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JAT

2010

2010

For r 3, n N and each 3-monotone continuous function f on [a, b] (i.e., f is such that its third divided differences [x0, x1, x2, x3] f are nonnegative for all choices of distinct points x0, . . . , x3 in [a, b]), we construct a spline s of degree r and of minimal defect (i.e., s Cr-1[a, b]) with n -1 equidistant knots in (a, b), which is also 3-monotone and satisfies f - sL[a,b] c4( f, n-1, [a, b]), where 4( f, t, [a, b]) is the (usual) fourth modulus of smoothness of f in the uniform norm. This answers in the affirmative the question raised in [8, Remark 3], which was the only remaining unproved Jackson-type estimate for uniform 3-monotone approximation by piecewise polynomial functions (ppfs) with uniformly spaced fixed knots. Moreover, we also prove a similar estimate in terms of the Ditzian

Related Content

Added |
19 May 2011 |

Updated |
19 May 2011 |

Type |
Journal |

Year |
2010 |

Where |
JAT |

Authors |
G. A. Dzyubenko, Kirill Kopotun, A. V. Prymak |

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