A short note on approximation properties of Stancu generalization of q-Durrmeyer operators
© Mishra and Patel; licensee Springer 2013
Received: 14 January 2013
Accepted: 19 March 2013
Published: 4 April 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 , .
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 , .
Using , we get , where , are constants independent of k. Hence, .
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.
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 :
The proof of the above lemma follows along the lines of , Theorem 3]; thus, we omit the details.
Dedicated to Prof. Hari M. Srivastava on the occasion of his 72th Birth Anniversary.
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