Three-monotone spline approximation

AbstractFor 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−s‖L∞[a,b]≤cω4(f,n−1,[a,b])∞, where ω4(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of the Ditzian–Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the Lp norm with p<∞. At the same time, positive results in the Lp 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”)

Pointwise estimates for 3-monotone approximation

AbstractWe prove that for a 3-monotone function F∈C[−1,1], one can achieve the pointwise estimates |F(x)−Ψ(x)|≤cω3(F,ρn(x)),x∈[−1,1], where ρn(x)≔1n2+1−x2n and c is an absolute constant, both with Ψ, a 3-monotone quadratic spline on the nth Chebyshev partition, and with Ψ, a 3-monotone polynomial of degree ≤n.The basis for the construction of these splines and polynomials is the construction of 3-monotone splines, providing appropriate order of pointwise approximation, half of which nodes are prescribed and the other half are free, but “controlled”