Fractional Convergence Theory of Positive Linear Operators

Abstract

In this chapter we study quantitatively with rates the weak convergence of a sequence of finite positive measures to the unit measure. Equivalently we study quantitatively the pointwise convergence of sequence of positive linear operators to the unit operator, all acting on continuous functions. From there we obtain with rates the corresponding uniform convergence of the latter. The inequalities for all of the above in their right hand sides contain the moduli of continuity of the right and left Caputo fractional derivatives of the involved function. From the uniform Shisha-Mond type inequality we derive the fractional Korovkin type theorem regarding the uniform convergence of positive linear operators to the unit.We give applications, especially to Bernstein polynomials for which we establish fractional quantitative results.