A ''K-attainable'' inequality related to the convergence of positive linear operators (Q1061963)

The convergence of a sequence of positive linear operators to the identity operator was studied deeply by Korovkin (1960). His main theorem was put in an inequality form by Shisha and Mond (1968), measuring the degree of this convergence. Later other authors gave similar, more general/flexible, quantitative results, e.g., Mond (1976). By the Riesz representation theorem, the above convergence is closely related to the weak convergence of a sequence of finite measures, to the unit (Dirac) measure at a fixed point. Introducing a new variation of the K-functional and using the terminology of moments, for convenience we consider only probability measures, we establish a sharp inequality which estimates the degree of the pointwise convergence of a sequence of positive linear operators to the identity operator, all acting on C([a,b]), where [a,b]\(\subset {\mathbb{R}}\).