## Abstract

Let I be a finite interval, s ∈_{0}, and r,ν,n ∈. Given a set M, of functions defined on I, denote by Δ_{+} ^{s}M the subset of all functions y M such that the s-difference Δ_{τ}^{s}y(̇) is nonnegative on I, ∀_{τ} > 0. Further, denote by W_{p}^{r} the Sobolev class of functions x on I with the seminorm ||x ^{(r)}||L_{p} ≤ 1. Also denote by ∑ _{ν,n}, the manifold of all piecewise polynomials of order ν and with n - 1 knots in I. If ν ≥ max {r,s}, 1 ≤ p,q ≤ ∞, and (r,p,q) ≠ (1,1,∞), then we give exact orders of the best unconstrained approximation E(Δ_{+}^{s} W_{p} ^{r}, Σ_{νn})_{Lq} and of the best s-monotonicity preserving approximation E(Δ_{+}^{s} W_{p} ^{r}, Σ_{νn})_{Lq}.

Original language | English |
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Pages (from-to) | 211-236 |

Number of pages | 26 |

Journal | Advances in Computational Mathematics |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2007 |

## Keywords

- Free-knot spline
- Order of approximation
- Shape preserving