Approximation properties of the modification of q-Stancu-Beta operators which preserve
© Cai; licensee Springer. 2014
Received: 5 April 2014
Accepted: 2 December 2014
Published: 12 December 2014
In this paper, we introduce a new kind of modification of q-Stancu-Beta operators which preserve based on the concept of q-integer. We investigate the moments and central moments of the operators by computation, obtain a local approximation theorem, and get the pointwise convergence rate theorem and also a weighted approximation theorem.
MSC:41A10, 41A25, 41A36.
Keywordsq-integer q-Stancu-Beta operators weighted approximation
for every , , . They estimated moments, established direct result in terms of modulus of continuity and present an asymptotic formula.
Since the types of operators which preserve are important in approximation theory, in this paper, we will introduce a modification of q-Stancu-Beta operators which will be defined in (5). The advantage of these new operators is that they reproduce not only constant functions but also .
provided the sums converge absolutely.
where , and , ().
2 Some preliminary results
In this section we give the following lemmas, which we need to prove our theorems.
Lemma 1 (see [, Lemma 1])
Lemma 2 is proved. □
Proof Since and from Lemma 2, we get Lemma 3 easily. □
3 Local approximation
In this section we establish direct local approximation theorem 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 .
This completes the proof of Theorem 1. □
4 Rate of convergence
Obviously, for a function , the modulus of continuity tends to zero as .
By taking , we get the assertion of Theorem 2. □
5 Weighted approximation
Now we will discuss the weighted approximation theorems.
Since and (see Lemma 2), (21) holds true for and .
since , we get and , so the second condition of (21) holds for as , then the proof of Theorem 3 is completed. □
The author thanks the editor and referee(s) for several important comments and suggestions, which improved the quality of the paper. This work is supported by the Educational Office of Fujian Province of China (Grant No. JA13269), the Startup Project of Doctor Scientific Research of Quanzhou Normal University, Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing, Fujian Province University.
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