Steckin-Marchaud-type inequalities in connection with Bernstein polynomials
Let \(0\leq \alpha \leq 2\), \(\phi (x)=(x(1-x))^{1/2}\), \(C_ 0=\{f\in C[0,1]:\) \(f(0)=f(1)=0\}\), \(C_{\alpha}=\{f\in C_ 0:\phi^{-\alpha}f\) continuous and bounded on (0,1)\(\}\). For \(f\in C_{\alpha}\) let \(\| f\|_{\alpha}\) be the uniform norm of \(\phi^{-\alpha}f\). Let \(\Delta^*_ hf(x)=f(x+h)-2f(x)+f(x-h)\) and \[ \omega_{\alpha}(t,f)=\sup \{\phi^{-\alpha}(x)| \Delta^*_{h\phi (x)}f(x)|: x,x\pm h\phi (x)\in [0,1],\quad 0<h\leq t\}. \] Let \(B_ k\) be the classical Bernstein operators on C[0,1]. The author proves that for \(f\in C_{\alpha}\) and \(n\in N\) one has \(\omega_{\alpha}(n^{-},f)\leq M_{\alpha}n^{- 1}\sum^{n}_{k=1}\| B_ kf-f\|_{\alpha}.\) Using this Steckin- Marchaud-type inequality a number of known inverse theorems for Bernstein polynomials are derived and improved and a problem posed by V. Totik in 1984 is solved.
inverse theorems for Bernstein polynomials
scientific article; zbMATH DE number 4004840