Asymptotics for central moments of modifications of operators similar to Bernstein polynomials (Q2387018)

Given a sequence \(\{L_n\}\) of linear positive operators (LPO), the functions \(L_n((t- x)^i; x)= S_i(L_n; x)\), \(i= 0,1,\dots\) are called the ``central moment of order \(i\)'' of the operator \(L_n\). On the other hand, from the sequence \(\{L_n\}\), new operators \(\{L_{n,m}\}\) can be recursively generated as follows, \[ \begin{gathered} L_{n,1}f= L_nf,\\ L_{n,m}f= L_nf- \sum^{m-1}_{i=1} L_{n,m-1} f^{(i)}S_i(L_n)/ i!,\;m= 2,3,\dots\;.\end{gathered} \] Here, the LPOs under consideration include the Bernstein, Kantorovich and other known polynomial operators as particular cases. As a continuation of previous papers of the present author, in this paper, asymptotics for the central moments of the operator \(L_{n,m}\) are first established. As a consequence, asymptotic theorems (Theorems 1 and 2) of the Voronosvskaya-Bernstein type are also deduced for \(L_{n,2m}\) and \(L_{n,2m-1}\). The title of the paper under review exactly coincides with an earlier work of the author [Automorphisms of 2-nondegenerate hypersurfaces in \(\mathbb{C}^3\), Math. Notes 69, No. 2, 188--195 (2001); translation from Mat. Zametki 69, No. 2, 214--222 (2001; Zbl 0997.32034)] whose content has become completely unreachable for this reviewer.