Processing math: 100%

ON WIDTHS OF SOME CLASSES OF ANALYTIC FUNCTIONS IN A CIRCLE

Mirgand Sh. Shabozov     (Tajik National University, 17 Rudaky Ave., Dushanbe, 734025, Tajikistan)
Muqim S. Saidusainov     (University of Central Asia, 155 Qimatsho Imatshoev, Khorog, GBAO, Tajikistan)

Abstract


We calculate exact values of some $n$-widths of the class W(r)q(Φ), r∈Z+, in the Banach spaces Lq,γ and Bq,γ, 1≤q≤∞, with a weight γ. These classes consist of functions f analytic in the unit circle, their rth order derivatives f(r) belong to the Hardy space Hq, 1≤q≤∞, and the averaged moduli of smoothness of boundary values of f(r) are bounded by a given majorant Φ at the system of points {π/(2k)}k∈N; more precisely,
kπ−2∫π/(2k)0ω2(f(r),2t)Hq,ρdt≤Φ(π2k)
for all k∈N, k>r.


Keywords


Modulus of smoothness, The best approximation, n-widths, The best linear method of approximation

Full Text:

PDF

References


  1. Ainulloev N., Taikov L.V. Best approximation in the sense of Kolmogorov of classes of functions analytic in the unit disc. Math. Notes, 1986. Vol. 40, No. 3. P. 699–705. DOI: 10.1007/BF01142473
  2. Babenko K.I. Best approximations for a class of analytic functions. Izv. Akad. Nauk SSSR. Ser. Mat., 1958. Vol. 22, No. 5. P. 631–640. (in Russian)
  3. Dveǐrin M.Z. Widths and ε-entropy of classes of analytic functions in the unit disc. Teor. Funkts., Funkts. Anal. Prilozh., 1975. Vol. 23, P. 32–46. (in Russian)
  4. Dvejrin M.Z., Chebanenko I.V. On the polynomial approximation in Banach spaces of analytic functions. In: Theory of mappings and approximation of functions. Collect. Sci. Works. Kiev: Naukova Dumka, 1983. P. 62–73. (in Russian)
  5. Farkov Yu.A. Widths of Hardy classes and Bergman classes on the ball in Cn. Russian Math. Surveys, 1990. Vol. 45, No. 5. P. 229–231. DOI: 10.1070/RM1990v045n05ABEH002677
  6. Farkov Yu.A. n-Widths, Faber expansion, and computation of analytic functions. J. Complexity, 1996. Vol. 12, No. 1. P. 58–79. DOI: 10.1006/jcom.1996.0007
  7. Fisher S.D., Stessin M.I. The n-width of the unit ball of Hq. J. Approx. Theory, 1991. Vol. 67, No. 3. P. 347–356. DOI: 10.1016/0021-9045(91)90009-Y
  8. Pinkus A. n-Widths in Approximation Theory. Heidelberg: Springer-Verlag, 1985. 252 p. DOI: 10.1007/978-3-642-69894-1
  9. Saidusainov M.S. On the best linear method of approximation of some classes analytic functions in the weighted Bergman space. Chebyshevskii Sb., 2016. Vol. 17, No. 1. P. 240–253. (in Russian)
  10. Saidusainov M.S. K-functionals and exact values of n-widths in the Bergman space. Ural Math. J., 2017. Vol. 3, No. 2. P. 74–81. DOI: 10.15826/umj.2017.2.010
  11. Saidusainov M.S. Analysis of a theorem on the Jackson–Stechkin inequality in the Bergman space B2. Trudy Inst. Mat. Mekh. UrO RAN, 2018. Vol. 24, No. 4. P. 217–224. (in Russian) DOI: 10.21538/0134-4889-2018-24-4-217-224
  12. Saidusainov M.S. Some inequalities between the best simultaneous approximation and the modulus of continuity in a weighted Bergman space. Ural Math. J., 2023. Vol. 9, No. 2. P. 165–174. DOI: 10.15826/umj.2023.2.014
  13. Shabozov M.Sh. Widths of some classes of analytic functions in the Bergman space. Dokl. Math., 2002. Vol. 65, No. 2. P. 194–197.
  14. Shabozov M.Sh. On the best simultaneous approximation of functions in the Hardy space. Trudy Inst. Mat. Mekh. UrO RAN, 2023. Vol. 29, No. 4. P. 283–291. (in Russian) DOI: 10.21538/0134-4889-2023-29-4-283-291
  15. Shabozov M.Sh. On the best simultaneous approximation in the Bergman space B2. Math. Notes, 2023. Vol. 114. P. 377–386. DOI: 10.1134/S0001434623090080
  16. Shabozov M.Sh., Saidusaynov M.S. Mean-square approximation of complex variable functions by Fourier series in the weighted Bergman space. Vladikavkazskii Mat. Zh., 2018. Vol. 20, No. 1. P. 86–97. (in Russian) DOI: 10.23671/VNC.2018.1.11400
