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Pointwise approximation for a type of Bernstein-Durrmeyer operators
Journal of Inequalities and Applications volume 2014, Article number: 106 (2014)
Abstract
We give the direct and inverse approximation theorems for a new type of Bernstein-Durrmeyer operators with the modulus of smoothness.
MSC:41A25, 41A27, 41A36.
1 Introduction
Durrmeyer [1] introduced the integral modification of the well-known Bernstein polynomials given by
where . Derriennic [2] established some direct results in ordinary and simultaneous approximation for Durrmeyer operators. Then Durrmeyer type operators were studied widely [3–5]. Recently Gupta et al. [6] considered a family of Durrmeyer type operators:
where with and for any , , is the falling factorial; and we get the rate of convergence for these operators for a function having derivatives of bounded variation and the result in the simultaneous approximation. In the present note our main aim is to get the direct and inverse approximation theorem for this type of operators. Here we shall utilize modulus of smoothness and K-functional as the tools, which are defined by [7]
where , , . It is well known that , where means that there exists some constant such that . We denote and state our main results as follows.
Theorem 1 For , , , one has
Theorem 2 Let , , , , , , for , and from
we get .
Throughout this paper and C denotes a positive constant independent of n and x not necessarily the same at each occurrence.
2 Lemmas
To prove the above theorems we need the following lemmas. First we define the moments, for any , .
Lemma 3 ([6])
The following claims hold.
-
(1)
For any , , the following recurrence relation is satisfied:
where for , we denote .
-
(2)
For any and ,
-
(3)
For any , , .
Remark For n sufficiently large and , it can be seen from Lemma 3 that
for any .
Lemma 4 For , , , , , , we have
Proof To complete the proof we consider two cases of and .
For , , . Using
where , , ; and , , one has
For , . From [7] we have
where a polynomial in of degree with non-constant bounded coefficients. Therefore,
By Holder’s inequality we get
Consequently , hence
From (2.3) and (2.4), (2.2) holds. □
Lemma 5 For , , , we have
Proof From , we have
Let . For one has
Noting that
and , we get . This completes the proof of Lemma 5. □
Lemma 6 ([8])
For , , , we have
3 Proof of the theorems
In this section we will give the proof of Theorem 1 and Theorem 2.
Proof of Theorem 1 By the definition of and the equivalence between and , for the fixed n and x, we can choose such that
We know that
and we have to estimate the second term on the right side of (3.2). By Taylor’s formula, , and Lemma 3 we have
We consider first. For ,
together with , one has
It is similar for .
Now we address . By the process of (9.6.1) in [7]
we get , and combining with (2.1) we deduce
By (3.2)-(3.5), we complete the proof of Theorem 1. □
Proof of Theorem 2 For convenience let . If , for every , we have
Combining Lemma 4, Lemma 5, and Lemma 6, we have
Utilizing (3.6), (3.7), and (3.8), choosing the appropriate g, we obtain
For every fixed and every , we can choose n such that . Then
So,
which yields the assertion of Theorem 2 by the Berens-Lorentz lemma. □
References
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Zeng XM, Chen W: On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation. J. Approx. Theory 2000, 102: 1-12. 10.1006/jath.1999.3367
Heiner G, Daniela K, Ioan R: The genuine Bernstein-Durrmeyer operators revisited. Results Math. 2012, 62: 295-310. 10.1007/s00025-012-0287-1
Gupta V: Approximation properties by Bernstein-Durrmeyer type operators. Complex Anal. Oper. Theory 2013, 7: 363-374. 10.1007/s11785-011-0167-9
Gupta V, López-Moreno AJ, Latorre-Palacios JM: On simultaneous approximation of the Bernstein Durrymeyer operators. Appl. Math. Comput. 2009, 213: 112-120. 10.1016/j.amc.2009.02.052
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Guo SS, Liu LX, Qi QL: Pointwise estimate for linear combinations of Bernstein-Kantorovich operators. J. Math. Anal. Appl. 2002, 265: 135-147. 10.1006/jmaa.2001.7700
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Liu, G. Pointwise approximation for a type of Bernstein-Durrmeyer operators. J Inequal Appl 2014, 106 (2014). https://doi.org/10.1186/1029-242X-2014-106
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DOI: https://doi.org/10.1186/1029-242X-2014-106