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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 732069, 8 pages
Degree of Approximation by Hybrid Operators
1Department of Applied Mathematics, Delhi Technological University (Formerly Delhi College of Engineering), Bawana Road, Delhi 110042, India
2Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea
3Department of Mathematics, Meijo University, Nagoya 468-8502, Japan
Received 8 May 2013; Accepted 12 July 2013
Academic Editor: Irena Rachůnková
Copyright © 2013 Naokant Deo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider hybrid (Szász-beta) operators, which are a general sequence of integral type operators including beta function, and we give the degree of approximation by these Szász-beta-Durrmeyer operators.
The Lupaş-Durrmeyer operators were introduced by Sahai and Prasad  who studied the asymptotic formula for simultaneous approximation, and many mathematicians have given different results for the Durrmeyer operators (see [2–6]). Now we consider here a sequence of linear positive operators, which was introduced by Gupta et al.  as follows. Let and be positive integers. For satisfying , where is a positive integer, Let . For a function on , we define the norm by Recently Jung and Sakai  investigated the Lupaş-Durrmeyer operators and studied the circumstances of convergence. Motivated with the idea of Jung and Sakai , we give the degree of approximation by Szász-Beta-Durrmeyer operators in this paper.
2. Basic Results
Lemma 1 (cf. ). Let , , , and be integers with , , and : Then one has(i) and ,(ii)for (iii)where is a polynomial of degree such that the coefficients of are bounded independently of .
Proof. Let . Then (i)
we see that
(ii) Using , we obtain
Since we know that
Then substituting (12) into (10), we consider the following:
Then since we see
Here the last equation follows from integration by parts. Furthermore, we easily see
Therefore, we conclude
Consequently, (ii) is proved.
(iii) For , (6) holds. Let us assume (6) for . We note So, we have, by the assumption of induction, Here, if is even, then and if is odd, then Hence we have and here we see that is a polynomial of degree such that the coefficients of are bounded independently of .
Lemma 2 (cf. ). Let , , and be integers with . Let satisfy for a positive integer Then one has, for , where
Proof. Using we have
3. Main Results
Theorem 3. Let , and let and be nonnegative integers. Let and be integers with . Let satisfy Then one has uniformly, for and ,
Proof. Let and . By the second inequality of (28), Let . Then using Lemma 2 and we have From (30) and Lemma 1, we have Next, we estimate . By the use of the first inequality in (28), we have Now using and the notation we have Then, with , Here for , we get because Finally we get From (32), If we put , then we get
In the following, we let ,.
Theorem 4. Let and be nonnegative integers. Let and be integers with . Let satisfy Then one has uniformly, for and ,
Let us define the weighted modulus of smoothness by where
Theorem 5. Let and be nonnegative integers. Let and be integers with . Then one has, for ,
To prove Theorem 5, we need the following theorem.
Theorem 6. Let and be nonnegative integers. Let and be integers with . Let satisfy Then one has uniformly, for , , and ,
The Steklov function for is defined as follows: Then for the Steklov function with respect to , we have the following properties.
Lemma 7 (see [8, Lemma 2.4]). Let , and let be a positive and nonincreasing function on . Then(i),(ii)(iii)(iv)
Now, we prove Theorem 5.
Proof of Theorem 5. We know that, for , Then first, we split it as follows: Then for the first term, we have, using Theorem 6, (62), and (65), Here, we suppose , and then we know that For the second term, from Theorem 4, (65), (63), and (64) of Lemma 7, Therefore, we have If we let , then because .
The authors thank the referees for many kind suggestions and comments.
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