Linear operators that preserve some test functions (Q885610)

The Korovkin Theorem states that if \(\{L_n\}\) is a sequence of positive operators such that \(L_n(x^i)\) converges to \(x^i\) for \(i=0,1,2\), then \(L_n(f)\) converges to \(f\) for all continuous functions on \([a,b]\). The author constructs positive operators \(L_n\) such that \(L_n(x^i)\) converges to \(x^i\) for \(i=0,2\). He then relates the convergence of \(L_n(f)\) to \(f\) to the rate of convergence of \(L_n(x)\) to \(x\) for various moduli of smoothness of \(f\).