla Akishev     (\(^1\)L.N. Gumilyov Eurasian National University, 2 Pushkin str., Nur-Sultan, 010008, Kazakhstan; \(^2\)Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russian Federation)


In this paper, we consider the anisotropic Lorentz space \(L_{\bar{p}, \bar\theta}^{*}(\mathbb{I}^{m})\) of periodic functions of \(m\) variables. The Besov space \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) of functions with logarithmic smoothness is defined. The aim of the paper is to find an exact order of the best approximation of functions from the class \(B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}\) by trigonometric polynomials under different relations between the parameters \(\bar{p}, \bar\theta,\) and \(\tau\).

The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function \(f\in L_{\bar{p}, \bar\theta^{(1)}}^{*}(\mathbb{I}^{m})\) to belong to the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\) in the case \(1{<\theta^{2}<\theta_{j}^{(1)}},$ ${j=1,\ldots,m},\) in terms of the best approximation and prove its unimprovability on the class \(E_{\bar{p},\bar{\theta}}^{\lambda}=\{f\in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\colon
{E_{n}(f)_{\bar{p},\bar{\theta}}\leq\lambda_{n},}\) \({n=0,1,\ldots\},}\) where \(E_{n}(f)_{\bar{p},\bar{\theta}}\) is the best approximation of the function \(f \in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\) by trigonometric polynomials of order \(n\) in each variable \(x_{j},\) \(j=1,\ldots,m,\) and \(\lambda=\{\lambda_{n}\}\) is a sequence of positive numbers \(\lambda_{n}\downarrow0\) as \(n\to+\infty\). In the second section, we establish order-exact estimates for the best approximation of functions from the class \(B_{\bar{p}, \bar\theta^{(1)}}^{(0, \alpha, \tau)}\) in the space \(L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})\).


Lorentz space, Nikol’skii–Besov class, Best approximation

Full Text:



