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

• Type: Journal
• Year: 2010
• Where: JAT
• Authors: G. A. Dzyubenko, Kirill Kopotun, A. V. Prymak
• Keywords: -1 Equidistant Knots, 3-monotone Approximation, Approximation, JAT 2010