A “ K-Attainable” inequality related to the convergence of positive linear operators

Abstract

The convergence of a sequence of positive linear operators to the identity operator was studied deeply by Korovkin [S]. His main theorem was put in an inequality form by Shisha and Mond [9], measuring the degree of this convergence. Later other authors gave similar, more general/flexible, quantitative results, e.g., Mond [7]. By 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 (see [3,4]), for convenience we consider only probability measures, and 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] c R.