## Abstract

Let _{E n} (f) denote the degree of approximation of f ∈ C [ - 1, 1] , by algebraic polynomials of degree <n, and assume that we know that for some α > 0 and N ≥ 2, nαEn(f)≤1,n≥N. Suppose that f changes its monotonicity s ≥ 1 times in [ - 1, 1] We are interested in what may be said about its degree of approximation by polynomials of degree <n that are comonotone with f. In particular, if f changes its monotonicity at _{Y s} {colon equals} {_{y1}, . . . , _{y s}} and the degree of comonotone approximation is denoted by _{E n} (f, _{Y s}) , we investigate when can one say that nαEn(f,Ys)≤c(α,s,N),n≥N*, for some ^{N *}. Clearly, ^{N *}, if it exists at all (we prove it always does), depends on α, s and N. However, it turns out that for certain values of α, s and N, ^{N *} depends also on _{Y s} and in some cases even on f itself. The results extend previous results in the case N = 1.

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

Pages (from-to) | 1-23 |

Number of pages | 23 |

Journal | Journal of Approximation Theory |

Volume | 179 |

DOIs | |

State | Published - Feb 2014 |

## Keywords

- Comonotone polynomial approximation
- Degree of approximation
- Uniform norm