Universal properties of approximation operators

We discuss universal properties of some operators Ln:C[0,1]→C[0,1]Ln:C[0,1]→C[0,1]. The operators considered are closely related to a theorem of Korovkin (1960) [4] which states that a sequence of positive linear operators LnLn on C[0,1]C[0,1] is an approximation process if Lnfi→fi(n→∞) uniformly for i=0,1,2i=0,1,2, where fi(x)=xifi(x)=xi. We show that LnfLnf may diverge in a maximal way if any requirement concerning LnLn in this theorem is removed. There exists for example a continuous function ff such that (Lnf)n∈N(Lnf)n∈N is dense in (C[0,1],‖.‖∞)(C[0,1],‖.‖∞), even if LnLn is positive, linear and satisfies LnP→P(n→∞) for all polynomials PP with P(0)=0P(0)=0.