Skip to main content

Sharp estimates of best approximations in terms of holomorphic functions of weierstrass-type operators

Estimates of the form

$$ {A_{\sigma }}{(f)_P}\leq KP\left( {\varPhi \left( \mathcal{W} \right)f} \right), $$

where W is a kernel of a special type summable on \( \mathbb{R} \), a function Φ is holomorphic in a neighborhood of the spectrum of W, and A σ(f ) P is the best approximation of a function f by entire functions of exponential type not greater than σ with respect to a seminorm P, are established. In some cases, for the uniform and integral norms the least possible constant K is found. The estimates are obtained by linear approximation methods. Bibliography: 13 titles.

References

  1. N. I. Akhiezer, Lectures in Approximation Theory [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  2. O. L. Vinogradov, “Sharp Jackson-type inequalities for approximations of classes of convolutions by entire functions of exponential type,” Algebra Analiz, 17, No, 4, 56–111 (2005).

    Google Scholar 

  3. O. L. Vinogradov, “Sharp estimates of best approximations by deviations of Weierstrass-type integrals,” Zap. Nauchn. Semin. POMI, 401, 53–70 (2012).

    Google Scholar 

  4. B. M. Makarov, M. G. Goluzina, A. A. Lodkin, and A. N. Podkorytov, Selected Problems in Real Analysis [in Russian], Nevskii Dialekt, BHV-Petersburg, St. Petersburg (2004).

    Google Scholar 

  5. A. G, Babenko and Yu. V. Kryakin, “Integral approximation of the characteristic: function of an interval and the Jackson inequality in \( C\left( \mathbb{T} \right) \),” Trudy Mat. Inst. Steklov, 15(1), 59–65 (2009).

    Google Scholar 

  6. O. L. Vinogradov and V. V. Zhuk, “Estimates for functionals with a known moment sequence in terms of deviations of Steklov type means,” Zap. Nauchn. Semin. POMI, 383, 5–32 (2010).

    Google Scholar 

  7. O. L. Vinogradov and V. V. Zhuk. “The rate of decrease of constants in Jackson type inequalities in dependence of the order of modulus of continuity,” Zap. Nauchn. Semin. POMI, 383, 33–52 (2010).

    Google Scholar 

  8. A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatlit, Moscow (1960).

    Google Scholar 

  9. N. P. Korneichuk, Sharp Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  10. I. F. Stefensen, Interpolation Theory [Russian translation], Moscow–Leningrad (1935).

  11. D. V. Widder, The Laplace Transform, Princeton (1946).

  12. V. V. Zhuk, Approximation of Periodic Functions [in Russian], Leningrad Univ., Leningrad (1982).

    Google Scholar 

  13. I. S. Gradshtein and I. M. Ryzhik, Tables of Intergrals, Sums, Series, and Products [in Russian], Fizmatlit, Moscow (1963).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to O. L. Vinogradov.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, V0l. 404, 2012, pp. 18–60.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Vinogradov, O.L. Sharp estimates of best approximations in terms of holomorphic functions of weierstrass-type operators. J Math Sci 193, 8–31 (2013). https://doi.org/10.1007/s10958-013-1429-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1429-z

Keywords

  • Approximation Method
  • Holomorphic Function
  • Linear Approximation
  • Entire Function
  • Exponential Type