Approximation of the derivatives of some combinations of operators of class B (Q913169)

The author investigates a rather general positive linear operator defined by \textit{Yu. I. Volkov} [Izv. Akad. Nauk SSSR, Ser. Mat. 47, No.3, 435-454 (1983; Zbl 0528.41013)], denoted by \(L_ n\). Volkov proved that if \(f\in C^{2k}\) then \(L_ n(f,x)=f(x)+\sum^{k}_{s=1}n^{-s}Y_ sf(x)+O(n^{-k})\) \((n\to \infty)\) where \(Y_ s\) are differential operators determined by the central moments of \(L_ n\). The author defines a sequence of differential operators \(A_ k\) by recursion. \(A_ 0\) is the identity operator and \(A_ k=-\sum^{k}_{r=1}A_{k-r}Y_ r\). After this he defines the operator \(L_{n,m}(f,x)=\sum^{m- 1}_{k=0}A_ kL_ n(f,x)n^{-k}\). Then he states, among other theorems, the following (Theorem 1): if \(f\in C^{2m+\nu}\) then \(\lim_{n\to \infty}n^ m(f(x)-L_{n,m}(f,x))^{(\nu)}=(A_ mf(x))^{(\nu)}.\)