The degree of approximation by positive operators on compact connected abelian groups

Abstract

In 1953 P. P. Korovkin proved that if (T n ) is a sequence of positive linear operators defined on the space C of continuous real 2w-periodic functions and lim T n f -f uniformly f o r / = 1, cos and sin, then lim T n f = f uniformly for all / e C. Quantitative versions of this result have been given, where the rate of convergence is given in terms of that of the test functions 1, cos and sin, and the modulus of continuity of /. We extend this result by giving a quantitative version of Korovkin's theorem for compact connected abelian groups. Throughout G will denote a compact Hausdorff abelian group, T its character group, and C(G) the space of continuous functions on G with the uniform norm II • ||. A linear operator T on C(G) will be called positive if Tf> 0 whenever / > 0. It is well known that such an operator takes real functions into real functions, and | 7/|<