Characterization Theorems for the Approximation by a Family of Operators

The intention of this paper is to study a family of positive linear approximation operators relating to most of the well known Bernstein-type operators. These operators depend on a parameter. We give some characterization theorems to show that the operators corresponding to different parameters can be quite different. The direct and converse results make use of the Ditzian-Totik modulus of smoothness.