A Korovkin-type theorem for sequences of positive linear operators on function spaces (Q2056250)

The author present some Korovkin-type theorems for special sequence of linear positive operators. The main result is the following Theorem. Let \(L_\infty\) be a positive linear operator on \(C[0, 1]\) and let \((L_n)_{n\geq 1}\) be a sequence of positive linear operators on \(C[0, 1]\) satisfying the following conditions: (i) \(\lim_{n\to \infty} L_n(e_2) = L_\infty(e_2)\) uniformly on \([0, 1]\); (ii) \(L_n( f ) \leq L_\infty( f )\) (or \(L_n( f ) \geq L_\infty ( f )\)) for every convex function \(f \in C[0, 1]\) and for every \(n \in \mathbb{}\). Then \(\lim_{n\to \infty} L_n( f ) = L_\infty( f )\) for all \(f \in C[0, 1]\). Several applications are given.