Asymptotic theorems for modifications of polynomials which are similar to the Baernstein polynomials
type: Item
instance of: scholarly article
title: Asymptotic theorems for modifications of polynomials which are similar to the Baernstein polynomials
zbMATH Open document ID: 0999.41008
author: T. V. Yershova
published in: Russian Mathematics
publication date: 2002-11-30T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 41A10
zbMATH DE Number: 1758693
zbMATH Keywords: Bernstein type operators
zbMATH Keywords: Voronovskaja formula
zbMATH Keywords: asymptotic expansion
reviewed by: Antonio Jesús López-Moreno
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1758693