Generalized Shift Invariant Multivariate Integral Operators

This chapter is a continuation and generalization of Chapters 11 and 14. Among others we further study global smoothness preservation over Rd,d ≥ 1. In particular, certain other similar to those in Chapter 11, but more general, multivariate integral operators are presented and studied. These operators come up naturally. And for all these are given sufficient conditions for multivariate: shift invariance, preservation of higher order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity with respect to uniform norm, shape preserving on Rd, and preservation of multivariate continuous probabilistic distribution functions. Several examples of diverse very general but specified multivariate integral operators fulfilling this theory are given at the end. Here we follow the basic study done by both authors in [25].