## Lucyna Rempulska, Karolina Tomczak

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

Comment.Math.Univ.Carolin. 49,4 (2008) 579-593. **Abstract:**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 $L_{n;r}$.

**Keywords:** positive linear operators, polynomial weighted space, degree of approximation

**AMS Subject Classification:** 41A25, 41A36