Interpolatory estimates for convex piecewise polynomial approximation

In this paper, among other things, we show that, given $r\in N$, there is a constant $c=c(r)$ such that if $f\in C^r[-1,1]$ is convex, then there is a number $\mathcal{N}=\mathcal{N}(f,r)$, depending on $f$ and $r$, such that for $n\ge \mathcal{N}$, there are convex piecewise polynomials $S$ of order $r+2$ with knots at the Chebyshev partition, satisfying $$\mathrm{\mid }f(x)-S(x)\mathrm{\mid }\le c(r){(\min \{1-x^2,n^{-1}\sqrt{1-x^2}\})}^r{\omega }_2(f^{(r)},n^{-1}\sqrt{1-x^2}),$$ for all $x\in [-1,1]$. Moreover, $\mathcal{N}$ cannot be made independent of $f$.