## Abstract

We prove that a convex function f ∈ L_{p}[-1, 1], 0 < p < ∞, can be approximated by convex polynomials with an error not exceeding C ω^{φ}_{3} (f, 1/n)_{p} where ω^{φ}_{3} (f, ·) is the Ditzian-Totik modulus of smoothness of order three of f. We are thus filling the gap between previously known estimates involving ω^{φ}_{2} (f, 1/n)_{p}, and the impossibility of having such estimates involving ω_{4}. We also give similar estimates for the approximation of f by convex C^{0} and C^{1} piecewise quadratics as well as convex C^{2} piecewise cubic polynomials.

Original language | English |
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Pages (from-to) | 409-422 |

Number of pages | 14 |

Journal | Constructive Approximation |

Volume | 12 |

Issue number | 3 |

DOIs | |

State | Published - 1996 |

## Keywords

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

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