Acta Mathematica Sinica, English Series

, Volume 29, Issue 3, pp 535–546 | Cite as

Average error bounds of trigonometric approximation on periodic Wiener spaces

Article

Abstract

In this paper, we study the approximation of identity operator and the convolution integral operator Bm by Fourier partial sum operators, Fejér operators, Vallée-Poussin operators, Cesáro operators and Abel mean operators, respectively, on the periodic Wiener space (C1(ℝ),W) and obtain the average error estimations.

Keywords

Average error bounds trigonometric polynomial approximation periodic Wiener spaces 

MR(2000) Subject Classification

28C20 41A35 42A10 42A85 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

10114_2012_1042_MOESM1_ESM.tex (31 kb)
Supplementary material, approximately 30.9 KB.

References

  1. [1]
    Jackson, D.: The Theory of Approximation, AMS Colloq. Publ., Amer. Math. Soc., Providence, RI, 1930MATHGoogle Scholar
  2. [2]
    Korneıčuk, N. P.: Extreme Problems in Approximation Theory, M. Nayka, Moscow, 1984Google Scholar
  3. [3]
    Lorentz, G. G.: Approximation of Functions, Holt, Rinehart & Winston, New York, 1966MATHGoogle Scholar
  4. [4]
    Kuo, H. H.: Gaussian Measures in Banach Spaces. Lecture Notes in Math., 463, Springer-Verlag, New York, 1975MATHGoogle Scholar
  5. [5]
    Lee, D.: Approximation of linear operators on a Wiener space. Rocky Mountain J. Math., 16, 641–659 (1986)MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Lee, D., Wasilkowski, G. W.: Approximation of linear operators on a Banach space with a Gauss measure. J. Complexity, 2, 12–43 (1986)MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Majorov, V. E.: Widths of Banach spaces with Gaussian measure. Dokl. Acad. Sci., 323, 233–237 (1994)Google Scholar
  8. [8]
    Michelli, C. A., Rivlin, T. J.: Optimal Estimation in Approximation Theory, Plenum Press, New York, 1977Google Scholar
  9. [9]
    Michelli, C. A., Rivlin, T. J.: Optimal Recovery of Best Approximations. IBM T. J. Watson Research Center Rep., Vol. 7071, 1978Google Scholar
  10. [10]
    Novak, E.: Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Math. 1349, Springer-Verlag, Berlin, 1988MATHGoogle Scholar
  11. [11]
    Novak, E.: Stochastic properties of quadrature firmulas. Number Math., 53, 609–620 (1988)MATHCrossRefGoogle Scholar
  12. [12]
    Ritter, K.: Approximation and optimization on the Wiener space. NumberMath. J. Complexity, 6, 337–364 (1990)MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Sun, Y. S.: Approximation of Functions I, Beijing Normal Univ. Press, Beijing, 1989Google Scholar
  14. [14]
    Sun, Y. S., Wang, C. Y.: µ-Average n-widths on the Wiener spaces. J. Complexity, 10, 428–436 (1994)MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Sun, Y. S., Wang, C. Y.: Average error bounds of best approximation of continuous functions on the Wiener spaces. J. Complexity, 11, 74–104 (1995)MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Sun, Y. S., Wang, C. Y.: Average error bounds of best approximation in a Banach space with Gaussian measure. East J. on Approx., 1, 431–439 (1995)MATHGoogle Scholar
  17. [17]
    Traub, J. F., Wozniakowski, H.: A General Theory of Optimal Algorithms, Academic Press, New York, 1980MATHGoogle Scholar
  18. [18]
    Traub, J. F., Wasilkowski, G. W., Wozniakowski, H.: Information-Based Complexity, Academic Press, New York, 1988MATHGoogle Scholar
  19. [19]
    Wang, C. Y.: Best Approximation Problems and Information-Based Complexity on AbstractWiener Space, Doctoral Dissertation, Beijing Normal Univ., 1994Google Scholar
  20. [20]
    Xu, G. Q.: The average error for Lagrange interpolation and Hermite-Fejér interpolation on the Wiener space. Acta Mathematica Sinica, Chinese Series, 50, 1281–1296 (2007)MathSciNetMATHGoogle Scholar
  21. [21]
    Xu, G. Q., Du, Y. F.: The average errors for trigonometric polynomial operators on the Brownian Bridge measure. Acta Mathematica Sinica, Chinese Series, 52(3), 109–120 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsBeijing JiaoTong UniversityBeijingP. R. China

Personalised recommendations