References

  1. Akishev G.A. The ortho-diameters of Nikol’skii and Besov classes in the Lorentz spaces. Russian Math. (Iz. VUZ) 2009, 53 (2), 21–29. doi:10.3103/S1066369X09020029 (translation of Izv. Vyssh. Uchebn. Zaved. Mat. 2009, 2, 25–33. (in Russian))
  2. Amanov T.I. Representation and embedding theorems for function spaces \(S^{(r)}_{p,\theta}B(\mathbb{R}_n)\) and \(S^{(r)}_{p,\theta^*}B\), \((0\leq x_j\leq2\pi\); \(j=1,\ldots,n)\). Tr. Mat. Inst. Steklova 1965, 77, 5–34. (in Russian)
  3. Andrianov A.V., Temlyakov V.N. On two methods of generalization of properties of univariate function systems to their tensor product. Proc. Steklov Inst. Math. 1997, 219, 25–35. (translation of Tr. Mat. Inst. Steklova 1997, 219, 32–43. (in Russian))
  4. Balgimbayeva S.A., Smirnov T.I. Estimates of the Fourier widths of the classes of periodic functions with given majorant of the mixed modulus of smoothness. Sib. Math. J. 2018, 59 (2), 217–230. doi:10.1134/S0037446618020040 (translation of Sibirsk. Mat. Zh. 2018, 59 (2), 277–292. doi:10.17377/smzh.2018.59.204 (in Russian))
  5. Bari N.K., Stechkin S.B. The best approximations and differential properties of two conjugate functions. Trans. Moscow Math. Soc. 1956, 5, 483–522. (in Russian)
  6. Bazarkhanov D.B. Estimates of the Fourier widths of classes of Nikol’skii-Besov and Lizorkin-Triebel types of periodic functions of several variables. Math. Notes 2010, 87 (1-2), 281–284. doi:10.1134/S0001434610010359 (translation of Mat. Zametki 2010, 87 (2), 305–308. doi:10.4213/mzm8592 (in Russian))
  7. Belinskii E.S. Approximation by a “floating” system of exponentials on classes of periodic functions with a bounded mixed derivative. Research on the theory of functions of many real variables: Proc. of Yaroslavl’ State University 1988, 16–33. (in Russian).
  8. Belinsky E.S. Estimates of entropy numbers and Gaussian measures for classes of functions with bounded mixed derivative. J. Approx. Theory 1998, 93, 114–127. doi:10.1006/jath.1997.3157
  9. Bernstein S.N. Collected work, Vol. II. Constructive theory of functions (1931-1953). Nauka, Moscow, 1954. (in Russian)
  10. D\(\rm\tilde{u}\)ng D. Approximation by trigonometric polynomials of functions of several variables on the torus. Sb. Math. 1988, 59 (1), 247–267. doi:10.1070/SM1988v059n01ABEH003134 (translation of Mat. Sb. 1986, 131(173) (2), 251–271. (in Russian))
