On the continuity in q of the family of the limit q-Durrmeyer operators

Abstract

Abstract This study deals with the one-parameter family { D q } q ∈ [ 0 , 1 ] {\left\{{D}_{q}\right\}}_{q\in \left[0,1]} of Bernstein-type operators introduced by Gupta and called the limit q q -Durrmeyer operators. The continuity of this family with respect to the parameter q q is examined in two most important topologies of the operator theory, namely, the strong and uniform operator topologies. It is proved that { D q } q ∈ [ 0 , 1 ] {\left\{{D}_{q}\right\}}_{q\in \left[0,1]} is continuous in the strong operator topology for all q ∈ [ 0 , 1 ] q\in \left[0,1] . When it comes to the uniform operator topology, the continuity is preserved solely at q = 0 q=0 and fails at all q ∈ ( 0 , 1 ] . q\in \left(0,1]. In addition, a few estimates for the distance between two limit q q -Durrmeyer operators have been derived in the operator norm on C [ 0 , 1 ] C\left[0,1] .