Convex polynomial and spline approximation in Lp, 0 < p < ∞

R. A. DeVore, Y. K. Hu, D. Leviatan

Research output: Contribution to journalArticlepeer-review

Abstract

We prove that a convex function f ∈ Lp[-1, 1], 0 < p < ∞, can be approximated by convex polynomials with an error not exceeding C ωφ3 (f, 1/n)p where ωφ3 (f, ·) is the Ditzian-Totik modulus of smoothness of order three of f. We are thus filling the gap between previously known estimates involving ωφ2 (f, 1/n)p, and the impossibility of having such estimates involving ω4. We also give similar estimates for the approximation of f by convex C0 and C1 piecewise quadratics as well as convex C2 piecewise cubic polynomials.

Original languageEnglish
Pages (from-to)409-422
Number of pages14
JournalConstructive Approximation
Volume12
Issue number3
DOIs
StatePublished - 1996

Keywords

  • Constrained approximation in L space
  • Degree of convex approximation
  • Polynomial approximation
  • Spline approximation

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