Linear operators that preserve some test functions.

Agratini, Octavian

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Bernstein type operators having 1 and x j as fixed points

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For certain generalized Bernstein operators {L n} we show that there exist no i, j ∈ {1, 2, 3,…}, i < j, such that the functions e i(x) = x i and e j (x) = x j are preserved by L n for each n = 1, 2,… But there exist infinitely many e i such that e 0(x) = 1 and e j (x) = x j are its fixed points.

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On approximation of functions by certain operators preserving $x^{2}$

Lucyna Rempulska, Karolina Tomczak (2008)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we extend the Duman-King idea of approximation of functions by positive linear operators preserving $e_{k} (x) = x^{k}$ , $k = 0, 2$ . Using a modification of certain operators $L_{n}$ preserving $e_{0}$ and $e_{1}$ , we introduce operators $L_{n}^{*}$ which preserve $e_{0}$ and $e_{2}$ and next we define operators $L_{n; r}^{*}$ for $r$ -times differentiable functions. We show that $L_{n}^{*}$ and $L_{n; r}^{*}$ have better approximation properties than $L_{n}$ and $L_{n; r}$ .