The best approximations  and widths of the classes of periodical functions of one and several variables in the space  $B_{\infty, 1}$

M. V. Hembarskyi; S. B. Hembarska; K. V. Solich

doi:10.15330/ms.51.1.74-85

We obtained exact-order estimates of the best approximations of classes $B^{\,\Omega}_{\infty,{\,\theta}}$ of periodic functions of many variables and for classes $B^{\omega}_{p,\theta}, \,\, 1 \leq p \le \infty$ of functions of one variable by trigonometric polynomials with corresponding spectra of harmonics in the metric of space $B_{\infty, 1}$. We also found exact orders for the Kolmogorov, linear and trigonometric widths of the same classes in space $B_{\infty, 1}$.

1. M.V. Hemarskyi, S.B. Hembarska, Width of classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables in the space $B_{1,1}$, Ukr. Math. Bull., 15 (2018), №1, 43-56. (in Ukrainian)

2. A.S. Romanyuk, V.S. Romanyuk, Approximating characteristics of the classes of periodic multivariate functions in the space $B_{\infty,1}$, Ukr. Mat. Zh., 71 (2019), №2, 271-282.

3. Dung D., Temlyakov V.N., Ullrich T. Hyperbolic Cross Approximation, Birkhauser, 2018, 218 p.

4. T.I. Amanov, Representation and embedding theorems for function spaces, Trudy Mat. Inst. Steklov., 77 (1965), 5-34. (in Russian)

5. P.I. Lizorkin, S.M. Nikolski.i, Function spaces of mixed smoothness from the decomposition point of view, Trudy Mat. Inst. Steklov, 187 (1989), 143-161. (in Russian)

6. N.N. Pustovoitov, Representation and approximation of periodic functions of several variables with given mixed modulus of continuity, Anal. Math., 20 (1994), 35-48. (in Russian)

7. Sun Yongsheng; Wang, Heping, Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness, Tr. Mat. Inst. Steklova, 219 (1997), Teor. Priblizh. Garmon. Anal., 356-377 (in Russian); En. transl: Proc. Steklov Inst. Math., 219 (1997), №4, 350-371.

8. S.N. Bernstein, Collected Works, V.II, Constructive theory of functions, 1931.1953, Moscow, Akademia Nauk SSSR, 1954.

9. S.B. Ste.ckin, On the order of the best approximations of continuous functions, Izvestiya Akad. Nauk SSSR. Ser. Mat., 15 (1951), №3, 219-242. (in Russian)

10. N.K. Bari, S.B. Ste.ckin, Best approximations and differential properties of two conjugate functions, Trudy Moskov. Mat. Ob.s.c., 5 (1956), 483-522. (in Russian)

11. S.M. Nikolski.i, Functions with dominatinf mixed derivative satisfying multiple Holders condition, Sibirsk. Mat. Zh., 4 (1963), №6, 1342-1364. (in Russian)

12. S.A. Stasyuk, O.V. Fedunyk, Approximation characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of many variables, Ukr. Matem. Zh., 58 (2006), №5, 692-704. (in Ukrainian); Engl. transl.: Ukr. Math. J., 58 (2006), №5, 779-793.

13. V.N. Temlyakov, Approximations of functions with bounded mixed derivative, Trudy Mat. Inst. Steklov., 178 (1986), 1-112. (in Russian)

14. Temlyakov V.N. Approximation of periodic functions, New York: Nova Sci. Publ. Inc., 1993, 419 p. 15. A.S. Romanyuk, Approximative characteristics of classes of periodic functions of several variables, Pratsi Inst. Math. Nats. Akad. Nauk Ukr., 93 (2012), 353 p. (in Russian)

16. Kolmoqoroff A. Uber die beste Annaaherung von Funktionen einer gegebenen Funktionenklasse, Ann. of Math. 37 (1936), №2, 107-111.

17. V.M. Tihomirov, Diameters of sets in functional spaces and the theory of best approximations, Uspehi Mat. Nauk, 15 (1960), №3, 81-120 (in Russian); Engl. transl.: Russian Math. Surveys, 15 (1960), №3, 75-111.

18. R.S. Ismagilov, Diameters of sets in normed linear spaces, and the approximation of functions by trigonometric polynomials, Uspehi Mat. Nauk, 29 (1974), №3(177), 161.178, (in Russian); Engl. transl.: Russian Math. Surveys, 29 (1974), №3, 16-186.

19. A.S. Romanyuk, On the best approximations and Kolmogorov widths of besov classes of periodic functions of many variables, Ukr. Mat. Zh., 47 (1995), №1, 79.92, (in Ukrainian); Engl. transl.: Ukr. Math. J., 47 (1995), №1, 91-106.

