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Positive linear operators which preserve ${x}^{2}$. (English) Zbl 1027.41028
The approximations of continuous functions $f$ on $\left[0,1\right]$ by a sequence of positive linear operators ${L}_{n}$ always converge to $f$ iff ${L}_{n}$ preserve the three functions ${e}_{i}\left(x\right)=x$, $i=0,1,2$ (Korovkin theorem). Replacing the variable $x$ in the Bernstein polynomials by some functions ${r}_{n}\left(x\right)$ the author defines the operators ${L}_{n}$ acting on $𝒞\left(\left[0,1\right]\right)$, 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 ${r}_{n}\left(x\right)$ and it is proved that $A$ preserves the limits of complex sequences provided ${lim}_{n\to \infty }{r}_{n}\left(x\right)=x$.

##### MSC:
 41A40 Saturation (approximations and expansions) 40G99 Special methods of summability