Pointwise estimates for 3-monotone approximation

We 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) := 1 n2 + √ 1−x2 n 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”. c ⃝ 2012 Elsevier Inc. All rights reserved.