q-analogue of a new sequence of linear positive operators
© Gupta et al.; licensee Springer 2012
Received: 7 May 2012
Accepted: 4 June 2012
Published: 22 June 2012
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.
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 .
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
where is the q-Baskakov operator defined by (1.3).
3 Direct theorems
Finally, taking the infimum over all and using the inequality , , we get the required result. This completes the proof of Theorem 1. □
We consider the following class of functions:
We observe that for function , the modulus of continuity tends to zero.
By taking we get the assertion of our theorem. □
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.
Theorem 3 Let, then the sequenceconverges to f uniformly onfor eachif and only if.
uniformly on any, .
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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