Shape preserving approximation in vector ordered spaces (Q812730)

Let \((X,\|\cdot\|,\leq_X)\) be a normed ordered real linear space. A generalised polynomial with coefficients in \(X\) is a function of the form \(P_n(x) =\sum^n_{k=0} c_k x^k\), where \(x\) is in an interval \([a, b]\). The paper considers generalised polynomial approximation to convex functions \(f: [-1, 1]\to X\). It is shown that there exists a constant \(C> 0\) such that for any convex function \(f: [-1, 1]\to X\) and each \(n\in\mathbb{N}\) there exists a convex generalised polynomial \(P_n\), of degree \(\leq n\), monotone if \(f\) is monotone, such that \(\| f- P_n\|_\infty\) and, for all \(x\in[-1,1]\), \(\| f(x)- P_n(x)\|\leq C\omega^2(f; \sqrt{1-x^n}/n)_\infty\), where \(\omega^2_\phi\) is the second Ditzian-Totik modulus of smoothness with \(\phi(x)= \sqrt{1- x^2}\), and \(\omega^2\) is the second uniform modulus of smoothness. The case \(X=\mathbb{R}\) of these results was obtained by \textit{D. Leviatan} [Proc. Am. Math. Soc. 98, 471--474 (1986; Zbl 0617.41009)]. The construction of the approximations \(P_n\) for the generalisation follows that for the case \(X= \mathbb{R}\); the proof of the estimates is a deduction by application of the estimates for the case \(X= \mathbb{R}\) to \(x^*\circ f[-1, 1]\to\mathbb{R}\) for each \(x^*\) in the unit ball of \(X^*\).