Three-monotone spline approximation

[摘要] For r >= 3, n is an element of N and each 3-monotone continuous function f on [a, b] (i.e., f is such that its third divided differences [x(0), x(1), x(2), x(3)] f are nonnegative for all choices of distinct points x(0),....,x(3) in [a, b]), we construct a spline s of degree r and of minimal defect (i.e., s is an element of Cr-1[a, b]) with n - 1 equidistant knots in (a, b), which is also 3-monotone and satisfies parallel to f - s parallel to(L infinity[a,b]) <= c omega(4)(f, n(-1,) [a,b])infinity, where omega(4)(f, t, [a, b])(infinity) 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of the Ditzian-Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the L-p norm with p < infinity. At the same time, positive results in the L-p case with p < infinity are still valid if one allows the knots of the approximating ppf to depend on f while still being controlled. These results confirm that 3-monotone approximation is the transition case between monotone and convex approximation (where most of the results are positive) and k-monotone approximation with k >= 4 (where just about everything is negative). (C) 2010 Elsevier Inc. All rights reserved.

[发布日期] 2010-12-01

[关键词] 3-monotone approximation by piecewise polynomials and splines;Degree of approximation;Jackson-type estimates