Abstract
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.
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1 Introduction
In 2012, Aral and Gupta [1] introduced the q analog of Stancu-Beta operators as
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 .
Firstly, we recall some concepts of q-calculus. All of the results can be found in [2]. For any fixed real number and each nonnegative integer k, we denote q-integers by , where
Also the q-factorial and q-binomial coefficients are defined as follows:
and
The q-improper integrals are defined as
provided the sums converge absolutely.
The q-Beta integral is defined as
where , and , ().
In particular for any positive integer n,
For , , and , we introduce the new modification of q-Stancu-Beta operators as
where
2 Some preliminary results
In this section we give the following lemmas, which we need to prove our theorems.
Lemma 1 (see [[1], Lemma 1])
The following equalities hold:
Lemma 2 Let , , we have
Proof From Lemma 1, we get and easily. Finally, we have
Lemma 2 is proved. □
Remark 1 Let and , then for every , by Lemma 2, we have
Lemma 3 For every and , we have
Proof Since and from Lemma 2, we get Lemma 3 easily. □
Remark 2 Let the sequence satisfy that and as , then for any fixed , by Lemma 3, we have
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 .
Further let us consider Peetre’s K-functional:
where and .
For , the modulus of continuity of second order is defined by
by [[3], p.177] there exists an absolute constant such that
For , the modulus of continuity is defined by
Our first result is a direct local approximation theorem for the operators .
Theorem 1 For , , , and , we have
Proof For , we define the auxiliary operators
Obviously, we have
Let , by Taylor’s expansion, we have
Using (14), we get
hence, by Lemma 3, we have
On the other hand, using (13) and Lemma 2, we have
Thus,
Hence taking the infimum on the right-hand side over all , we get
By (11), for every , we have
This completes the proof of Theorem 1. □
4 Rate of convergence
Let be the set of all functions f defined on satisfying the condition , where is a constant depending only on f. We denote the subspace of all continuous functions belonging to by . Also, let be the subspace of all functions for which is finite. The norm on is . We denote the usual modulus of continuity of f on the closed interval () by
Obviously, for a function , the modulus of continuity tends to zero as .
Theorem 2 Let , and be the modulus of continuity on the finite interval , where . Then we have
Proof For and , we have . Hence . Thus . Hence, we obtain
For and , we have
From (17) and (18), we get
For and , by Schwarz’s inequality, Lemma 2, and Lemma 3, we have
By taking , we get the assertion of Theorem 2. □
5 Weighted approximation
Now we will discuss the weighted approximation theorems.
Theorem 3 Let the sequence satisfy and as , for , we have
Proof By using the Korovkin theorem in [4], we see that it is sufficient to verify the following three conditions:
Since and (see Lemma 2), (21) holds true for and .
Finally, for , we have
since , we get and , so the second condition of (21) holds for as , then the proof of Theorem 3 is completed. □
References
Aral A, Gupta V: On the q analogue of Stancu-Beta operators. Appl. Math. Lett. 2012,25(1):67–71. 10.1016/j.aml.2011.07.009
Kac VG, Cheung P Universitext. In Quantum Calculus. Springer, New York; 2002.
DeVore RA, Lorentz GG: Constructive Approximation. Springer, Berlin; 1993.
Gadjiev AD: Theorems of the type of P.P. Korovkin type theorems. Mat. Zametki 1976,20(5):781–786. (English translation: Math. Notes 20(5–6), 996–998 (1976))
Acknowledgements
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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Cai, QB. Approximation properties of the modification of q-Stancu-Beta operators which preserve . J Inequal Appl 2014, 505 (2014). https://doi.org/10.1186/1029-242X-2014-505
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DOI: https://doi.org/10.1186/1029-242X-2014-505