  11. D\(\rm\tilde{u}\)ng D., Temlyakov V.N., Ullrich T. Hyperbolic Cross Approximation. Birkhauser, Basel, 2018.
  12. Fedunyk-Yaremchuk O.V., Hembars’ka S.B. Estimates of approximative characteristics of the classes \(B^{\Omega}_{p,\theta}\) of periodic functions of several variables with given majorant of mixed moduli of continuity in the space \(L_{q}\). Carpathian Math. Publ. 2019, 11 (2), 281–295. doi:10.15330/cmp.11.2.281-295
  13. Fedunyk-Yaremchuk O.V., Hembars’kyi M.V., Hembars’ka S.B. Approximative characteristics of the Nikol’skii-Besov-type classes of periodic functions in the space \(B_{\infty,1}\). Carpathian Math. Publ. 2020, 12 (2), 376–391. doi:10.15330/cmp.12.2.376-391
  14. Fedunyk-Yaremchuk O.V., Solich K.V. Estimates of approximative characteristics of the classes \(B^{\Omega}_{p,\theta}\) of periodic functions of many variables with given majorant of mixed continuity moduli in the space \(L_{\infty}\). J. Math. Sci. (N.Y.) 2018, 231 (1), 28–40. doi:10.1007/s10958-018-3803-3 (translation of Ukr. Mat. Visn. 2017, 14 (3), 345–360. (in Ukrainian))
  15. Galeev E.M. Orders of the orthoprojection widths of classes of periodic functions of one and of several variables. Math. Notes 1988, 43 (2), 110–118. doi:10.1007/BF01152547 (translation of Mat. Zametki 1988, 43 (2), 197–211. (in Russian))
  16. Hembars’kyi M.V., Hembars’ka S.B. Approximate characteristics of the classes \(B^{\Omega}_{p,\theta}\) of periodic functions of one variable and many ones. J. Math. Sci. (N.Y.) 2019, 242 (6), 820–832. doi:10.1007/s10958-019-04518-0 (translation of Ukr. Mat. Visn. 2019, 16 (1), 88–104. (in Ukrainian))
  17. Hembars’kyi M.V., Hembars’ka S.B., Solich K.V. The best approximations and widths of the classes of periodic functions of one and several variables in the space \(B_{\infty,1}\). Mat. Stud. 2019, 51 (1), 74–85. doi:10.15330/ms.51.1.74-85 (in Ukrainian)
  18. Konograi A.F., Stasyuk S.A. Best trigonometric approximations of the classes \(B^{\Omega}_{p,\theta}\) of periodic functions of many variables. Approx. Theory of Functions and Related Problems: Proc. Inst. Math. NAS Ukr. 2007, 4 (1), 151–171. (in Ukrainian)
  19. Lizorkin P.I., Nikol’skii S.M. Function spaces of mixed smoothness from the decomposition point of view. Proc. Steklov Inst. Math. 1990, 187, 163–184. (translation of Tr. Mat. Inst. Steklova 1989, 187, 143–161. (in Russian))
  20. Nikol’skii S.M. Functions with dominant mixed derivative, satisfying a multiple Holder condition. Sibirsk. Mat. Zh. 1963, 4 (6), 1342–1364. (in Russian)
  21. Pustovoitov N.N. Representation and approximation of periodic functions of several variables with given mixed modulus of continuity. Anal. Math. 1994, 20, 35–48. doi:10.1007/BF01908917 (in Russian)
  22. Pustovoitov N.N. On the widths of multivariate periodic classes of functions whose mixed moduli of continuity are bounded by a product of power-and logarithmic-type functions. Anal. Math. 2008, 34, 187–224. doi:10.1007/s10476-008-0303-6 (in Russian)
  23. Romanyuk A.S. Approximative characteristics of the classes of periodic functions of many variables. Proc. Inst. Math. NAS Ukr., Kiev, 2012, 93. (in Russian)
  24. Romanyuk A.S. Approximation of ñlasses of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials. Ukrainian Math. J. 1996, 48 (1), 90–100. doi:10.1007/BF02390986 (translation of Ukrain. Mat. Zh. 1996, 48 (1), 80–89. (in Russian))
  25. Romanyuk A.S. Approximation of classes of periodic functions in several variables. Math. Notes 2002, 71 (1), 98–109. doi:10.1023/A:1013982425195 (translation of Mat. Zametki 2002, 71 (1), 109–121. doi:10.4213/mzm332 (in Russian))
