References

  1. Belinsky E.S. Approximation by a “floating” system of exponentials on classes of periodic functions with bounded mixed derivative. In: Studies in the Theory of Functions of Several Real Variables, Yaroslav State Univ., Yaroslavl, 1988, 16–33. (in Russian)
  2. 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
  3. Besov O.V. Investigation of a class of function spaces in connection with imbedding and extension theorems. Tr. Mat. Inst. Steklova 1961, 60, 42–81. (in Russian)
  4. DeVore R.A., Temlyakov V.N. Nonlinear approximation by trigonometric sums. J. Fourier Anal. Appl. 1995, 2 (1), 29–48. doi:10.1007/s00041-001-4021-8
  5. Düng D., Temlyakov V., Ullrich T. Hyperbolic cross approximation. Adv. Courses Math. Birkhauser, CRM Barselona, 2018. doi:10.1007/978-3-319-92240-9
  6. Dung D., Thanh V.Q. On nonlinear \(n\)-widths. Proc. Amer. Math. Soc. 1996, 124 (9), 2757–2765. doi:10.1090/S0002-9939-96-03337-0
  7. 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
  8. Fedunyk-Yaremchuk O.V., Hembars'ka S.B. Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables. Carpathian Math. Publ. 2022, 14 (1), 171–184. doi:10.15330/cmp.14.1.171-184
  9. Ismagilov R.S. Widths of sets in normed linear spaces and the approximation of functions by trigonometric polynomials. Russian Math. Surveys 1974, 29 (3), 169–186. doi:10.1070/RM1974v029n03ABEH001287 (translation of Uspekhi Mat. Nauk 1974, 29 (3(177)), 161–178 (in Russian))
  10. Jiang Y., Yongping L. Average widths and optimal recovery of multivariate Besov classes in \(L_p(\mathbb{R}^d)\). J. Approx. Theory 2000, 102 (1), 155–170. doi:10.1006/jath.1999.3384
  11. Kashin B.S., Temlyakov V.N. Best \(m\)-term approximations and the entropy of sets in the space \(L_1\). Math. Notes 1994, 56 (5-6), 1137–1157. doi:10.1007/BF02274662 (translation of Mat. Zametki 1994, 56 (5), 57–86. (in Russian))
  12. Kolmogorov A.N. Über die beste Annaherung von Funktionen einer gegebenen Funktionklasse. Ann. of Math.(2) 1936, 37 (1), 107–110. doi:10.2307/1968691
  13. Lizorkin P.I. Generalized Holder spaces \(B^{(r)}_{p,\theta}\) and their correlations with the Sobolev spaces \(L^{(r)}_p\). Sibirsk. Mat. Zh. 1968, 9 (5), 1127–1152. (in Russian)
  14. Nikol'skii S.M. Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables. Tr. Mat. Inst. Steklova 1951, 38, 244–278. (in Russian)
  15. 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))
  16. 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))
  17. Romanyuk A.S. Approximation of the isotropic classes \(\mathbf{B}^r_{p,\theta}\) of periodic functions of several variables in the space \(L_q\). Approx. Theory of Functions and Related Problems: Proc. Inst. Math. NAS Ukr. 2008, 5 (1), 263–278. (in Russian)
  18. Romanyuk A.S. Approximative characteristics of the isotropic classes of periodic functions of many variables. Ukrainian Math. J. 2009, 61 (4), 613–626. doi:10.1007/s11253-009-0232-y (translation of Ukraı̈n. Mat. Zh. 2009, 61 (4), 513–523. (in Russian))
  19. Romanyuk A.S. Bilinear approximations and Kolmogorov widths of periodic Besov classes. Theory of Operators, Differential Equations, and the Theory of Functions: Proc. Inst. Math. NAS Ukr. 2009, 6 (1), 222–236. (in Russian)
