Skip to main content
SpringerLink
Log in

On the jackson theorem for periodic functions in spaces with integral metric

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We consider the approximation of periodic functions by trigonometric polynomials in metric (not normed) spaces that are generalizations of the spaces L p , 0 < p < 1, and L 0. In particular, we prove the multidimensional Jackson theorem in L p (Tm), 0 < p < 1.

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. D. Jackson, über die Genauigkeit der Anndherung stetiger Functionen durch gauze rationale Functionen gegebenen Grades undtrigonometrische Summen gegebener Ordnung, Dissertation, Gottingen f 1911).

  2. E. Quade, “Trigonometric approximation in the mean,” Duke Math. J., 3, 37–49 (1937).

    Article  MathSciNet  Google Scholar 

  3. A. F. Timan, Theory of Approximation of Functions of a Real Variables [in Russian], Fi/matgiz, Moscow (1960).

    Google Scholar 

  4. N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian] Nauka, Moscow 1987.

    Google Scholar 

  5. S. M. Nikol’skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian] Nauka, Moscow 1969.

    Google Scholar 

  6. V. A. Yudin, “Multidimensional Jackson theorem,” Mat. Zametki. 20, No. 3, 439–444 (1976).

    MATH  MathSciNet  Google Scholar 

  7. V. N. Temlyakov, “On approximation of functions of many variables by trigonometric polynomials with harmonics from hyperboliccrosses,” Ukr. Mat. Zh., 41, No. 4, 518–524 (1989).

    Article  MathSciNet  Google Scholar 

  8. E. A. Storozhenko, P. Osval’d, and V. G. Krotov, “Direct and inverse theorems of the Jackson type in the spaces L p , 0 < p < 1,” Mat. Sb., 98, No. 3, 395–115 (1975).

    MathSciNet  Google Scholar 

  9. V. I. Ivanov, “Direct and inverse theorems of approximation theory in the metric of L p for 0 < p < 1,” Mat. Zametki, 16, No. 5, 641–658 (1975).

    Google Scholar 

  10. A. A. Kornicnko, “On the Jackson theorem in the space with convergence in measure,” E. J. Approxim., 4, No. 1, 15–24 (1998).

    Google Scholar 

  11. K. V. Runovskii, “On approximation by families of linear polynomial operators in the spaces L p , 0 < p < 1,” Mat. Sb., 185, No. 8, 81–102 (1994).

    Google Scholar 

  12. E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton 1971.

    MATH  Google Scholar 

Download references

Authors
  1. S. A. Pichugov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pichugov, S.A. On the jackson theorem for periodic functions in spaces with integral metric. Ukr Math J 52, 133–147 (2000). https://doi.org/10.1007/BF02514142

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02514142

Keywords

Search

Navigation