Convergence of the q-Stancu-Szász-Beta type operators
© Dinlemez; licensee Springer 2014
Received: 2 April 2014
Accepted: 27 August 2014
Published: 24 September 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 .
Keywordsq-Stancu type operators Szász-Beta type operators weighted approximation
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.
and so we have the proof of (iii). □
To obtain our main results we need to compute the second moment.
And the proof of Lemma 2 is now finished. □
3 Direct estimates
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.
And the proof of the Lemma 4 is now completed. □
Taking the infimum over on the right-hand side of the above inequality and using the inequality (3.2), we get the desired result. □
And then .
we get . Thus, from AD Gadzhiev’s theorem in , we obtain the desired result of Theorem 2. □
The author would like thank the referee for many helpful comments.
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