## Abstract

Let [-1,1] be the space of continuous functions on [-,1], and denote by Δ^{2} the set of convex functions f [-,1]. Also, let E _{n} (f) and E _{n} ^{(2)} (f) denote the degrees of best unconstrained and convex approximation of f Δ^{2} by algebraic polynomials of degree < n, respectively. Clearly, En (f) E _{n} ^{(2)} (f), and Lorentz and Zeller proved that the inverse inequality E _{n} ^{(2)} (f) cE _{n} (f) is invalid even with the constant c = c(f) which depends on the function f Δ^{2}. In this paper we prove, for every α > 0 and function f Δ^{2}, that ⊃ α En(2) (f):n q c(α)⊃ α En (f):n N, where c(α) is a constant depending only on α. Validity of similar results for the class of piecewise convex functions having s convexity changes inside (-1,1) is also investigated. It turns out that there are substantial differences between the cases 1 and s 2.

Original language | English |
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Pages (from-to) | 273-290 |

Number of pages | 18 |

Journal | Acta Mathematica Hungarica |

Volume | 123 |

Issue number | 3 |

DOIs | |

State | Published - May 2009 |

## Keywords

- (co)convex approximation by polynomials
- Degree of approximation