  17. Shabozov M.Sh., Saidusaynov M.S. Upper bounds for the approximation of certain classes of functions of a complex variable by Fourier series in the space L2 and n-widths. Math. Notes, 2018. Vol. 103. P. 656–668. DOI: 10.1134/S0001434618030343
  18. Shabozov M.Sh., Saidusainov M.S. Mean-square approximation of functions of a complex variable by Fourier sums in orthogonal systems. Trudy Inst. Mat. Mekh. UrO RAN, 2019. Vol. 25, No. 2. P. 258–272. (in Russian) DOI: 10.21538/0134-4889-2019-25-2-258-272
  19. Shabozov M.Sh., Saidusaynov M.S. approximation of functions of a complex variable by Fourier sums in orthogonal systems in L2. Russ. Math., 2020. Vol. 64. P. 56–62. DOI: 10.3103/S1066369X20060080
  20. Shabozov M.Sh., Saidusainov M.S. Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space B2,γ. Chebyshevskii Sb., 2022. Vol. 23, No. 1. P. 167–182. (in Russian) DOI: 10.22405/2226-8383-2022-23-1-167-182
  21. Shabozov M.Sh., Shabozov O.Sh. Widths of some classes of analytic functions in the Hardy space H2. Math. Notes, 2000. Vol. 68, No. 5–6. P. 675–679. DOI: 10.1023/A:1026692112651
  22. Shabozov M.Sh., Shabozov O.Sh. On the best approximation of some classes of analytic functions in weighted Bergman spaces. Dokl. Math., 2007. Vol. 75, No. 1. P. 97–100. DOI: 10.1134/S1064562407010279
  23. Shabozov M.Sh., Yusupov G.Y. Best approximation and width of some classes of analytic functions. Dokl. Math., 2002. Vol. 65, No. 1. P. 111–113.
  24. Shabozov M.Sh., Yusupov G.A. On the best polynomial approximation of functions in the Hardy space Hq,R(1≤q≤∞,R≥1). Chebyshevskii Sb., 2023. Vol. 24, No. 1. P. 182–193. (in Russian)  DOI: 10.22405/2226-8383-2023-24-1-182-193
  25. Shikhalev N.I. An inequality of Bernstein-Markov kind for analytic functions. Dokl. Akad. Nauk Azerb. SSR, 1975. Vol. 31, No. 8. P. 9–14. (in Russian)
  26. Smirnov V.I., Lebedev N.A. A Constructive Theory of Functions of a Complex Variable. Moscow-Leningrad: Nauka. 1964. 440 p.
  27. Taikov L.V. On the best approximation in the mean of certain classes of analytic functions. Math. Notes, 1967. Vol. 1. P. 104–109. DOI: 10.1007/BF01268058
  28.  Taikov L.V. Diameters of certain classes of analytic functions. Math. Notes, 1977. Vol. 22. P. 650–656. DOI: 10.1007/BF01780976
  29. Tikhomirov V.M. Diameters of sets in function spaces and the theory of best approximations. Russian Math. Surveys, 1960. Vol. 15, No. 3. P. 75–111. DOI: 10.1070/rm1960v015n03abeh004093
  30. Tikhomirov V.M. Approximation Theory. In: Analysis II. Convex Analysis and Approximation Theory, Gamkrelidze R.V. eds. Encyclopaedia Math. Sci., vol. 14. Berlin, Heidelberg: Springer, 1990. P. 93–243. DOI: 10.1007/978-3-642-61267-1_2
  31. Vakarchuk S.B. Widths of certain classes of analytic functions in the Hardy space H2. Ukr. Math. J., 1989. Vol. 41, No. 6. P. 686–689. DOI: 10.1007/BF01060570
  32. Vakarchyuk S.B. Best linear methods of approximation and widths of classes of analytic functions in a disk. Math. Notes, 1995. Vol. 57. P. 21–27. DOI: 10.1007/BF02309390
  33. Vakarchuk S.B. Exact values of widths for certain functional classes. Ukr. Math. J., 1996. Vol. 48. P. 151–153. DOI: 10.1007/BF02390993
  34. Vakarchyuk S.B. Exact values of widths of classes of analytic functions on the disk and best linear approximation methods. Math. Notes, 2002. Vol. 72. P. 615–619. DOI: 10.1023/A:1021496620022
  35. Vakarchuk S.B. On some extremal problems of approximation theory in the complex plane. Ukr. Math. J., 2004. Vol. 56. P. 1371–1390. DOI: 10.1007/s11253-005-0122-x
  36. Vakarchuk S.B., Zabutnaya V.I. Best linear approximation methods for functions of Taikov classes in the Hardy spaces Hq,ρ, q≥1, 0<ρ≤1. Math. Notes, 2009. Vol. 85. P. 322–327. DOI: 10.1134/S000143460903002X




DOI: http://dx.doi.org/10.15826/umj.2024.2.011

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.