Constructive Approximation

, Volume 8, Issue 2, pp 187–201

Polynomial approximation inLp (0<p<1)

  • Ronald A. DeVore
  • Dany Leviatan
  • Xiang Ming Yu


We prove that forfLp, 0<p<1, andk a positive integer, there exists an algebraic polynomialPn of degree ≤n such that
$$\left\| {f - P_n } \right\|_p \leqslant C\omega _k^\varphi \left( {f,\frac{1}{n}} \right)_p $$
whereωkϕ(f,t)p is the Ditzian-Totik modulus of smoothness off inLp, andC is a constant depending only onk andp. Moreover, iff is nondecreasing andk≤2, then the polynomialPn can also be taken to be nondecreasing.

AMS classification

41A25 41A20 

Key words and phrases

Degree of approximation Monotone approximation Polynomials 


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Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Ronald A. DeVore
    • 1
  • Dany Leviatan
    • 2
  • Xiang Ming Yu
    • 1
  1. 1.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  2. 2.Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

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