Uniform estimates for monotone approximation (Q1336001)

Let \(C\) be the space of continuous functions on the interval \([-1,1]\) with uniform norm \(\|\cdot\|\), and \(C^ r\) the set of functions having \(r\)th continuous derivative \(f^{(r)}\). The author studies the quantity \(E_ n^{(1)} (f)= \inf\| f-P\|\) with the infimum taken over all polynomials \(p\) of degree \(\leq n\) satisfying the monotonicity condition \(p'>0\). He investigates the validity of estimates of the form \(E_ n^{(1)} (f)\leq cn^{-r} \varphi(1/n)\) for nondecreasing functions \(f\in C^ r\) where \(\varphi\) is a higher order modulus of continuity of \(f^{(r)}\). This estimate is proved for \(r\geq 3\) for general \(\varphi\) and for \(r=2\) with an additional condition on \(\varphi\).