Pointwise estimates for 3-monotone approximation

We prove that for a 3-monotone function F is an element of C[-1, 1], one can achieve the pointwise estimates [F(x) - Psi(x)] <= c omega(3)(F.rho(n)(x)). x is an element of [-1,1] where rho(n)(x) := 1/n(2) + root 1-x(2)/n and c is an absolute constant, both with Psi, a 3-monotone quadratic spline on the nth Chebyshev partition, and with Psi, 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.