The order of comonotone approximation of differentiable periodic functions (Q2146229)

The authors prove the following theorem: For any set \(Y\), there is a function \(f_{Y}=f\in C^{(1)}\cap \Delta ^{(1)}(Y)\) such that \[ \lim_{n\rightarrow \infty }\sup \frac{nE_{n}^{(1)}\left( f\right) }{\omega _{4}(f^{\prime },\pi /n)}=\infty, \] where \(\Lambda ^{(1)}(Y)\) is a set of all continuous comonotone functions, \(E_{n}^{(1)}\left( f\right) \) is the error of the best uniform commonote approximation of the function \(f\) by a trigonometric polynomial and \(\omega _{k}(f,\mathit{\cdot })\) is the modulus of smoothness of the \(k\)-th order.