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

Abstract

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

Keywords

• q-durrmeyer operator
• q-bernstein operator
• operator norm
• strong operator topology
• uniform operator topology
• 47b38
• 41a36