Comparison of Bernstein polynomials with the metrical means of Kantorovitch (Q1092347)

Let \(B_ nf=\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)x^ k(1-x)^{n-k}f(k/n)\) be Bernstein polynomials and \(B^*_ nf\), modified Bernstein polynomials in the sense of Kantorovich, given by \(B_ nf=\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)x^ k(1-x)^{n-k}M(E_{kn},f),\) \(M(E,f)=\sup \{u\in R| meas\{v\in E| f(v)\geq u\}\geq meas E\}\), \(E_{kn}=[k/n-\delta_ n,k/n+\delta_ n]\cap [0,1],\delta_ n=3n^{-1/3}\). If \(f\in C_{[0,1]}\) and \(\omega\) is the continuity modulus of f, then \[ | (B^*_ nf-f)(x)| \leq | (B_ nf-f)(x)+w(\delta_ n,f),\quad | (B_ nf-f)(x)| \leq | (B_ n^*f-f)(x)| +w(\delta_ n,f) \] hold. These inequalities cannot be strengthened to \[ | (B^*_ nf- f)(x)| \leq C_{x,f}| (B_ nf-f)(x)| \leq C^*_{x,f}| (B^*_ nf-f)(x)| \] for some constants \(C_{x,f}\) and \(C^*_{x,f}\) independent of \(n\in N\).