An operator version of the Korovkin theorem for continuous periodic functions

Abstract

The following operator version of the Korovkin theorem for continuous 2π-periodic functions is proved: Let Ln:C2π(R)→C2π(R) be a sequence of linear positive operators and U:C2π(R)→C2π(R) a linear positive operator such that [U(sin⁡)]2+[U(cos⁡)]2=[U(1)]2. If limn→∞⁡Ln(1)=U(1), limn→∞⁡Ln(sin⁡)=U(sin⁡), limn→∞⁡Ln(cos⁡)=U(cos⁡) all uniformly on R, then for every φ∈C2π(R) we have limn→∞⁡[Ln(φ)U(1)−Ln(1)U(φ)]=0 uniformly on R. If in addition, U(1)(x)>0 for every x∈R then for every φ∈C2π(R), limn→∞⁡Ln(φ)=U(φ) uniformly on R. As application we give various concrete examples.