20. A.S. Romanyuk, Linear Widths of the Besov Classes of Periodic Functions of Many Variables. I, Ukr. Mat. Zh., 53 (2001), №5, 647.661, (in Ukrainian); Engl. transl.: Ukr. Math. J., 53 (2001), №5, 744-761.

21. A.S. Romanyuk, Linear Widths of the Besov Classes of Periodic Functions of Many Variables. I, Ukr. Mat. Zh., 53 (2001), №6, 820.829, (in Ukrainian); Engl. transl.: Ukr. Math. J., 53 (2001), №6, 965-977.

22. A.S. Romanyuk, Kolmogorov widths of the Besov classes $B^r_{p,\theta}$ in the metric of the space $L_{\infty}$, Ukr. Mat. Visn., 2 (2005), №2, 201.218, (in Russian); Engl. transl.: Ukr. Math. Bull., 2 (2005), №2, 205-222.

23. A.S. Romanyuk, Kolmogorov and trigonometric widths of the Besov classes $B^r_{p,\theta}$ of multivariate periodic functions, Mat. Sb., 197 (2006), №1, 71.96, (in Russian); Engl. transl.: Sb. Math., 197 (2006), №1, 69-93.

24. A.S. Romanyuk, Best approximations and widths of classes of periodic functions of several variables, Mat. Sb., 199 (2008), №2, 93.114; (in Russian), Engl. transl.: Sb. Math., 199 (2008), №2, 253-275.

25. A.S. Romanyuk, Diameters and best approximation of the classes B$B^r_{p,\theta}$ of periodic functions of several variables, Anal. Math., 37 (2011), 181-213. (in Russian)

26. A.F. Konograi, Kolomogorov widths of classes $B^{\Omega}_{p,\theta}$ in the space $L_\infty$, Zb. Pr. Inst. Mat. NAN Ukr., 3 (2006), №4, 181-197. (in Ukrainian)

