Three-monotone spline approximation

For r≥3r≥3, n∈Nn∈N and each 3-monotone continuous function ff on [a,b][a,b] (i.e.  , ff is such that its third divided differences [x0,x1,x2,x3]f[x0,x1,x2,x3]f are nonnegative for all choices of distinct points x0,…,x3x0,…,x3 in [a,b][a,b]), we construct a spline ss of degree rr and of minimal defect (i.e.  , s∈Cr−1[a,b]s∈Cr−1[a,b]) with n−1n−1 equidistant knots in (a,b)(a,b), which is also 3-monotone and satisfies ‖f−s‖L∞[a,b]≤cω4(f,n−1,[a,b])∞,‖f−s‖L∞[a,b]≤cω4(f,n−1,[a,b])∞, where ω4(f,t,[a,b])∞ω4(f,t,[a,b])∞ is the (usual) fourth modulus of smoothness of ff 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of the Ditzian–Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the LpLp norm with p<∞p<∞. At the same time, positive results in the LpLp case with p<∞p<∞ are still valid if one allows the knots of the approximating ppf to depend on ff 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 kk-monotone approximation with k≥4k≥4 (where just about everything is “negative”).