## Abstract

Let I be a finite interval, r, n ∈ N, s ∈ N_{0} and 1 {less-than or slanted equal to} p {less-than or slanted equal to} ∞. 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 )} ∥_{Lp} {less-than or slanted equal to} 1. We obtain the exact orders of the Kolmogorov and the linear widths, and of the shape-preserving widths of the classes Δ_{+}^{s} W_{p}^{r} in L_{q} for s > r + 1 and ( r, p, q ) ≠ ( 1, 1, ∞ ). We show that while the widths of the classes depend in an essential way on the parameter s, which characterizes the shape of functions, the shape-preserving widths of these classes remain asymptotically ≈ n^{- 2}.

Original language | English |
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Pages (from-to) | 101-126 |

Number of pages | 26 |

Journal | Journal of Approximation Theory |

Volume | 140 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2006 |

## Keywords

- Kolmogorov width
- Linear width
- Sobolev-type classes
- s-monotone functions