  1. Akishev G.A. On imbedding of some classes of functions of several variables into the Lorentz space. Izv. Akad. Nauk Kaz. SSR, Ser. Fiz.-Mat., 1982. No. 3. P. 47–51. (in Russian)
  2. Akishev G. The estimates of approximations classes in the Lorentz space. AIP Conf. Proc., 2015. Vol. 1676, No. 1. Art. no. 020027. P. 1–4. DOI: 10.1063/1.4930453
  3. Akishev G. An inequality of different metric for multivariate generalized polynomials. East J. Approx., 2006. Vol. 12, No. 1. P. 25–36.
  4. Akishev G.A. Estimates for best approximations of functions from the logarithmic smoothness class in the Lorentz space. Trudy Inst. Mat. Mekh. UrO RAN, 2017. Vol. 23, No. 3. P. 3–21. DOI: 10.21538/0134-4889-2017-23-3-3-21 (in Russian)
  5. Andrienko V.A. The imbedding of certain classes of functions. Math. USSR-Izv., 1967. Vol. 1, No. 6. P. 1255–1270.
  6. Bekmaganbetov K.A. About order of approximation of Besov classes in metric of anisotropic Lorentz spaces. Ufimsk. Mat. Zh., 2009. Vol. 1, No. 2. P. 9–16. (in Russian)
  7. Bekmaganbetov K.A. Order of approximation of Besov classes in the metric of anisotropic Lorentz spaces. In: Methods of Fourier Analysis and Approximation Theory. Ruzhansky M., Tikhonov S. (eds.) Appl. Numer. Harmon. Anal. Cham: Birkhäuser, 2016. P. 149–158. DOI: 10.1007/978-3-319-27466-9_10
  8. Blozinski A.P. Multivariate rearrangements and Banach function spaces with mixed norms. Trans. Amer. Math. Soc., 1981. Vol. 263, No. 1. P. 149–167. DOI: 10.2307/1998649
  9. Cobos F., Domínguez Ó. On Besov spaces of logarithmic smoothness and Lipschitz spaces. J. Math. Anal. Appl., 2015. Vol. 425, No. 1. P. 71–84. DOI: 10.1016/j.jmaa.2014.12.034
  10. Cobos F., Domínguez Ó. Approximation spaces, limiting interpolation and Besov spaces. J. Approx. Theory, 2015. Vol. 189. P. 43–66. DOI: 10.1016/j.jat.2014.09.002
  11. DeVore R.A., Lorentz G.G. Constructive Approximation. Grundlehren Math. Wiss., vol. 33. Berlin, Heidelberg: Springer–Verlag, 1993. 453 p.
  12. Dinh Dũng, Temlyakov V.N., Ullrich T. Hyperbolic Cross Approximation. 2016. 154 p. arXiv:1601.03978v1 [math.NA]
  13. Domínguez Ó., Tikhonov S. Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations. 2018. 162 p. arXiv:1811.06399v2 [math.FA]
  14. Johansson H. Embedding of \(H_{p}^{\omega}\) in Some Lorentz Spaces. Research Reports. Univ. Umeå, 1975. Report No. 6. 26 p.
  15. Kolyada V.I. Imbedding theorems and inequalities in various metrics for best approximations. Mat. Sb. (N.S.), 1977. Vol. 102 (144), No. 2. P. 195–215. (in Russian)
  16. Kolyada V.I. Rearrangements of functions and embedding theorems. Russian Math. Surveys, 1989. Vol. 44, No. 5 P. 73–117. DOI: 10.1070/RM1989v044n05ABEH002287
  17. Nikol’skii S.M. Priblizhenie funkcij mnogih peremennyh i teoremy vlozheniya [Approximation of Function of Several Variables and Imbedding Theorems]. Moscow: Nauka, 1977. 456 p. (in Russian)
  18. Nursultanov E.D. Interpolation theorems for anisotropic function spaces and their applications. Dokl. Akad. Nauk RAN, 2004. Vol. 394, No. 1. P. 22–25.
  19. Nursultanov E.D. Nikol’skii’s inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space. Proc. Steklov Inst. Math., 2006. Vol. 255. P. 185–202. DOI: 10.1134/S0081543806040158
  20. Pietsch A., Approximation spaces. J. Approx. Theory, 1981. Vol. 32, No. 2. P. 115–134. DOI: 10.1016/0021-9045(81)90109-X
  21. Romanyuk A.S. Approximation of the isotropic classes \(B_{p, \theta}^{r}\) of periodic functions of several variables in the space \(L_{q}\). Zb. Pr. Inst. Mat. NAN Ukr., 2008. Vol. 5, No. 1. P. 263–278. (in Russian)
  22. Sherstneva L.A. On the properties of best Lorentz approximations, and certain embedding theorems. Soviet Math. (Iz. VUZ), 1987. Vol. 31, No. 10. P. 62–73.
  23. Smailov E.S., Akishev G. Embedding theorems in the Lorentz space and their applications. Izv. Akad. Nauk Kaz. SSR, Ser. Fiz.-Mat., 1984. No. 1. P. 66–70. (in Russian)
  24. Stasyuk S.A. Approximating characteristics of the analogs of Besov classes with logarithmic smoothness. Ukr. Math. J., 2014. Vol. 66, No. 4, P. 553–560. DOI: 10.1007/s11253-014-0952-5
  25. Stasyuk S.A. Kolmogorovwidths for analogsof the Nikol’skii–Besovclasses with logarithmic smoothness. Ukr. Math. J., 2015. Vol. 67, No. 11. P. 1786–1792. DOI: 10.1007/s11253-016-1190-9
  26. Storozhenko È.A. Embedding theorems and best approximations. Math. USSR-Sb., 1975. Vol. 26, No. 2. P. 213–224. DOI: 10.1070/SM1975v026n02ABEH002477
  27. Temirgaliev N. Embedding in some Lorentz spaces. Izv. Vyssh. Uchebn. Zaved. Mat., 1980. No. 6. P. 83–85. (in Russian)
  28. Temirgaliev N. Embeddings of the classes \(H_{p}^{\omega}\) in Lorentz spaces. Sib. Math. J., 1983. Vol. 24, No. 2. P. 287–298. DOI: 10.1007/BF00968743
  29. Temlyakov V. Multivariate Approximation. Cambridge University Press. 2018. 551 p. DOI: 10.1017/9781108689687
  30. Ul’janov P.L. The imbedding of certain function classes \(H_{p}^{\omega}\). Math. USSR-Izv., 1968. Vol. 2, No. 3. P. 601–637. DOI: 10.1070/IM1968v002n03ABEH000650


Article Metrics

Metrics Loading ...


  • There are currently no refbacks.