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In this paper we extend the Duman-King idea of approximation of functions by positive linear operators preserving $e_k\left(x\right)=x^k$, $k=0,2$. 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 $L_{n;r}$.

We consider the Picard operators ${𝒫}_n$ and ${𝒫}_{n;r}$ in exponential weighted spaces. We give some elementary and approximation properties of these operators.