  20. Romanyuk A.S. Approximate characteristics of classes of periodic functions. Proc. of the Institute of Mathematics of the NAS of Ukraine, Kiev, 2012, 93. (in Russian)
  21. Romanyuk A.S. Best trigonometric and bilinear approximations of classes of functions of several variables. Math. Notes 2013, 94 (3), 379–391. doi:10.1134/S0001434613090095 (translation of Mat. Zametki 2013, 94 (3), 401–415. doi:10.4213/mzm8892 (in Russian))
  22. Romanyuk A.S. Entropy numbers and widths for the classes \(B^{r}_{p,\theta}\) of periodic functions of many variables. Ukrainian Math. J. 2017, 68 (10), 1620–1636. doi:10.1007/s11253-017-1315-9 (translation of Ukraı̈n. Mat. Zh. 2016, 68 (10), 1403–1417. (in Russian))
  23. Romanyuk A.S., Romanyuk V.S. Trigonometric and orthoprojection widths of classes of periodic functions of many variables. Ukrainian Math. J. 2009, 61 (10), 1589–1609. doi:10.1007/s11253-010-0300-3 (translation of Ukraı̈n. Mat. Zh. 2009, 61 (10), 1348–1366. (in Ukrainian))
  24. 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 Ukraı̈n. Mat. Zh. 2019, 71 (2), 271–281. (in Ukrainian))
  25. 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 Ukraı̈n. Mat. Zh. 2019, 71 (8), 1102–1115 (in Ukrainian))
  26. Romanyuk A.S., Romanyuk V.S. Approximative characteristics and properties of operators of the best approximation of classes of functions from the Sobolev and Nikol'skii-Besov spaces. J. Math. Sci. (N.Y.) 2021, 252 (4), 508–525. doi:10.1007/s10958-020-05177-2 (translation of Ukr. Mat. Visn. 2020, 17 (3), 372–395 (in Ukrainian))
  27. Romanyuk A.S., Yanchenko S.Ya. Approximation of the classes of periodic functions of one and many variables from the Nikol'skii-Besov and Sobolev spaces. Ukrainian Math. J. 2022, 74 (6), 967–980. doi:10.1007/s11253-022-02110-5 (translation of Ukraı̈n. Mat. Zh. 2022, 74 (6), 844–855. doi:10.37863/umzh.v74i6.7141 (in Ukrainian))
  28. Stasyuk S.A. Best \(m\)-term trigonometric approximation of periodic functions of several variables from Nikol'skii-Besov classes for small smoothness. J. Approx. Theory 2014, 177, 1–16. doi:10.1016/j.jat.2013.09.006
  29. Stechkin S.B. On absolute convergence of orthogonal series. Dokl. Akad. Nauk SSSR 1955, 102 (2), 37–40. (in Russian)
  30. Stepanyuk T.A. Order estimates of best orthogonal trigonometric approximations of classes of infinitely differentiable functions. In: Raigorodskii A., Rassias, M. (Eds.) Trigonometric Sums and Their Applications. Springer, Cham., 2020, 273–287. doi:10.1007/978-3-030-37904-9_13
  31. 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))
  32. Temlyakov V.N. Approximation of periodic functions. Nova Sci. Publ., New York, 1993.
  33. Temlyakov V.N. Greedy algorithm and \(m\)-term trigonometric approximation. Constr. Approx. 1998, 14 (4), 569–587. doi:10.1007/s003659900090
  34. Temlyakov V.N. Multivariate approximation. Cambridge University Press, 2018.
  35. Tikhomirov V.M. Widths of sets in function spaces and the theory of best approximations. Russian Math. Surveys 1960, 15 (3), 75–111. doi:10.1070/RM1960v015n03ABEH004093 (translation of Uspekhi Mat. Nauk 1960, 15 (3(93)), 81–120. (in Russian))
  36. Trigub R.M., Belinsky E.S. Fourier Analysis and Approximation of Functions. Kluwer Academic Publishers, Dordrecht, 2004.