Positive linear operators which preserve \(x^2\) (Q1416129)

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 \(\mathcal C([0,1])\), 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 \(\lim_{n\to \infty}r_n(x)=x\).