On the convergence of a kind of q-gamma operators
© Cai and Zeng; licensee Springer. 2013
Received: 15 November 2012
Accepted: 25 February 2013
Published: 18 March 2013
In this paper, we introduce a kind of q-gamma operators based on the concept of a q-integral. We estimate moments of these operators and establish direct and local approximation theorems of the operators. The estimates on the rate of convergence and weighted approximation of the operators are obtained, a Voronovskaya asymptotic formula is also presented.
MSC:41A10, 41A25, 41A36.
In recent years, the applications of q-calculus in the approximation theory is one of the main areas of research. After q-Bernstein polynomials were introduced by Phillips  in 1997, many researchers have performed studies in this field; we mention some of them [1–4].
In 2009, Karsli, Gupta and Izgi  gave an estimate of the rate of pointwise convergence of these operators (1) on a Lebesgue point of bounded variation function f defined on the interval . In 2010, Karsli and Özarslan  obtained some direct local and global approximation results and gave a Voronoskaya-type theorem for the operators (1). As the application of q-calculus in approximation theory is an active field, it seems there are no papers mentioning the q analogue of these operators defined in (1). Inspired by Aral and Gupta , they defined a generalization of q-Baskakov type operators using q-Beta integral and obtained some important approximation properties, which motivates us to introduce the q analogue of this kind of gamma operators.
provided the sums converge absolutely.
where and , .
Note that for , become the gamma operators defined in (1).
2 Some preliminary results
In order to obtain the approximation properties of the operators , we need the following lemmas.
Lemma 1 is proved. □
Proof From Lemma 1, taking , we get (6) and (7). Since and , using (6), (7), we obtain (8) and (9) easily. □
which is the moments and central moments of the operators defined in (1).
3 Local approximation
In this section we establish direct and local approximation theorems in connection with the operators .
We denote the space of all real-valued continuous bounded functions f defined on the interval by . The norm on the space is given by .
where and .
Our first result is a direct local approximation theorem for the operators .
where and are defined in (12). This completes the proof of Theorem 1. □
Remark 2 Let be a sequence satisfying and , we have and , these give us the pointwise rate of convergence of the operators to .
4 Rate of convergence
Obviously, for function , the modulus of continuity tends to zero.
By taking , we get the assertion of Theorem 2. □
5 Weighted approximation and Voronovskaya-type asymptotic formula
Now we will discuss the weighted approximation theorem.
Since and , (20) holds true for and .
since , we get , so the condition of (21) holds for as , then the proof of Theorem 3 is completed. □
Finally, we give a Voronovskaya-type asymptotic formula for by means of the second and fourth central moments.
for every , .
Theorem 4 is proved. □
This work is supported by the National Natural Science Foundation of China (Grant No. 61170324), the Natural Science Foundation of Fujian Province of China (Grant No. 2010J01012) and the Project of the Educational Office of Fujian Province of China (Grant No. JK2011041).
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