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.
Hinweise
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.
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
(1.1)
where , , , . Obviously is bounded and positive in the space . When , is the well-known Durrmeyer operator
Anzeige
To describe our results, we give the definitions of the first order modulus of smoothness and the K-functional (cf. [9]). For (), ,
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 2.1For (), , we have
(2.7)
Proof By the definition of and the relation (1.2), for fixed n, we can choose such that
Since
(2.8)
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 .
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 3.1For (), , , we have
(3.1)
Proof We will show (3.1) for the two cases of and . Since
Now we estimate the last part of (3.9) in four phases:
For , , , noting , we have
(3.10)
Since and , we have
(3.11)
To estimate , we will need the relation [[9], p.129, (9.4.15)]
By the Hölder inequality and (2.3), we get
(3.12)
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
(3.13)
For , using the mean value theorem, we know
where and , then
For , we get from the procedure of (3.13)
(3.14)
Combining (3.13) and (3.14), we get for
(3.15)
From (3.8)-(3.12) and (3.15), we obtain
(3.16)
By (3.7) and (3.16), Lemma 3.1 holds. □
Lemma 3.2For , , , we have
(3.17)
Proof By the Riesz-Thorin theorem, we shall prove Lemma 3.2 for and . For and noting that , we have
Then
(3.18)
By (2.9) and (2.10) we have
hence
(3.19)
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
(3.20)
Noting that , by (2.5) we have
(3.21)
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
(3.22)
So we get
(3.23)
For , using (2.1), (2.4), (2.5), and integration by parts, we have
Let
(3.24)
For , noting that and , we have
(3.25)
For , we estimate
Using the Hölder inequality and (), we have
and
Therefore we have
(3.26)
By (3.24)-(3.26) we have
(3.27)
By (3.23) and (3.27), Lemma 3.2 holds. □
Using Lemmas 3.1 and 3.2, we can prove the inverse theorem.
Theorem 3.3For (), , ,
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
(3.28)
From (1.2) and (3.28), we see that the proof of Theorem 3.3 is completed. □
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.
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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.