Paltanea type theorems on estimation by positive discrete functionals (Q6634785)

Let \(F\) be a positive linear functional of the form \(F(f)=\sum_{y\in Y}\gamma(y)f(y)\), where \(Y\) has no point of accumulation, \(\gamma(y):Y\to(0,\infty)\), \(f:I\to \mathbb{R}\), \(I=[\min Y,\max Y]\) is such that \(\sum_{y\in Y}\gamma(y)|f(y)|<\infty\). Let \(x\in I\) be fixed and suppose that \(F(e_1-xe_0)=0\), where \(e_0(t)=1\), \(e_1(t)=t\), \(t\in \mathbb{R}\). The second order modulus of continuity of a function \(f\) on \(I\) is \(\omega_2(f,h)=\sup\{|f(t+\rho)-2f(t)+f(t-\rho)|,\;t\pm\rho\in I, |\rho|\le h\}\), \(h>0\). \N\NIn the book [\textit{R. Păltănea}, Approximation theory using positive linear operators. Boston, MA: Birkhäuser (2004; Zbl 1154.41013)], conditions are found in which we have the inequality \N\[\N|F(f)-F(e_0)f(x)|\le F(e_0)\omega_2(f,h),\text{ for }f:I\to\mathbb{R},\ h>0 \tag{\(\star\)}, \N\]\Nin order to prove the optimal estimate \(|B_n(f)(x)-f(x)|\le \omega_2(f,n^{-1/2})\), \(f\in B[0,1]\), \(x\in [0,1]\). where \(B_n\) are Bernstein's operators. \N\NHere, similar results are obtained, in a more general framework. We quote the main result: \N\NSuppose that there exist the sets \(Y_+\subset Y\cap ([x-h,x-(3/4)h]\cup [x-(1/2)h,\infty))\) and \(Y_-\subset Y\cap[(-\infty,x+(1/2)h]\cup[x+(3/4)h,x+h])\), such that \N\begin{itemize}\N\item[i)] \(\sum_{y\in Y_+}\gamma(y)(x-y)>0\); \N\item[ii)] \(\sum_{y\in Y_+\cap[\max\{(z-h,x\},\infty)}\gamma(y)(z-y)\ge 0\), for \(z\in[x+(3/4)h,x+(5/4)h]\) and \N\item[iii)] \(\sum_{y\in Y_+\cap [z-h/2,\infty)}\gamma(y)(z-y)\ge 0\), for \(z\in [x+(5/4)h,\sup Y)\), as well the symmetrical conditions for the set \(Y_-\), \N\end{itemize}\Nthen inequality (\(\star\)) holds.