Affiliation:
1. Department of Mathematics, Recep Tayyip Erdogan University , 53100 , Rize , Turkey
2. Department of Mathematics, Atilim University , Incek 06830 , Ankara , Turkey
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]
.
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