Global smoothness preservation and the variation-diminishing property

In the center of our paper are two counterexamples showing the independence of the concepts of global smoothness preservation and variation diminution for sequences of approximation operators. Under certain additional assumpt ions it is shown that the variation-diminishing property is the stronger on e. It is also demonstrated, however, that there are positive linear operato rs giving an optimal pointwise degree of approximation, and which preserve global smoothness, monotonicity and convexity, but are not variation-dimini shing.