Three-monotone spline approximation

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Three-monotone spline approximation
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
G. A. Dzyubenko, Kirill Kopotun, A. V. Prymak
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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