Constructive Approximation

, Volume 8, Issue 2, pp 187–201

Polynomial approximation inLp (0<p<1)

Authors

  • Ronald A. DeVore
    • Department of MathematicsUniversity of South Carolina
  • Dany Leviatan
    • Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv University
  • Xiang Ming Yu
    • Department of MathematicsUniversity of South Carolina
Article

DOI: 10.1007/BF01238268

Cite this article as:
DeVore, R.A., Leviatan, D. & Yu, X.M. Constr. Approx (1992) 8: 187. doi:10.1007/BF01238268

Abstract

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

41A2541A20

Key words and phrases

Degree of approximationMonotone approximationPolynomials

Copyright information

© Springer-Verlag New York Inc. 1992