Pointwise estimates for 3-monotone approximation

We prove that for a 3-monotone function F∈C[−1,1]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 cc is an absolute constant, both with ΨΨ, a 3-monotone quadratic spline on the nnth Chebyshev partition, and with ΨΨ, a 3-monotone polynomial of degree ≤n≤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”.