Uniform Approximation by General Multivariate Singular Integral Operators

Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In this chapter, we present the uniform approximation properties of general multivariate singular integral operators over \({\mathbb{R}}^{N}\), N ≥ 1. We give their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss-Weierstrass, Poisson Cauchy and trigonometric singular integral operators where this theory can be applied directly. This chapter relies on [2].

References

  1. 1.
    G. Anastassiou, Basic Convergence with rates of smooth Picard singular integral operators, J. Comput. Anal. Appl, 8(2006), 313-334.MATHMathSciNetGoogle Scholar
  2. 2.
    G. Anastassiou, General Uniform Approximation Theory by Multivariate Singular Integral Operators, submitted, 2011.Google Scholar
  3. 3.
    G. Anastassiou and S.S. Dragomir, On some estimates of the remainder in Taylor’s formula, J. of Math. Anal. and Appl., Vol 263, issue 1, pp. 246-263, (2001).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    G. Anastassiou and O. Duman, Statistical Approximation by double Picard singular integral operators, Studia Univ. Babes-Bolyai Math., 55(2010), 3-20.MATHMathSciNetGoogle Scholar
  5. 5.
    G. Anastassiou and O. Duman, Uniform Approximation in statistical sense by double Gauss-Weierstrass singular integral operators, Nonlinear Funct. Anal. Appl. (accepted for publication, 2009).Google Scholar
  6. 6.
    G. Anastassiou and S. Gal, Approximation Theory, Birkhaüser, Boston, Basel, Berlin, 2000.MATHCrossRefGoogle Scholar
  7. 7.
    G. Anastassiou and R. Mezei, Uniform convergence with rates of smooth Gauss-Weierstrass singular integral operators, Applicable Analysis, 88, 7(2009), 1015-1037.Google Scholar
  8. 8.
    G. Anastassiou and R. Mezei, Uniform convergence with rates of smooth Poisson-Cauchy type singular integral operators, Mathematical and Computer Modelling, 50(2009), 1553-1570.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    G. Anastassiou and R. Mezei, Uniform convergence with rates of general singular operators, CUBO, accepted, 2011.Google Scholar
  10. 10.
    J. Edwards, A treatise on the integral calculus, Vol II, Chelsea, New York, 1954.Google Scholar
  11. 11.
    S.G. Gal, Remark on the degree of approximation of continuous functions by singular integrals, Math. Nachr., 164(1993), 197-199.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    R.N. Mohapatra and R.S. Rodriguez, On the rate of convergence of singular integrals for Hölder continuous functions, Math. Nachr., 149(1990), 117-124.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    D. Zwillinger, CRC standard Mathematical Tables and Formulae, 30th edition, Chapman & Hall/CRC, Boca Raton, 1995.CrossRefGoogle Scholar

Copyright information

© George A. Anastassiou 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

Personalised recommendations