Coconvex Approximation

Abstract. Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials which are coconvex with it, namely, polynomials that change their convexity exactly at the points where f does. We discuss some Jackson type estimates where the constants involved depend on the location of the points of change of convexity. We also show that in some cases the constants may be taken independent of the points of change of convexity, but that in other cases this dependence is essential. But mostly we obtain such estimates for functions f that themselves are continuous piecewise polynomials on the Chebyshev partition, which form a single polynomial in a small neighborhood of each point of change of convexity. These estimates involve the k modulus of smoothness of the piecewise polynomials when they themselves are of degree k − 1.