Convergence of the q-Stancu-Szász-Beta type operators
Journal of Inequalities and Applications volume 2014, Article number: 354 (2014)
In this paper, we study on q-Stancu-Szász-Beta type operators. We give these operators convergence properties and obtain a weighted approximation theorem in the interval .
In , Mahmudov constructed q-Szász operators and obtained rate of global convergence in the frame of weighted spaces and a Voronovskaja type theorem for these operators. In , Gupta and Mahmudov studied on the q-analog of the Szász-Beta type operators. In , Yüksel and Dinlemez gave a Voronovskaja type theorem for q-analog of a certain family Szász-Beta type operators. In , Govil and Gupta introduced the q-analog of certain Beta-Szász-Stancu operators. They estimated the moments and established direct results in terms of modulus of continuity and an asymptotic formula for the q-operators. In [5–14], interesting generalization about q-calculus were given. Our aims are to give approximation properties and a weighted approximation theorem for q-Stancu-Szász-Beta type operators. We use without further explanation the basic notations and formulas, from the theory of q-calculus as set out in [15–19]. Let and f be a real valued continuous function defined on the interval . For , q-Stancu-Szász-Beta type operators are defined as
2 Auxiliary results
For the sake of brevity, the notation and will be used throughout the article. Now we are ready to give the following lemma for the Korovkin test functions.
Lemma 1 Let , , we get
Then, using (2.1), for , we get
and the proof of (i) is finished. With a direct computation, we obtain (ii) as follows:
Using the equality
and so we have the proof of (iii). □
To obtain our main results we need to compute the second moment.
Lemma 2 Let and . Then we have the following inequality:
Proof From the linearity of the operators and Lemma 1, we write the second moment as
From (2.2), we have
And the proof of Lemma 2 is now finished. □
3 Direct estimates
Now in our considerations, denotes the set of all bounded-continuous functions from to ℝ. is a normed space with the norm . We denote the first modulus of continuity on the finite interval , ,
The Peetre K-functional is defined by
where . By Theorem 2.4 in , p.177, there exists a positive constant C such that
Gadzhiev proved the weighted Korovkin-type theorems in . We give the Gadzhiev results in weighted spaces. Let and the weighted spaces denote the space of all continuous functions f, satisfying , where is a constant depending only on f. is a normed space with the norm and denotes the subspace of all functions for which exists finitely.
Thus we are ready to give direct results. The following lemma is routine and its proof is omitted.
Lemma 3 Let
Then the following assertions hold for the operators (3.3):
Lemma 4 Let and . Then for every and , we have the inequality
Proof Using Taylor’s expansion
and Lemma 3, we obtain
Then, using Lemma 1 and the inequality
And the proof of the Lemma 4 is now completed. □
Theorem 1 Let a sequence such that as . Then for every , and , we have the inequality
Proof Using (3.3) for any , we obtain the following inequality:
From Lemma 4, we get
By using equality (3.1) we have
Taking the infimum over on the right-hand side of the above inequality and using the inequality (3.2), we get the desired result. □
Theorem 2 Let a sequence such that as . Then , and we have
Proof From Lemma 1, it is obvious that . Since and is positive and bounded from above for each , we obtain
And then .
Similarly for every , we write
we get . Thus, from AD Gadzhiev’s theorem in , we obtain the desired result of Theorem 2. □
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The author would like thank the referee for many helpful comments.
The author declares to have no competing interests.
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Dinlemez, Ü. Convergence of the q-Stancu-Szász-Beta type operators. J Inequal Appl 2014, 354 (2014). https://doi.org/10.1186/1029-242X-2014-354