Constructive Approximation

, Volume 1, Issue 1, pp 323–331 | Cite as

Pointwise estimates for monotone polynomial approximation

  • Ronald A. DeVore
  • Xiang Ming Yu
Article

Abstract

We prove that iff is increasing on [−1,1], then for eachn=1,2,... there is an increasing algebraic polynomialP n of degreen such that |f(x)−P n (x)|≤cω2(f,√1−x2/n), whereω2 is the second-order modulus of smoothness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we characterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.

1980 AMS(MOS) subject classifications

41 A 10 41 A 25 41 A 29 

Key words and phrases

Monotone approximation by algebraic polynomials Degree of approximation Pointwise estimates 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. Beatson (preprint):Shape preserving convolution operators.Google Scholar
  2. 2.
    R. DeVore (1976):Degree of approximation. In: Approximation Theory II (G. G. Lorentz, C. K. Chui, L. Schumaker, eds.). New York: Academic Press, pp. 117–162.Google Scholar
  3. 3.
    R. DeVore (1977):Monotone approximation by polynomials. SIAM J. Math. Anal.,8:906–921.Google Scholar
  4. 4.
    G. G. Lorentz (1966): Approximation of Functions. New York: Holt.Google Scholar
  5. 5.
    G. G. Lorentz (1972):Monotone approximation. In: Inequalities III (O. Shisha, ed.). New York: Academic Press, pp. 201–215.Google Scholar
  6. 6.
    G. G. Lorentz, K. Zeller (1969):Degree of approximation by monotone polynomials, II. J. Approx. Theory,2:265–269.Google Scholar
  7. 7.
    S. Teljakovskii (1966):Two theorems on the approximation of functions by algebraic polynomials. Mat. Sbornik,70(112):252–265.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1985

Authors and Affiliations

  • Ronald A. DeVore
    • 1
  • Xiang Ming Yu
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of South Carolina Columbia
  2. 2.Department of MathematicsNanjing Normal UniversityNanjingPeople's Republic of China

Personalised recommendations