Shift Invariant Multivariate Integral Operators

Here among other topics we further study global smoothness preservation over Rd,d ≥ 1. This is a generalization of Chapter 10 in the multivariate case. Namely, a general positive linear multivariate integral type operator is given through a convolution-like iteration of another general positive linear multivariate operator with a multivariate scaling type function. For this sufficient conditions are given for shift invariance, global smoothness preservation and its sharpness, convergence to the unit with rates, shape preserving and preservation of continuous probabilistic distribution functions. Additionally, four examples of general specialized multivariate operators are given fulfilling all the above properties; especially, the inequalities for global smoothness preservation are sharp. In this chapter global smoothness preservation and convergence to the unit with rates involve a naturally arising suitable multivariate modulus of continuity. Here we follow the basic study done by the first author and Gonska [29]. Chapter 8 is also related here.