Skip to main content
SpringerLink
Log in

Smoothness of functions in the metric spaces L ψ

  • Published:
Ukrainian Mathematical Journal Aims and scope

Let L 0 (T ) be the set of real-valued periodic measurable functions, let ψ : R + → R + be a modulus of continuity (ψ ≠ 0) , and let

$$ {L_{\uppsi }}\equiv {L_{\uppsi }}(T)=\left\{ {f\in {L_0}(T):{{{\left\| f \right\|}}_{\uppsi }}:=\int\limits_T {\uppsi \left( {\left| {f(x)} \right|} \right)dx<\infty } } \right\}. $$

The following problems are investigated: the relationship between the rate of approximation of f by trigonometric polynomials in L ψ and the smoothness in L 1, the relationship between the moduli of continuity of f in L ψ and L 1 and the imbedding theorems for the classes Lip(α, ψ) in L 1, and the structure of functions from the class Lip(1, ψ).

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. S. A. Pichugov, “On the Jackson theorem for periodic functions in spaces with integral metric,” Ukr. Mat. Zh., 52, No. 1, 122–133 (2000); English translation: Ukr. Math. J., 52, No. 1, 133–147 (2000).

    Article  MathSciNet  Google Scholar 

  2. S. A. Pichugov, “On the Jackson theorem for periodic functions in metric spaces with integral metric. II,” Ukr. Mat. Zh., 63, No. 11, 1524–1533 (2011); English translation: Ukr. Math. J., 63, No. 11, 1733–1744 (2011).

    Article  MathSciNet  Google Scholar 

  3. S. A. Pichugov, “Inequalities for trigonometric polynomials in spaces with integral metric,” Ukr. Mat. Zh., 63, No. 12, 1657–1571 (2011); English translation: Ukr. Math. J., 63, No. 12, 1883–1899 (2011).

    Article  Google Scholar 

  4. S. A. Pichugov, “Inverse Jackson theorems in spaces with integral metric,” Ukr. Mat. Zh., 64, No. 3, 351–362 (2012); English translation: Ukr. Math. J., 64, No. 3, 394–407 (2012).

    Article  MATH  Google Scholar 

  5. E. A. Storozhenko, P. Oswald, and V. G. Krotov, “Direct and inverse the Jackson-type theorems in the spaces L p , 0 < p < 1,” Mat. Sb., 98, No. 3, 395–415 (1975).

    MathSciNet  Google Scholar 

  6. V. I. Ivanov, “Direct and inverse theorems of the approximation theory in the metric of L p for 0 < p < 1,” Mat. Zametki, 18, No. 5, 641–658 (1975).

    MathSciNet  MATH  Google Scholar 

  7. V. V. Arestov, “On the integral inequalities for trigonometric polynomials and their derivatives,” Izv. Akad. Nauk SSSR, Ser. Mat., 45, No. 1, 3–22 (1982).

    MathSciNet  Google Scholar 

  8. S. M. Nikol’skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  9. E. A. Storozhenko, “On some imbedding theorems,” Mat. Zametki, 19, No. 2, 187–200 (1976).

    MathSciNet  MATH  Google Scholar 

  10. V. G. Krotov, “On the differentiability of functions from L p , 0 < p < 1,” Mat. Sb., 117, No. 1, 95–113 (1982).

    MathSciNet  Google Scholar 

  11. G. H. Hardy and J. Littlewood, “Some properties of functional integrals. I,” Math. Z., 27, 565–606 (1928).

    Article  MathSciNet  MATH  Google Scholar 

  12. A. F. Timan, Theory of Approximation of Functions of Real Variable [in Russian], Fizmatgiz, Moscow (1960).

    Google Scholar 

  13. C. Bennett and R. Sparsley, Interpolation of Operators, Academic Press, New York (1988).

    MATH  Google Scholar 

  14. S. A. Pichugov, “Approximation of measurable periodic functions in measure by step functions,” Ukr. Mat. Zh., 48, No. 5, 711–715 (1996); English translation: Ukr. Math. J., 48, No. 5, 795–800 (1996).

    Article  MathSciNet  Google Scholar 

  15. S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dnepropetrovsk National University of Railway Transport, Dnepropetrovsk, Ukraine

    S. A. Pichugov

Authors
  1. S. A. Pichugov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 9, pp. 1214–1232, September, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pichugov, S.A. Smoothness of functions in the metric spaces L ψ . Ukr Math J 64, 1382–1402 (2013). https://doi.org/10.1007/s11253-013-0723-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-013-0723-8

Keywords

Search

Navigation