Convex Polynomial Approximation in the Uniform Norm: Conclusion

http://dx.doi.org/10.4153/CJM-2005-049-6
Canad. J. Math. 57(2005), 1224-1248

  Published:2005-12-01
 Printed: Dec 2005

Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each $r$th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian-Totik $k$th modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs $(k,r)$ we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.