A Voronovskaya-type theorem for a general class of discrete operators
A general class of discrete, not necessarily positive operators is studied that acts on functions defined on an interval of the real line and has the form \[ (S_nf)(t)=\sum _{k=0}^\infty K_n(t,\nu_{n,k})f(\nu_{n,k}),\quad n\in\mathbb N,\;t\in I, \] where \(I\) is a fixed interval (bounded or not) in \(\mathbb R\) and, for every \(n\in\mathbb N\), \((\nu_{n,k})_{k\in\mathbb N_0}\subset I\) is a sequence satisfying suitable assumptions. Under specific assumptions on the ``kernel functions'' \(K_n\) and for a function \(f\) such that \(f''(t)\) exists at a point \(t\in I\), a general Voronovskaya-type theorem is established. The obtained asymptotic formula is used for the well-known discrete operators of Bernstein, Szász-Mirak'jan, Chlodovsky, Baskakov, Meyer-König and Zeller, Bleimann-Butzer-Hahn and their various extensions. Some of the obtained results coincide with the classical ones. Finally, an extension of the proved asymptotic formula to a discrete nonlinear operator is given.
scientific article; zbMATH DE number 5614517
scientific article