One example in monotone approximation
It is readily seen that a monotone function \(f\in C^1[-1,1]\) can be approximated by monotone polynomials of degree not exceeding \(n\) at the rate \(E_{n-1}(f')\), where \(E_n(g)\) is the degree of unconstrained approximation of \(g\in C[-1,1]\) by polynomials of degree \(\leq n\). It is also quite easy and well-known that for \(f\in C^1[-1,1]\), we have the estimate \(E_n(f)\leq{C\over n}E_{n-1}(f')\). It has been known for many years that the degree of monotone approximation of a monotone function is not as small as the degree of unconstrained approximation. But it is quite natural to ask if we can still improve the estimate for monotone approximation, for instance by having an estimate of the type \(n^{-\alpha}E_{n-1}(f')\), for some \(\alpha>0\). The author shows that this is impossible by proving that for every \(n>0\) there is a nondecreasing \(f_n\in C^1[-1,1]\) such that \[ {1\over 200}< {E^{(1)}_n(f)\over E_{n-1}(f')}< 2. \] Here \(E^{(1)}_n(f)\) denote the degree of monotone approximation by polynomials of degree \(\leq n\).
scientific article; zbMATH DE number 938741