  26. Romanyuk A.S. Best trigonometric approximations for some classes of periodic functions of several variables in the uniform metric. Math. Notes 2007, 82 (2), 216–228. doi:10.1134/S0001434607070279 (translation of Mat. Zametki 2007, 82 (2), 247–261. doi:10.4213/mzm3797 (in Russian))
  27. Romanyuk A.S. Bilinear and trigonometric approximations of periodic functions of several variables of Besov classes \(B^{r}_{p,\theta}\). Izv. Math. 2006, 70 (2), 277–306. doi:10.1070/IM2006v070n02ABEH002313 (translation of Izv. Ross. Akad. Nauk Ser. Mat. 2006, 70 (2), 69–98. doi:10.4213/im558 (in Russian))
  28. Romanyuk A.S. Diameters and best approximation of the classes \(B^r_{p,\theta}\) of periodic functions of several variables. Anal. Math. 2011, 37, 181–213. doi:10.1007/s10476-011-0303-9 (in Russian)
  29. Romanyuk A.S. Estimates for approximation characteristics of the Besov classes \(B^{r}_{p,\theta}\) of periodic functions of many variables in the space \(L_q\). I. Ukrainian Math. J. 2001, 53 (9), 1473–1482. doi:10.1023/A:1014314708184 (translation of Ukrain. Mat. Zh. 2001, 53 (9), 1224–1231. (in Russian))
  30. Romanyuk A.S. Estimates for approximation characteristics of the Besov classes \(B^{r}_{p,\theta}\) of periodic functions of many variables in the space \(L_q\). II. Ukrainian Math. J. 2001, 53 (10), 1703–1711. doi:10.1023/A:1015200128349 (translation of Ukrain. Mat. Zh. 2001, 53 (10), 1402–1408. (in Russian))
  31. Romanyuk A.S., Romanyuk V.S. Approximating characteristics of the classes of periodic multivariate functions in the space \(B_{\infty,1}\). Ukrainian Math. J. 2019, 71 (2), 308–321. doi:10.1007/s11253-019-01646-3 (translation of Ukrain. Mat. Zh. 2019, 71 (2), 271–282. (in Ukrainian))
  32. Romanyuk A.S., Romanyuk V.S. Estimation of some approximating characteristics of the classes of periodic functions of one and many variables. Ukrainian Math. J. 2020, 71 (8), 1257–1272. doi:10.1007/s11253-019-01711-x (translation of Ukrain. Mat. Zh. 2019, 71 (8), 1102–1115. (in Ukrainian))
  33. Romanyuk A.S., Yanchenko S. Ya. Approximation of classes of periodic functions of one and many variables from the Nikol’skii-Besov and Sobolev spaces. Ukrain. Mat. Zh. 2022, 74 (6), 857–868. (in Ukrainian)
  34. Stasyuk S.A., Fedunyk O.V. Approximation characteristics of the classes \(B^{\Omega}_{p,\theta}\) of periodic functions of many variables. Ukrainian Math. J. 2006, 58 (5), 779–793. doi:10.1007/s11253-006-0101-x (translation of Ukrain. Mat. Zh. 2006, 58 (5), 692–704. (in Ukrainian))
  35. Stechkin S.B. On the order of the best approximations of continuous functions. Izv. Ross. Akad. Nauk Ser. Mat. 1951, 15 (3), 219–242. (in Russian)
  36. Temlyakov V.N. Approximation of Periodic Functions. Nova Science Publishers Inc., New York, 1993.
  37. Temlyakov V.N. Approximation of functions with bounded mixed derivative. Proc. Steklov Inst. Math. 1989, 178, 1–121. (translation of Tr. Mat. Inst. Steklova 1986, 178, 3–113. (in Russian))
  38. Temlyakov V.N. Diameters of some classes of functions of several variables. Dokl. Akad. Nauk 1982, 267 (2), 314–317. (in Russian)
  39. Temlyakov V.N. Estimates of the asymptotic characteristics of classes of functions with bounded mixed derivative or difference. Proc. Steklov Inst. Math. 1990, 189, 161–197. (translation of Tr. Mat. Inst. Steklova 1989, 189, 138–168. (in Russian))
  40. Yongsheng S., Heping W. Representation and approximation of multivariate periodic functions with bounded mixed moduli of smoothness. Tr. Mat. Inst. Steklova 1997, 219, 356–377.
  41. Zaderey P.V., Hembars’ka S.B. Best orthogonal trigonometric approximations of the Nikol’skii-Besov-type classes of periodic functions in the space \(B_{\infty,1}\). Ukrain. Mat. Zh. 2022, 74 (6), 784–795. (in Ukrainian)