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



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Demonstratio Mathematica


The present paper deals with the one-parameter family {Dq}q∈[0,1] of Bernstein-type operators introduced by Gupta, and called the limit q-Durrmeyer operators. The continuity of this family with respect to the parameter q 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] is continuous in the strong operator topology for all q ∈ [0, 1]. When it comes to the uniform operator topology, the continuity is preserved solely at q = 0, and fails at all q ∈ (0, 1]. In addition, a few estimates for the distance between two limit q-Durrmeyer operators have been derived in the operator norm on C[0, 1].


Open Access; Published by Demonstratio Mathematica;; Övgü Gürel Yılmaz, Recep Tayyip Erdogan University, Department of Mathematics, 53100, Rize, Turkey; Sofiya Ostrovska, Mehmet Turan, Atilim University, Department of Mathematics, Incek 06830, Ankara, Turkey.


q-Durrmeyer operator, q-Bernstein-operator, operator norm, strong operator topology, uniform operator topology, 47B38, 41A36