Generalized Shift Invariant Univariate Integral Operators

Abstract

This chapter is a continuation and generalization of Chapter 10. Among others we further study global smoothness preservation over R. In particular, certain others, similar to those in Chapter 10, but more general integral operators are presented and studied. These operators arise in a natural way. And for all these are given sufficient conditions for: shift invariance, preservation of higher order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity, shape preserving and preservation of continuous probabilistic distribution functions. Several examples of diverse, very general specialized operators are given fulfilling all the above listed properties. Here we follow the basic study done by both authors in [24].