CONVEX POLYNOMIAL AND SPLINE APPROXIMATION IN L(P),0-LESS-THAN-P-LESS-THAN-INFINITY

We prove that a convex function f epsilon L(p)[-1, 1], 0 < p < infinit y, can be approximated by convex polynomials with an error not exceedi ng C omega(3)(phi)(f, 1/n)p where omega(3)(phi)(f,.) is the Ditzian-To tik modulus of smoothness of order three of f. We are thus filling the gap between previously known estimates involving omega(2)(phi)(f, 1/n )(p), and the impossibility of having such estimates involving omega(4 ). We also give similar estimates for the approximation of f by convex C-0 and C-1 piecewise quadratics as well as convex C-2 piecewise cubi c polynomials.