Interpolatory pointwise estimates for convex polynomial approximation.

This paper deals with approximation of smooth convex functions f on an interval by convex algebraic polynomials which interpolate f and its derivatives at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of our main theorem is the following result on approximation of f ∈ Δ (2) , the set of convex functions, from W r , the space of functions on [ - 1 , 1 ] for which f (r - 1) is absolutely continuous and ‖ f (r) ‖ ∞ : = ess sup x ∈ [ - 1 , 1 ] | f (r) (x) | < ∞ : For any f ∈ W r ∩ Δ (2) , r ∈ N , there exists a number N = N (f , r) , such that for every n ≥ N , there is an algebraic polynomial of degree ≤ n which is in Δ (2) and such that f - P n φ r ∞ ≤ c (r) n r ‖ f (r) ‖ ∞ , where φ (x) : = 1 - x 2 . For r = 1 and r = 2 , the above result holds with N = 1 and is well known. For r ≥ 3 , it is not true, in general, with N independent of f. [ABSTRACT FROM AUTHOR]