This paper deals with Durrmeyer type generalization of q-Baskakov type operators using the concept of q-integral, which introduces a new sequence of positive q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous functions defined on the interval . An estimate for the rate of convergence and weighted approximation properties are also obtained.
MSC: 41A25, 41A36.
Keywords:Durrmeyer type operators; weighted approximation; rate of convergence; q-integral
In the year 2003 Agrawal and Mohammad  introduced a new sequence of linear positive operators by modifying the well-known Baskakov operators having weight functions of Szasz basis function as
It is observed in  that these operators reproduce constant as well as linear functions. Later, some direct approximation results for the iterative combinations of these operators were studied in .
A lot of works on q-calculus are available in literature of different branches of mathematics and physics. For systematic study, we refer to the work of Ernst , Kim [10,11], and Kim and Rim . The application of q-calculus in approximation theory was initiated by Phillips , who was the first to introduce q-Bernstein polynomials and study their approximation properties. Very recently the q-analogues of the Baskakov operators and their Kantorovich and Durrmeyer variants have been studied in [2,3] and  respectively. We recall some notations and concepts of q-calculus. All of the results can be found in  and . In what follows, q is a real number satisfying .
The q-binomial coefficients are given by
The q-Beta integral is defined by 
which satisfies the following functional equation:
For and each positive integer n, the q-Baskakov operators  are defined as
Remark 1 The first three moments of the q-Baskakov operators are given by
As the operators have mixed basis functions in summation and integration and have an interesting property of reproducing linear functions, we were motivated to study these operators further. Here we define the q-analogue of the operators as
In case , the above operators reduce to the operators (1.1). In the present paper, we estimate a local approximation theorem and the rate of convergence of these new operators as well as their weighted approximation properties.
2 Moment estimation
Lemma 1The following equalities hold:
Next, we have
Remark 2 If we put , we get the moments of a new sequence considered in  as operators as
3 Direct theorems
The Peetre’s K-functional is defined by
where . By , pp.177], there exists a positive constant such that and the second order modulus of smoothness is given by
Applying Lemma 2, we obtain
Using Lemma 1, we have
We consider the following class of functions:
Let be the set of all functions f defined on satisfying the condition , where is a constant depending only on f. By , we denote the subspace of all continuous functions belonging to . Also, let be the subspace of all functions , for which is finite. The norm on is . We denote the modulus of continuity of f on closed interval , as by
From (3.1) and (3.2) we can write
4 Higher order moments and an asymptotic formula
Lemma 3 ()
Now, we present higher order moments for the operators (1.4).
The proof of Lemma 4 can be obtained by using Lemma 3.
We consider the following classes of functions:
By the Cauchy-Schwarz inequality, we have
Then, we get the following
The authors declare that they have no competing interests.
All authors contributed in preparing the manuscript equally.
The work was done while the first author visited Division of General Education-mathematics, Kwangwoon University, Seoul, South Korea for collaborative research during June 15-25, 2010.
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