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A short note on approximation properties of Stancu generalization of q-Durrmeyer operators
Fixed Point Theory and Applications volume 2013, Article number: 84 (2013)
In the present paper, we introduce a simple Stancu generalization of q-analogue of well-known Durrmeyer operators. We first estimate moments of q-Durrmeyer-Stancu operators. We also establish the rate of convergence as well as Voronovskaja type asymptotic formula for q-Durrmeyer-Stancu operators.
In the last decade, the applications of q-calculus in the approximation theory is one of the main areas of research. To approximate Lebesgue integrable functions on the interval , Durrmeyer introduced the integral modification of the well-known Bernstein polynomials. In 1981, Derriennic  first studied these operators in detail. After the q-analogue of Bernstein polynomials by Phillips , Gupta and Heping  introduced q-Durrmeyer operators. Several other researchers have studied in this direction and obtained different approximation properties of many operators [4, 5]. In the present article, we propose the q-analogue of the Stancu generalization of Durrmeyer operators and study the convergence behavior. We have used notations of q-calculus as given in [6–8].
We set , .
Phillips  proposed the following q-Bernstein polynomials, which for each positive integer n and are defined as
In 2005, Derriennic  introduced a q-analogue of the Durrmeyer operators and has established some approximation properties of the q-Durrmeyer operators. After couple of years, Gupta  studied q-analogue of Durrmeyer operators and discussed approximation properties of the following q-Durrmeyer operators: For ,
Recently, Ibrahim et al. [11, 12] introduced Stancu generalization of certain operators and discussed its approximation properties. Motivated by such type operators, we introduce the Stancu type generalization of the q-Durrmeyer operators (1) for , which is defined as follows:
It can be easily verified that in case , and , the operators defined in (2) reduce to the well-known Durrmeyer operators as defined in . Throughout the present manuscript, the expression means uniform convergence of a sequence to .
The present note deals with the study of q-Durrmeyer-Stancu operators for . First, we estimate the moments for q-Durrmeyer-Stancu operators. We also study the rate of convergence as well as asymptotic formula for these operators . We establish a direct results in terms of .
2 Estimation of moments
In this section, we shall obtain , .
Note that for and by the definition of q-Beta function , we have
Lemma 1 We have
Lemma 2 We have
Remark 1  By simple computation, it can easily be verified that
Using , we get , where , are constants independent of k. Hence, .
Remark 2 For all , , we have the following recursive relation for the images of the monomials under in terms of ; , as
Since for and is a polynomial of degree (see ), we get is a polynomial of degree .
3 Convergence of q-Durrmeyer-Stancu operators
Theorem 1 Let . Then the sequence convergence to f uniformly on for each if and only if .
The proof of the above theorem follows along the lines of , Theorem 2], thus we omit the details.
Let be fixed. We define and for
Using the fact that , we have
Using (3) and (4), it is easy to prove that
For , , we define the modulus of continuity as follows: . We shall show the following theorem.
Theorem 2 Let then for each the sequence converges to uniformly on . Furthermore, .
The proof of the above theorem follows along the lines of , Theorem 3], thus we omit the details.
Remark 3 We may observe that, for , we have , where means that and , and means that there exists a positive constant C independent of n such that . Hence, the estimate of Theorem 2 is sharp in the following sense: the sequence in Theorem 2 cannot be replaced by any other sequence decreasing to zero more rapidly as .
Lemma 3 
Let L be a positive linear operator on , which reproduces linear functions. If , then if and only if f is linear.
Remark 4 Since for consequence of Lemma 3 we have the following:
Theorem 3 Let be fixed and let . Then for all if and only if f is linear.
Remark 5 Let be fixed and let . Then the sequence does not approximate unless f is linear. This is completely in contrast to the classical Bernstein polynomials, by which approximates for any .
Theorem 4 For any , converges to f uniformly on as .
Next, we establish a Voronovskaja type asymptotic formula for the operators :
Theorem 5 Let f be bounded and integrable on the interval , second derivative of f exists at a fixed point and such that as , then
The proof of the above lemma follows along the lines of , Theorem 3]; thus, we omit the details.
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Dedicated to Prof. Hari M. Srivastava on the occasion of his 72th Birth Anniversary.
The authors declare that they have no competing interests.
PP computed the moments of the modified operators, established the rate of convergence and Voronovskaja type asymptotic formula. VM conceived of the study and participated in its design and coordination. VM and PP contributed equally to this work. All the authors read and approved the final manuscript.
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Mishra, V.N., Patel, P. A short note on approximation properties of Stancu generalization of q-Durrmeyer operators. Fixed Point Theory Appl 2013, 84 (2013). https://doi.org/10.1186/1687-1812-2013-84
- Asymptotic Formula
- Recursive Relation
- Approximation Property
- Convergence Behavior
- Integral Modification