Liouville–Weyl Derivatives of Double Trigonometric Series

  • Ainur Jumabayeva
  • Boris Simonov
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


We obtain estimates of norms and best approximations of the generalized Liouville–Weyl derivative in the two-dimensional case.


Best approximation Moduli of smoothness Liouville–Weyl derivative General monotone sequences 

Mathematics Subject Classification (2000)

42A10 41A25 26A48 



This research was partially supported by the RFBR (grant N 18-01-00281), AP 05132590.


  1. 1.
    S. Bernstein, On the best approximation of continuous functions by polynomials of a given degree. Commun. Soc. Math. Kharkow Ser. 13(2), 49–194 (1912)Google Scholar
  2. 2.
    R. DeVore, G.G. Lorentz, Constructive Approximation (Springer, Berlin, 1993)CrossRefGoogle Scholar
  3. 3.
    M. Dyachenko, S. Tikhonov, Convergence of trigonometric series with general monotone coefficients. C. R. Acad. Sci. Paris 345(3), 123–126 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. Dyachenko, S. Tikhonov, A Hardy–Littlewood theorem for multiple series. J. Math. Anal. Appl. 339, 503–510 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Jumabayeva, Liouville–Weyl derivatives, best approximations, and moduli of smoothness. Acta Math. Hungar. 145(2), 369–391 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A.A. Konyushkov, Best approximations by trigonometric polynomials and Fourier coefficients. Mat. Sb. (N.S.), 44(86), 53–84 (1958)Google Scholar
  7. 7.
    E. Liflyand, S. Tikhonov, A concept of general monotonicity and application. Math. Nachr. 284(8–9), 1083–1098 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    S.M. Nikolskii, Approximation of Functions of Many Variables and Imbedding Theorems. Nauka, M., 1969. English translation: S.M. Nikolskii, Approximation of Functions of Several Variables and Imbedding Theorems (Springer, New York, 1975)Google Scholar
  9. 9.
    M.K. Potapov, On Approximation by “angle” in Proceedings of the Conference on Constructive Theory of Functions. Approximation Theory (1969) (Akad. Kiadó, Budapest, 1972), pp. 371–399Google Scholar
  10. 10.
    M.K. Potapov, B.V. Simonov, S. Tikhonov, Relations between mixed moduli of smoothness and embedding theorems for Nikol’skii classes. Proc. Steklov Inst. Math. 269, 197–207 (2010); translation from Russian: Trudy Matem. Inst. V. A. Steklova 269, 204–214 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    M.K. Potapov, B.V. Simonov, S. Tikhonov, Mixed moduli of smoothness in L p, 1 < p < : a survey. Surveys Approx. Theory 8, 1–57 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    M.K. Potapov, B.V. Simonov, S. Tikhonov, Fractional Moduli of Smoothness (Maks Press, Moscow, 2016) (in Russian)Google Scholar
  13. 13.
    B. Simonov, S. Tikhonov, Embedding theorems in constructive approximation. Sb. Math. 199(9), 1367–1407 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    S.B. Steckin, On best approximation of conjugate functions by trigonometric polynomials. Izv. Akad. Nauk SSSR Ser. Mat. 20, 197–206 (1956)MathSciNetGoogle Scholar
  15. 15.
    S. Tikhonov, Embedding results in questions of strong approximation by Fourier series. Acta Sci. Math. (Szeged) 72, 117–128 (2006); published first as S. Tikhonov, Embedding theorems of function classes, IV. CRM preprint (2005)Google Scholar
  16. 16.
    S. Tikhonov, Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326, 721–735 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    S. Tikhonov, Trigonometric series of Nikol’skii classes. Acta Math. Hungar. 114(1–2), 61–78 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    M.F. Timan, The imbedding of the \(L_p^{(k)}\) classes of functions. Izv. Vyssh. Uchebn. Zaved. Mat. 10, 61–74 (1974)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ainur Jumabayeva
    • 1
  • Boris Simonov
    • 2
  1. 1.L.N. Gumilyov Eurasian National UniversityNur-SultanKazakhstan
  2. 2.Volgograd State Technical UniversityVolgogradRussia

Personalised recommendations