Regular and Convergent Korovkin Sequences

Abstract

Let C_2π denote the space of 2π-periodic continuous functions with the uniform norm. Then Korovkin’s theorem states that if {L_n} is a sequence of monotone (positive) linear operators C_2π → C_2π then necessary and sufficient conditions for L_nf → f for each f ∈ C_2π are that the convergence should take place for the functions 1, cosine and sine. A proof will be found in Korovkin [6], a rather more general one is given by Price [7]. The monograph by De Vore [4] is devoted to an examination of positive operators and further references will be found there.