Interpolatory estimates for convex piecewise polynomial approximation

In this paper, among other things, we show that, given r is an element of N, there is a constant c= c(r) such that if f is an element of C-r[-1,1] is convex, then there is a number N = N(f, r), depending on f and r, such that for n >= N, there are convex piecewise polynomials S of order r + 2 with knots at the nth Chebyshev partition, satisfying vertical bar f (x) - S(x)vertical bar <= c(r) (min {1 - x(2), n(-1) root 1-x(2)})(r) omega(2) (f((r)), n(-1) root 1-x(2)) for all x is an element of [-1, 1]. Moreover, N cannot be made independent of f. (C) 2019 Elsevier Inc. All rights reserved.