Constructive Approximation

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

Pointwise estimates for monotone polynomial approximation

  • Ronald A. DeVore
  • Xiang Ming Yu


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 


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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

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