Skip to main content
SpringerLink
Log in

Inequalities for moduli of smoothness of functions and their Liouville–Weyl derivatives

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We obtain the Lp inequalities for moduli of smoothness of the generalized Liouville–Weyl derivative in terms of moduli of smoothness of a function itself. We study the limiting cases L1 and L∞.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bari, N.K., Steckin, S.B.: Best approximations and differential properties of two conjugate functions. Trudy Moskov. Mat. Obšč. 5, 483–522 (1956). (in Russian)

    MathSciNet  Google Scholar 

  2. N. K. Bari, Trigonometric Series, Nauka (Moscow, 1961).

  3. Butzer, P.L., Dyckhoff, H., Gorlich, E., Stens, R.L.: Best trigonometric approximation, fractional order derivatives and Lipschitz classes. Canad. J. Math. 29, 781–793 (1977)

    Article  MathSciNet  Google Scholar 

  4. R. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag (Berlin, 1993).

  5. Ditzian, Z., Tikhonov, S.: Moduli of smoothnes of functions and their derivatives. Studia Math. 180, 143–160 (2007)

    Article  MathSciNet  Google Scholar 

  6. Ditzian, Z., Tikhonov, S.: Ul’yanov and Nikol’skii-type inequalities. J. Approx. Theory 133, 100–133 (2005)

    Article  MathSciNet  Google Scholar 

  7. H. Johnen and K. Scherer, On the equivalence of the \(K\)-functional and moduli of continuity and some applications, in: Constructive Theory of Functions of Several Variables (Proc. Conf., Math. Res. Inst., Oberwolfach, (1976)), Lecture Notes in Math. 571, Springer-Verlag (Berlin–Heidelberg, 1977), pp. 119–140.

  8. Jumabayeva, A.: Liouville-Weyl derivatives, best approximations, and moduli of smoothness. Acta Math. Hungar. 145, 369–391 (2015)

    Article  MathSciNet  Google Scholar 

  9. Jumabayeva, A.: Sharp Ul’yanov inequalities for generalized Liouville-Weyl derivatives. Analysis Math. 43, 279–302 (2017)

    Article  MathSciNet  Google Scholar 

  10. Y. Kolomoitsev and S. Tikhonov, Hardy–Littlewood and Ulyanov inequalities, Memoirs of the AMS (to appear), arXiv:1711.08163.

  11. Liflyand, E., Tikhonov, S.: A concept of general monotonicity and application. Math. Nachr. 284, 1083–1098 (2011)

    Article  MathSciNet  Google Scholar 

  12. Potapov, M.K., Simonov, B.V., Tikhonov, S.: Mixed moduli of smoothness in \(L_p\), \(1 < p < \infty \): a survey. Surv. Approx. Theory 8, 1–57 (2013)

    MathSciNet  MATH  Google Scholar 

  13. M. K. Potapov, B. V. Simonov and S. Tikhonov, On the Besov and the Besov–Nikolskii classes and estimates for mixed moduli of smoothness of fractional derivatives, Tr. Mat. Inst. Steklova, 243 (2003), 244–256 (in Russian); translation in Proc. Steklov Inst. Math., 243 (2003), 234–246.

  14. Salem, R.: Sur les transformations des séries de Fourier. Fund. Math. 33, 108–114 (1939)

    Article  MathSciNet  Google Scholar 

  15. Simonov, B., Tikhonov, S.: On embedings of functional classes defined by constructive characteristics. Banach Center Publ. 72, 285–307 (2006)

    Article  Google Scholar 

  16. B. Simonov and S. Tikhonov, Embedings theorems in the constructive theory of approximations, Mat. Sb., 199 (2008), 107–148 (in Russian); translation in Sb. Math., 199 (2008), 1367–1407.

  17. Simonov, B., Tikhonov, S.: Sharp Ul’yanov-type inequalities using fractional smoothness. J. Approx. Theory 162, 1654–1684 (2010)

    Article  MathSciNet  Google Scholar 

  18. S. B. Steckin, On the problem of multipliers for trigonometric polynomials, Dokl. Akad. Nauk SSSR (N.S.), 75 (1950), 165–168 (in Russian).

  19. Steckin, S.B.: On best approximation of conjugate functions by trigonometric polynomials. Izv. Akad. Nauk SSSR Ser. Mat. 20, 197–206 (1956). (in Russian)

    MathSciNet  Google Scholar 

  20. Taberski, R.: Differences, moduli and derivatives of fractional orders. Comment. Math. Prace Mat. 19, 389–400 (1977)

    MathSciNet  MATH  Google Scholar 

  21. Tikhonov, S.: Weak type inequalities for moduli of smoothness: the case of limit value parameters. J. Fourier Anal. Appl. 16, 590–608 (2010)

    Article  MathSciNet  Google Scholar 

  22. Tikhonov, S.: Trigonometric series with general monotone coefficients. J. Math. Anal. Appl. 326, 721–735 (2007)

    Article  MathSciNet  Google Scholar 

  23. S. Tikhonov, Embedding results in questions of strong approximation by Fourier series, Acta Sci. Math. (Szeged), 72 (2006), 117–128; published first as S. Tikhonov, Embedding theorems of function classes, IV., CRM preprint (November 2005).

  24. Tikhonov, S., Trebels, W.: Ulyanov-type inequalities and generalized Liouville derivatives. Proc. Roy. Soc. Edinburgh Sect. A 141, 205–224 (2011)

    Article  MathSciNet  Google Scholar 

  25. Trebels, W.: Inequalities for moduli of smoothness versus embeddings of function spaces. Arch. Math. 94, 155–164 (2010)

    Article  MathSciNet  Google Scholar 

  26. Ul’yanov, P.L.: The embedding of certain classes H\(^{\omega }_p\) of functions. Izv. Akad. Nauk SSSR Ser. Mat. 2, 601–637 (1968). (in Russian)

    Google Scholar 

  27. Uno, Y.: Lipschitz functions and convolution. Proc. Japan Acad. 50, 785–788 (1974)

    Article  MathSciNet  Google Scholar 

  28. A. Zygmund, Trigonometric Series, Vols. I, II, 3rd edition, Cambridge University Press (Cambridge, 2002).

Download references

Author information

Authors and Affiliations

  1. Gumilyov Eurasian National University, Munaitpasov 7, 010010, Astana, Kazakhstan

    A. Jumabayeva

  2. Universitat Autonoma de Barcelona, and Centre de Recerca Matematica, Campus de Bellaterra, Edifici C, Barcelona, 08193, Spain

    A. Jumabayeva

  3. Volgograd State Technical University, Volgograd, 400131, Russia

    B. Simonov

Authors
  1. A. Jumabayeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Simonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Jumabayeva.

Additional information

This research was partially supported by the MTM201459174P and RFBR (grant No. 160100350), AP 05132590.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jumabayeva, A., Simonov, B. Inequalities for moduli of smoothness of functions and their Liouville–Weyl derivatives. Acta Math. Hungar. 156, 1–17 (2018). https://doi.org/10.1007/s10474-018-0867-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-018-0867-y

Key words and phrases

Mathematics Subject Classification

Search

Navigation