Asymptotic theorems for modifications of polynomials which are similar to the Baernstein polynomials
In order to study the behavior of the derivatives of the classical Bernstein operators on \([0,1]\), V. S. Videnskij defines the modified Bernstein operators given, for \(f:[0,1]\rightarrow \mathbb R\), \(n\in\mathbb N\) and \(x\in [0,1]\), by \[ U_nf(x)={n \over{x(1-x)}} \sum_{k=0}^n \left( {k\over n}-x\right)^2 f\left( {k\over n}\right) \left(\begin{matrix} n\\ k\end{matrix}\right) x^k(1-x)^{n-k}. \] The author studies the modification of the \(U_n\) operators obtained by applying the technique of augmenting the order of approximation introduced by Videnskij and T. P. Pendina. More precisely, he consideres the operators \(U_{n,\nu}\) defined by means of the recurrence process: \[ \begin{matrix} U_{n,1}f=U_nf, \cr \cr U_{n,\nu}f(x)=U_nf(x)- \sum_{k=1}^{\nu-1} {U_n\left((t-x)^k\right)(x) \over k!} U_{n,\nu-k}f^{(k)}(x) \;\text{ for } \nu\geq 2,\end{matrix} \] where \(t:z\ni [0,1]\mapsto t(z)=z\in [0,1]\) is the identity map on \([0,1]\). The paper is devoted to the study of the asymptotic behavior of these operators and their central moments as \(n\) tends to infinity. The author first analyzes the central moments \(U_{n,2m-1}\left((t-x)^{2m-1} \right)(x)\) by computing the limit \[ \lim_{n\rightarrow \infty} { n^m U_{n,2m-1}\left((t-x)^{2m-1} \right)(x) \over (2m-1)!} \] explicitly (the calculation for the moment \(U_{n,2m}\left((t-x)^{2m} \right)(x)\) is made by the author in a preceding paper). With the aid of these results Voronovskaya type formulae and explicit expressions of the remainder term of higher order asymptotic expansions are obtained. In this way, in the main result the limit \[ \lim_{n\rightarrow \infty} n^{m+l}\left( U_{n,2m}f(x)-f(x) -\sum_{k=0}^{2l-2} {U_{n,2m+k}\left((t-x)^{2m+k}\right)(x)\over (2m+k)!} f^{(2m+k)}(x) \right) \] is computed (it is indicated that the analogous result for \(U_{n,2m-1}\) can be proved in a similar way).
scientific article; zbMATH DE number 1758693