G. Akishev. Estimates for best approximations of functions from the logarithmic smoothness class in the Lorentz space ... P. 3-21.

The Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$ of periodic functions of $m$ variables is considered. The Besov space $B_{p, \tau, \theta}^{0, \alpha}$ of functions with logarithmic smoothness is defined. The aim of the paper is to find the exact order of the best approximation of functions from the class $B_{p, \tau, \theta}^{0, \alpha}$ under different relations between the parameters $p$, $\tau$, and $\theta$. The paper consists of three sections. In the first section, known facts necessary for the proof of the main results are given and several auxiliary statements are proved. In the second section, order-exact estimates for the best approximation of functions from the class $B_{p, \tau, \theta}^{0, \alpha}$ are established in the space $L_{p,\tau}(\mathbb{T}^{m})$. In the third section, an inequality for different metrics of trigonometric polynomials is proved and a sufficient condition for the belonging of a function $f\in L_{p,\tau_{1}}(\mathbb{T}^{m})$ to the space $L_{p,\tau_{2}}(\mathbb{T}^{m})$ in terms of the best approximation is established in the case $1<\tau_{2}<\tau_{1}$. In contrast to anisotropic Lorentz spaces, the condition is independent of the number $m$ of the variables. Order-exact estimates for the best approximation of functions from the Besov class $B_{p, \tau_{1}, \theta}^{0, \alpha}$ by trigonometric polynomials $L_{p,\tau_{2}}(\mathbb{T}^{m})$ are obtained in the case $1<\tau_{2}<\tau_{1}$.

Keywords: Lorentz space, Besov class, best approximation, logarithmic smoothness.

The paper was received by the Editorial Office on June 28, 2017.

Gabdolla Akishev,  Dr. Phys.-Math. Sci., Prof., RSE Academician E. A. Buketov Karaganda State University, the Republic of Kazakhstan, 100028;  Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: akishev_g@mail.ru


1.   Stein E., Weiss G. Introduction to Fourier analysis on Euclidean spaces. Princeton, Princeton University Press, 1971, 312 p. ISBN: 9780691080789 . Translated to Russian under the title Vvedenie vgarmonicheskii analiz na evklidovykh prostranstvakh. Moscow, Mir Publ., 1974, 332 p.

2.   Kashin B.S., Temlyakov V.N. Ob odnoi norme i approksimatsionnykh kharakteristikakh klassov funktsii mnogikh peremennykh [On a norm and approximation characteristics of classes of functions of severalvariables]. In: Metricheskaya teoriya funktsii i smezhnye voprosy analiza [Metric theory of functions and related problems in analysis], edited by Nikol’skii, Izd. Nauchno-Issled. Aktuarno-Finans. Tsentra(AFTs), Moscow, 1999, pp. 69–99, ISBN: 5-93379-002-8 .

3.   DeVore R.A., Riemenschneider S.D., Sharpley R.C. Weak interpolation in Banach spaces. Jour. Func. Anal., 1979, vol. 33, pp. 58–94. doi: 10.1016/0022-1236(79)90018-1 .

4.   Cobos F., Milman M. On a limit class of approximation spaces. Numer. Funct. Anal. Optim., 1990, vol. 11, no. 1-2, pp. 11–31. doi: 10.1080/01630569008816358 .

5.   Cobos F., Dominguez O. On Besov spaces of logarithmic smoothness and Lippschitz spaces. J. Math. Anal. Appl., 2015, vol. 425, pp. 71–84. doi: 10.1016/j.jmaa.2014.12.034 .

6.   Romanyuk A.S. Approximation of the isotropic classes Bp,θr of periodic functions of several variables in the space Lq. Zb. Pr. Inst. Mat. NAN Ukr., 2008, vol. 5, no. 1, pp. 263–278 (in Russian).

7.   Stasyuk S.A. Approximating characteristics of the analogs of Besov classes with logarithmic smoothness. Ukr. Math. J., 2014, vol. 66, no. 4, pp. 553–560. doi: 10.1007/s11253-014-0952-5 .

8.   Stasyuk S.A. Kolmogorov widths for analogs ofthe Nikol’skii–Besov classes with logarithmic smoothness. Ukr. Math. Jour., 2015, vol. 67, no. 11, pp. 1786–1792. doi: 10.1007/s11253-016-1190-9 .

9.   Dinh Dung, Temlyakov V.N., Ullrich T. Hyperbolic cross approximation. Available at: https://arxiv.org/pdf/1601.03978.pdf . 154 p. [arXiv: 1601. 03978v1[math.NA] 15 Jan. 2016.]

10.   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, pp. 47–51 (in Russian).

11.   Sherstneva L.A. On the properties of best Lorentz approximations and certain embedding theorems. Soviet Math. (Iz. VUZ), 1987, vol. 31, no. 10, pp. 62–73.

12.   Lizorkin P.I. Generalized Holder spaces Bp,θ(r) and their relations with the Sobolev spaces Lp(r). Sib. Mat. Zh., 1968, vol. 9, no. 5, pp. 1127–1152 (in Russian).

13.   Janson S. On the interpolation of sublinear operators. Studia Math., 1982, vol. 75, pp. 51–53.

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

15.   Nikol’skii S.M. Approximation of functions of several variables and embedding theorems. New York, Springer-Verlag, 1975, 418 p. ISBN: 9780387064420 . Original Russian text (2nd ed.) published inNikol’skii S.M. Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya. 2-e izd. Moscow, Nauka Publ. 1977, 455 p.

16.   Akishev G. On the orders of M-terms approximations of classes of functions of the symmetrical space. Mat. Zh., 2014, vol. 14, no. 4, pp. 46–71 (in Russian).

17.   Ditzian Z., Prymak A. Nikol’skii inequalities for Lorentz spaces. Rocky Mountain Jour. Math., 2010, vol. 40, no. 1, pp. 209–223. doi: 10.1216/RMJ-2010-40-1-209 .

18.   Johansson H. Embedding of Hpω in some Lorentz spases. Research Report University UmeАa, 1975, vol. 6, pp. 1–36.

19.   Temlyakov V.N. Approximation of functions with a bounded mixed derivative. Proc. Steklov Inst. Math., Providence, American Mathematical Society (AMS), 1989, vol. 178. 121 p. Original Russian text published in Temlyakov V.N. Priblizhenie funktsii s ogranichennoi smeshannoi proizvodnoi, Tr. MIAN SSSR, vol. 178, ed. S.M. Nikol’skii, 1986, 113 p.

20.   Akishev G. The estimates of approximations classes in the Lorentz space. AIP Conf. Proc., vol. 1676, 020027, pp. 1–4. Internat. Conf. Advancements in Math. Sci. (5-7 November, 2015). Antalya, 2015.
doi: 10.1063/1.4930453 .

21.   Akishev G. Estimates of the best approximation of functions of the class with logarithmic smoothness in the Lorentz space. Proc. Intern. Conf. “Voronezhskaya zimnyaya matematicheskaya shkola” [Voronezh Winter Mathematical School’] (January 26 – February 1), Voronezh, 2017, pp. 12–14 (in Russian). 
ISBN: 978-5-9273-2415-6 .