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)‖∞:=esssupx∈[-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-Pnφr∥∞≤c(r)nr‖f(r)‖∞,where φ(x):=1-x2. 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.