Shift Invariant Univariate Integral Operators

In this chapter among other topics, we further study global smoothness over R. A very general positive linear integral type operator is given through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and preservation of continuous probabilistic distribution functions. Furthermore, examples of very general specialized operators are presented fulfilling all the above properties; especially, the inequalities for global smoothness preservation are proven to be sharp. Here we follow the basic study done by the first author and Gonska [30].