	The best approximations and widths of the classes of periodical functions of one and several variables in the space $B_{\infty, 1}$ (in Ukrainian)
Author	M. V. Hembarskyi¹, S. B. Hembarska², K. V. Solich³ gembarskaya72@gmail.com¹ Lesya Ukrainka Eastern European National University, Lutsk, Ukraine
Abstract	We obtained exact-order estimates of the best approximations of classes $B^{\,\Omega}_{\infty,{\,\theta}}$ of periodic functions of many variables and for classes $B^{\omega}_{p,\theta}, \,\, 1 \leq p \le \infty$ of functions of one variable by trigonometric polynomials with corresponding spectra of harmonics in the metric of space $B_{\infty, 1}$. We also found exact orders for the Kolmogorov, linear and trigonometric widths of the same classes in space $B_{\infty, 1}$.
Keywords	periodical function; exact-order estimate; best approximantion; linear width; trigonometric width; the Kolmogorov width; several variables
DOI	doi:10.15330/ms.51.1.74-85
Reference	1. M.V. Hemarskyi, S.B. Hembarska, Width of classes $B^{\Omega}_{p,\theta}$ of periodic functions of several variables in the space $B_{1,1}$, Ukr. Math. Bull., 15 (2018), №1, 43-56. (in Ukrainian) 2. A.S. Romanyuk, V.S. Romanyuk, Approximating characteristics of the classes of periodic multivariate functions in the space $B_{\infty,1}$, Ukr. Mat. Zh., 71 (2019), №2, 271-282. 3. Dung D., Temlyakov V.N., Ullrich T. Hyperbolic Cross Approximation, Birkhauser, 2018, 218 p. 4. T.I. Amanov, Representation and embedding theorems for function spaces, Trudy Mat. Inst. Steklov., 77 (1965), 5-34. (in Russian) 5. P.I. Lizorkin, S.M. Nikolski.i, Function spaces of mixed smoothness from the decomposition point of view, Trudy Mat. Inst. Steklov, 187 (1989), 143-161. (in Russian) 6. N.N. Pustovoitov, Representation and approximation of periodic functions of several variables with given mixed modulus of continuity, Anal. Math., 20 (1994), 35-48. (in Russian) 7. Sun Yongsheng; Wang, Heping, Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness, Tr. Mat. Inst. Steklova, 219 (1997), Teor. Priblizh. Garmon. Anal., 356-377 (in Russian); En. transl: Proc. Steklov Inst. Math., 219 (1997), №4, 350-371. 8. S.N. Bernstein, Collected Works, V.II, Constructive theory of functions, 1931.1953, Moscow, Akademia Nauk SSSR, 1954. 9. S.B. Ste.ckin, On the order of the best approximations of continuous functions, Izvestiya Akad. Nauk SSSR. Ser. Mat., 15 (1951), №3, 219-242. (in Russian) 10. N.K. Bari, S.B. Ste.ckin, Best approximations and differential properties of two conjugate functions, Trudy Moskov. Mat. Ob.s.c., 5 (1956), 483-522. (in Russian) 11. S.M. Nikolski.i, Functions with dominatinf mixed derivative satisfying multiple Holders condition, Sibirsk. Mat. Zh., 4 (1963), №6, 1342-1364. (in Russian) 12. S.A. Stasyuk, O.V. Fedunyk, Approximation characteristics of the classes $B^{\Omega}_{p,\theta}$ of periodic functions of many variables, Ukr. Matem. Zh., 58 (2006), №5, 692-704. (in Ukrainian); Engl. transl.: Ukr. Math. J., 58 (2006), №5, 779-793. 13. V.N. Temlyakov, Approximations of functions with bounded mixed derivative, Trudy Mat. Inst. Steklov., 178 (1986), 1-112. (in Russian) 14. Temlyakov V.N. Approximation of periodic functions, New York: Nova Sci. Publ. Inc., 1993, 419 p. 15. A.S. Romanyuk, Approximative characteristics of classes of periodic functions of several variables, Pratsi Inst. Math. Nats. Akad. Nauk Ukr., 93 (2012), 353 p. (in Russian) 16. Kolmoqoroff A. Uber die beste Annaaherung von Funktionen einer gegebenen Funktionenklasse, Ann. of Math. 37 (1936), №2, 107-111. 17. V.M. Tihomirov, Diameters of sets in functional spaces and the theory of best approximations, Uspehi Mat. Nauk, 15 (1960), №3, 81-120 (in Russian); Engl. transl.: Russian Math. Surveys, 15 (1960), №3, 75-111. 18. R.S. Ismagilov, Diameters of sets in normed linear spaces, and the approximation of functions by trigonometric polynomials, Uspehi Mat. Nauk, 29 (1974), №3(177), 161.178, (in Russian); Engl. transl.: Russian Math. Surveys, 29 (1974), №3, 16-186. 19. A.S. Romanyuk, On the best approximations and Kolmogorov widths of besov classes of periodic functions of many variables, Ukr. Mat. Zh., 47 (1995), №1, 79.92, (in Ukrainian); Engl. transl.: Ukr. Math. J., 47 (1995), №1, 91-106. 20. A.S. Romanyuk, Linear Widths of the Besov Classes of Periodic Functions of Many Variables. I, Ukr. Mat. Zh., 53 (2001), №5, 647.661, (in Ukrainian); Engl. transl.: Ukr. Math. J., 53 (2001), №5, 744-761. 21. A.S. Romanyuk, Linear Widths of the Besov Classes of Periodic Functions of Many Variables. I, Ukr. Mat. Zh., 53 (2001), №6, 820.829, (in Ukrainian); Engl. transl.: Ukr. Math. J., 53 (2001), №6, 965-977. 22. A.S. Romanyuk, Kolmogorov widths of the Besov classes $B^r_{p,\theta}$ in the metric of the space $L_{\infty}$, Ukr. Mat. Visn., 2 (2005), №2, 201.218, (in Russian); Engl. transl.: Ukr. Math. Bull., 2 (2005), №2, 205-222. 23. A.S. Romanyuk, Kolmogorov and trigonometric widths of the Besov classes $B^r_{p,\theta}$ of multivariate periodic functions, Mat. Sb., 197 (2006), №1, 71.96, (in Russian); Engl. transl.: Sb. Math., 197 (2006), №1, 69-93. 24. A.S. Romanyuk, Best approximations and widths of classes of periodic functions of several variables, Mat. Sb., 199 (2008), №2, 93.114; (in Russian), Engl. transl.: Sb. Math., 199 (2008), №2, 253-275. 25. A.S. Romanyuk, Diameters and best approximation of the classes B$B^r_{p,\theta}$ of periodic functions of several variables, Anal. Math., 37 (2011), 181-213. (in Russian) 26. A.F. Konograi, Kolomogorov widths of classes $B^{\Omega}_{p,\theta}$ in the space $L_\infty$, Zb. Pr. Inst. Mat. NAN Ukr., 3 (2006), №4, 181-197. (in Ukrainian)
Pages	74-85
Volume	51
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

The best approximations and widths of the classes of periodical functions of one and several variables in the space $B_{\infty, 1}$ (in Ukrainian)