Fractional Trigonometric Convergence Theory of Positive Linear Operators

Abstract

In this chapter we study quantitatively with rates the trigonometric weak convergence of a sequence of finite positive measures to the unit measure. Equivalently we study quantitatively the trigonometric pointwise convergence of sequence of positive linear operators to the unit operator, all acting on continuous functions on [ − π,π]. From there we obtain with rates the corresponding trigonometric 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 these uniform trigonometric Shisha-Mond type inequality we derive the trigonometric fractional Korovkin type theorem regarding the trigonometric uniform convergence of positive linear operators to the unit. We give applications, especially to Bernstein polynomials over [ − π,π] for which we establish fractional trigonometric quantitative results. This chapter relies on [46].