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

${e}_{i}\left(x\right)=x$,

$i=0,1,2$ (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

$\mathcal{C}\left(\right[0,1\left]\right)$, 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

${lim}_{n\to \infty}{r}_{n}\left(x\right)=x$.