## Abstract

We prove that if a function f ∈ C [0, 1] changes sign finitely many times, then for any n large enough the degree of copositive approximation to f by quadratic spliners with n-1 equally spaced knots can be estimated by Cω_{2}(f, 1/n), where C is an absolute constant. We also show that the degree of copositive polynomial approximation to f ∈ C^{1}[0, 1] can be estimated by Cn^{-1}ω_{r}(f′, 1/n), where the constant C depends only on the number and position of the points of sign change. This improves the results of Leviatan (1983, Proc. Amer. Math. Soc.88, 101-105) and Yu (1989, Chinese Ann. Math.10, 409-415), who assumed that for some r ≥ 1, f ∈ C^{r}[0, 1]. In addition, the estimates involved Cn^{-r}ω(f^{r}, 1/n) and the constant C dependended on the behavior of f in the neighborhood of those points. One application of the results is a new proof to our previous ω_{2} estimate of the degree of copositive polynomia approximation of f ∈ C[0, 1], and another shows that the degree of copositive spline approximation cannot reach ω_{4}, just as in the case of polynomials.

Original language | English |
---|---|

Pages (from-to) | 204-218 |

Number of pages | 15 |

Journal | Journal of Approximation Theory |

Volume | 80 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1995 |