CONVEX POLYNOMIAL AND SPLINE APPROXIMATION IN C[-1,1]

We prove that a convex function f is-an-element-of C[-1, 1] can be approximated by convex polynomials p(n) of degree n at the rate of omega3(f, 1/n). We show this by proving that the error in approximating f by C2 convex cubic splines with n knots is bounded by omega3(f, 1/n) and that such a spline approximant has an L(infinity) third derivative which is bounded by n3omega3(f, 1/n). Also we prove that if f is-an-element-of C2[-1, 1], then it is approximable at the rate of n-2omega(f'', 1/n) and the two estimates yield the desired result.