Three-monotone spline approximation

• Published in: Journal of approximation theory 2010-12, Vol.162 (12), p.2168-2183
• Main Authors: Dzyubenko, G.A., Kopotun, K.A., Prymak, A.V.
• Format: Article
• Language: English
• Subjects: 3-monotone approximation by piecewise polynomials and splines; Degree of approximation; Jackson-type estimates
• ISSN: 0021-9045; 1096-0430
• DOI: 10.1016/j.jat.2010.07.004

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 [ 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 ∈ C r − 1 [ a , b ] ) with n − 1 equidistant knots in ( a , b ) , which is also 3-monotone and satisfies ‖ f − s ‖ L ∞ [ a , b ] ≤ c ω 4 ( f , n − 1 , [ a , b ] ) ∞ , where ω 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of the Ditzian–Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the L p norm with p < ∞ . At the same time, positive results in the L p case with p < ∞ 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”).