Skip to main content
SpringerLink
Log in

Quantitative Approximations by Convolution Polynomials in Bergman Spaces

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

In this paper we construct concrete convolution polynomials for which quantitative results in approximation of functions in Bergman spaces on the unit disk are obtained, with the estimates expressed in terms of higher order \(L^{p}\)-moduli of smoothness and in terms of the \(L^{p}\)-best approximation quantity, \(0<p<+\infty \).

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. Chen, Y., Ren, G.: Jackson’s theorem in \(Q_{p}\) spaces. Sci. China Math. 53(2), 367–372 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Colzani, L.: Jackson theorems in Hardy spaces and approximation by Riesz means. J. Approx. Theory 49(3), 240–251 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  4. Duren, P., Schuster, A.: Bergman Spaces, Mathematical Surveys and Monographs, vol. 100. American Mathematical Society, Rhode Island (2004)

    MATH  Google Scholar 

  5. Gal, S.G.: Convolution-type integral operators in complex approximation. Comput. Methods Funct. Theory 1(2), 417–432 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gal, S.G.: Geometric and approximate properties of convolution polynomials in the unit disk. Bull. Inst. Math. Acad. Sinica (N.S.) 1(2), 307–336 (2006)

  7. Gal, S.G.: Shape Preserving Approximation by Real and Complex Polynomials. Birkhäuser, Boston (2008)

    Book  MATH  Google Scholar 

  8. Gal, S.G., Sabadini, I.: Approximation in compact balls by convolution operators of quaternion and paravector variable. Bull. Belg. Math. Soc. Simon Stevin 20, 481–501 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Gal, S.G., Sabadini, I.: Approximation by polynomials on quaternionic compact sets. Math. Meth. Appl. Sci 38(14), 3063–3074 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lorentz, G.G.: Approximation of Functions. Chelsea Publ. Comp, New York (1986)

    MATH  Google Scholar 

  11. Oswald, P.: On some approximation properties of real Hardy spaces (\(0 <p \le 1\)). J. Approx. Theory 40, 45–65 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ren, G., Wang, M.: Holomorphic Jackson’s theorems in polydiscs. J. Approx. Theory 134(2), 175–198 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ren, G., Chen, Y.: Gradient estimates and Jackson’s theorem in \(Q_{\mu }\) spaces related to measures. J. Approx. Theory 155(2), 97–110 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Storozhenko, E.A.: On sums of the Bernstein-Rogosinski and de la Vallée-Poussin type in \(H_{p}\), In: Approximation and function spaces (Gdańsk, 1979), pp. 712–730. North-Holland, Amsterdam (1981)

  15. Storozhenko, E.A.: Theorems of Jackson type in \(H_{p}\), 0< p< 1. Math. USSR-Izvestiya 17(1), 203–218 (1981)

  16. Zhong, L.: Interpolation polynomial approximation in Bergman spaces (Chinese). Acta Math. Sin. 34(1), 56–66 (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Oradea, Str. Universitatii No. 1, 410087, Oradea, Romania

    Sorin G. Gal

Authors
  1. Sorin G. Gal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sorin G. Gal.

Additional information

Communicated by Irene Sabadini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gal, S.G. Quantitative Approximations by Convolution Polynomials in Bergman Spaces. Complex Anal. Oper. Theory 12, 355–364 (2018). https://doi.org/10.1007/s11785-016-0601-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-016-0601-0

Keywords

Mathematics Subject Classification

Search

Navigation