Abstract
In this paper we give the direct approximation theorem, the inverse theorem, and the equivalence theorem for Szász-Durrmeyer-Bézier operators in the space () with Ditzian-Totik modulus.
MSC:41A25, 41A27, 41A36.
Similar content being viewed by others
1 Introduction
In 1972 Bézier [1] introduced the Bézier basic function and Bézier-type operators are the generalized types of the original operators. The introduction of these operators should have some background. Some properties of the convergence and approximation for some Bézier-type operators have been studied (cf. [2–6]), but there are other aspects that have not yet been considered. For more information as regards the development of the study on this topic or related field, the interested readers can consult the monograph [7] and the paper [8]. In this paper we will consider the direct, inverse and equivalence theorems for the Szász-Durrmeyer-Bézier operator, which is defined by
where , , , . Obviously is bounded and positive in the space . When , is the well-known Durrmeyer operator
To describe our results, we give the definitions of the first order modulus of smoothness and the K-functional (cf. [9]). For (), ,
where .
It is well known that (cf. [9])
here means that there exists such that .
Now we state our equivalence theorem as follows.
Theorem For (), , , we have
Throughout this paper, C denotes a constant independent of n and x, but it is not necessarily the same in different cases.
2 Direct theorem
For convenience, we list some basic properties which will be used later and can be found in [9] and [5] or obtained by simple computation:
-
(1)
(2.1)
-
(2)
(2.2)
-
(3)
(2.3)
-
(4)
(2.4)
-
(5)
(2.5)
-
(6)
(2.6)
where .
Now we give the direct theorem.
Theorem 2.1 For (), , we have
Proof By the definition of and the relation (1.2), for fixed n, we can choose such that
Since
we only need to estimate the second term in the above relation. By the Riesz-Thorin theorem (cf. [[10], Theorem 3.6]), we separate the proof of the assertions for and .
I. . Noting that , we write
Since
and , using the Hölder inequality, we have
By (2.2) and (2.6) we have
Then
Then, by (2.8) and (2.11), we get
II. . By (2.2) and the Fubini theorem, we have
Now we estimate , by using and :
Since
we have
Using the equation and (2.4), we have
So
Since
we can write
Using the result of [[5], Lemma 3]
we get
Consequently
By (2.8) and (2.13) we have
From (2.12) and (2.14), (2.7) is obtained. □
Remark 1 In [11] we show that the second order modulus cannot be used for the Baskakov-Bézier operators. Similarly in (2.7) cannot be used instead of .
3 Inverse theorem
To prove the inverse theorem, we need the following lemmas.
Lemma 3.1 For (), , , we have
Proof We will show (3.1) for the two cases of and . Since
using , we have
For , , by (2.1) and (2.3) we get
here we used (cf. [[9], p.128, Lemma 9.4.3])
For , , by (2.4) we have
By (3.3) and (3.4), we get
Noting , we have
then
Combining (3.2), (3.5), and (3.6) we get for
For , we have
Hence we can write
Now we estimate the last part of (3.9) in four phases:
For , , , noting , we have
Since and , we have
To estimate , we will need the relation [[9], p.129, (9.4.15)]
By the Hölder inequality and (2.3), we get
In order to estimate , we consider the two cases of and (when , ).
For , . Using integration by parts, we can deduce
Noting that and , and from (2.3), we have
For , using the mean value theorem, we know
where and , then
For , we get from the procedure of (3.13)
Combining (3.13) and (3.14), we get for
From (3.8)-(3.12) and (3.15), we obtain
By (3.7) and (3.16), Lemma 3.1 holds. □
Lemma 3.2 For , , , we have
Proof By the Riesz-Thorin theorem, we shall prove Lemma 3.2 for and . For and noting that , we have
Then
By (2.9) and (2.10) we have
hence
For , and by (2.1) and (2.4) we have
By (2.6) we get
For , we write
First, using (2.6), , and the Hölder inequality, we have
Next, for , , and , we have
Then we get
Noting that , by (2.5) we have
For , , using (2.3), and the Hölder inequality, we have
Noting that , one has
The third term of the above is , denotes the difference of the front two terms, and we need only to consider . By (2.1), (2.4), (2.5), and integration by parts, we have
Thus
So we get
For , using (2.1), (2.4), (2.5), and integration by parts, we have
Let
For , noting that and , we have
For , we estimate
Using the Hölder inequality and (), we have
and
Therefore we have
By (3.24)-(3.26) we have
By (3.23) and (3.27), Lemma 3.2 holds. □
Using Lemmas 3.1 and 3.2, we can prove the inverse theorem.
Theorem 3.3 For (), , ,
implies .
Proof Using Lemmas 3.1 and 3.2, for a suitable function g, we have
which by the Berens-Lorentz lemma (cf. [[9], Lemma 9.3.4]) implies that
From (1.2) and (3.28), we see that the proof of Theorem 3.3 is completed. □
References
Bézier P: Numerical Control: Mathematics and Applications. Wiley, London; 1972.
Chang G: Generalized Bernstein-Bézier polynomials. J. Comput. Math. 1983, 4: 322–327.
Liu Z: Approximation of continuous functions by the generalized Bernstein-Bézier polynomials. Approx. Theory Appl. 1986,4(2):105–130.
Zeng X, Piriou A: On the rate of convergence of two Bernstein-Bézier type operators for bounded variation functions. J. Approx. Theory 1998, 95: 369–387. 10.1006/jath.1997.3227
Zeng X: On the rate of the convergence of the generalized Szász type operators for functions of bounded variation. J. Math. Anal. Appl. 1998, 226: 309–325. 10.1006/jmaa.1998.6063
Zeng X, Gupta V: Rate of convergence of Baskakov-Bézier type operators for locally bounded functions. Comput. Math. Appl. 2002, 44: 1445–1453. 10.1016/S0898-1221(02)00269-9
Gupta V, Agarwal RP: Convergence Estimates in Approximation Theory. Springer, Cham; 2014. xiv+361 pp.
Verma DK, Gupta V, Agrawal PN: Some approximation properties of Baskakov-Durrmeyer-Stancu operators. Appl. Math. Comput. 2012,218(11):6549–6556. 10.1016/j.amc.2011.12.031
Ditzian Z, Totik V: Moduli of Smoothness. Springer, New York; 1987.
Bennett C, Sharpley R Pure and Applied Mathematics 129. In Interpolation of Operator. Academic Press, New York; 1988.
Guo S, Qi Q, Liu G: The central approximation theorems for Baskakov-Bézier operators. J. Approx. Theory 2007, 147: 112–124. 10.1016/j.jat.2005.02.010
Acknowledgements
Xiuzhong Yang was supported by the National Natural Science Foundation of China (grant No. 11371119) and both authors were supported by the Natural Science Foundation of Education Department of Hebei Province (grant No. Z2014031). The authors express their thanks to the referees for their constructive suggestions in improving the quality of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors conceived of the study, participated its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Liu, G., Yang, X. On the approximation for generalized Szász-Durrmeyer type operators in the space . J Inequal Appl 2014, 447 (2014). https://doi.org/10.1186/1029-242X-2014-447
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-447