## Abstract

The main results are as follows. (1) Let f ∈ C[0, 1] change its sign a finite number of times; then the degree of copositive approximation of f by splines with n equally spaced knots is bounded by C ω_{3}(f, 1/n) for n large enough. This rate is the best in the sense that ω_{3} cannot be replaced by ω_{4}. (2) An algorithm is developed based on the proof. (3) The first result above holds for a copositive polynomial approximation of f. (4) If f ∈ C^{1}[0, 1], then the degree of approximation by copositive splines of order r is bounded by Cn^{-1}ω_{r-1}(f′, 1/n). The results on f ∈ C[0, 1] fill a gap left by S. P. Zhou [Israel J. Math., 78 (1992), pp. 75-83], and Y. K. Hu, D. Leviatan, and X. M. Yu [J. Anal., 1 (1993), pp. 85-90; J. Approx. Theory, 80 (1995), pp. 204-218].

Original language | English |
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Pages (from-to) | 388-398 |

Number of pages | 11 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1996 |

## Keywords

- Computer algorithm
- Degree of copositive approximation
- Polynomial approximation
- Spline approximation
- Splines with equally spaced knots