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Positive linear operators which preserve $x^2$. (English) Zbl 1027.41028
The approximations of continuous functions $f$ on $[0,1]$ by a sequence of positive linear operators ${L_n}$ always converge to $f$ iff $L_n$ preserve the three functions $e_i(x)=x$, $i=0,1,2$ (Korovkin theorem). Replacing the variable $x$ in the Bernstein polynomials by some functions $r_n(x)$ the author defines the operators $L_n$ acting on $\Cal C([0,1])$, satisfying the Korovkin condition and leading to the order of approximation of $f$ at least as good as the order of approximation by Bernstein polynomials. The summability matrix $A$ is defined by means of the functions $r_n(x)$ and it is proved that $A$ preserves the limits of complex sequences provided $\lim_{n\to \infty}r_n(x)=x$.
Reviewer: Jaczek Gilewicz (Les Arcs sur Argens)

41A40Saturation (approximations and expansions)
40G99Special methods of summability
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