## Abstract

In this paper, among other things, we show that, given r∈N, there is a constant c=c(r) such that if f∈C ^{r} [−1,1] is convex, then there is a number N=N(f,r), depending on f and r, such that for n≥N, there are convex piecewise polynomials S of order r+2 with knots at the nth Chebyshev partition, satisfying |f(x)−S(x)|≤c(r)(min{1−x ^{2} ,n ^{−1} 1−x ^{2} }) ^{r} ω _{2} (f ^{(r)} ,n ^{−1} 1−x ^{2} ), for all x∈[−1,1]. Moreover, N cannot be made independent of f.

Original language | English |
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Pages (from-to) | 467-479 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 474 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jun 2019 |

## Keywords

- Convex approximation by polynomials
- Degree of approximation
- Jackson-type interpolatory estimates