Abstract
In this paper, we give a direct approximation theorem, inverse theorem, and equivalent theorem for a generalization of Bernstein operators in the space \(L_{p}[0,1]\) (\(1\leq p \leq\infty\)).
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1 Introduction
The Sikkema-Bézier-type generalization of Bernstein-Kantorovich operators is given by
where \(J_{n,k}(x)=\sum^{n}_{j=k}p_{n,j}(x)\) are Bézier basic functions, \(p_{n,k}(x)={n\choose k}x^{k}(1-x)^{n-k}\), \(\alpha\geq1\), and \(s_{n}\) is a bounded sequence of natural numbers. If \(\alpha=1\) and \(s_{n}=0\), then \(S_{n,\alpha}(f,x)\) are just the well-known Bernstein-Kantorovich operators [1]
Bézier-type operators were introduced by Chang [2], later many results were given in [3–8], and more recent approximation results can be found in [9]. Most of these results are on the rate of convergence of some Bézier-type operators for functions of bounded variation, whereas in the present paper, we give direct, inverse, and equivalent approximation theorems for a generalization of Bézier-type operators in \(L_{p}\) spaces. We showed [5] that for Bézier-type operators, the second-order modulus cannot be used, so here we shall use the first-order modulus too. For Sikkema-type operators, we can also see many investigations (see [10]). Next, we state the central approximation theorem for \(S_{n,\alpha}(f,x)\) in the spaces \(L_{p}[0,1]\) (\(1\leq p \leq\infty\)), which will be proved in Sections 2 and 3.
Theorem 1.1
For \(f\in L_{p}[0, 1]\) (\(1\le p\leq\infty\)), \(\varphi(x)=\sqrt{x(1-x)}\), and \(0< \beta<1\), we have
In this theorem, we use the first-order modulus defined by
which is equivalent to the K-functionals
and
where \(W_{p}=\{f| f\in \mathit{A.C.}_{\mathrm{loc}}, \|f'\|_{p}<\infty\}\). It is well known that [1]
where \(a\sim b\) means that there exists \(C>0\) such that \(C^{-1}a\le b\le Ca\).
Throughout this paper, C denotes a constant independent of n and x, but it is not necessarily the same in different cases.
Remark
In [8], we obtained a pointwise approximation for \(S_{n,\alpha}(f,x)\). In Theorem 1 of [8], \(\lambda=1\), which is the case where \(p=\infty\) in (1.1). So in the present paper, by the Riesz-Thorin theorem, we shall only need to prove the case where \(p=1\).
2 Direct theorem
To prove the direct theorem, we need the following convergence property of Bernstein-Kantorovich-Bézier operators defined by
Lemma 2.1
For \(f\in L_{p}[0, 1]\) (\(1\le p\leq\infty\)), we have
Proof
For \(p=1\), we will have to split estimate (2.1) into estimates on two domains, that is, \(x\in E_{n}^{c}=[0, \frac{1}{n}]\cup[1-\frac{1}{n}, 1]\) and \(x\in E_{n}=( \frac{1}{n}, 1-\frac{1}{n})\).
First, we choose \(g=g_{n}\) such that
For \(x\in E_{n}^{c}\), we have
Noting that
we get
Since
we obtain
For \(x\in E_{n}\), let \(\vert \int^{\frac{k^{*}}{n+1}}_{x}\varphi(u) |g'(u)|\,du\vert =\max_{j=k,k+1}\vert \int^{\frac{j}{n+1}}_{x}\varphi(u) |g'(u)|\,du\vert \), where \(k^{*}\) is either k or \(k+1\). Then we have
To estimate \(R_{1}\) and \(R_{2}\), we follow [3], pp. 146-147, with a similar method. We now define
Rewrite \(R_{1}\) as follows:
Similarly to [3], (9.6.11), for \(x\in E_{n} \), we have
We now define
and by the procedure of [3], p.147, obtain
On the other hand, for \(R_{2}\), we have
Similarly, we get
Using (2.4) and (2.8), we complete the proof of Lemma 2.1. □
Theorem 2.2
For \(f\in L_{1}[0,1] \), we have
Proof
By Lemma 2.1 we have
and, by (3.1.5) in [1],
The proof is complete. □
3 Inverse theorem
Lemma 3.1
For \(f\in L_{1}[0, 1]\), \(\varphi(x)=\sqrt {x(1-x)}\), and \(\delta_{n}(x)=\varphi(x)+\frac{1}{\sqrt{n}}\), we have
and, furthermore, for \(f\in W_{1}\),
Proof
First, we prove (3.1), that is,
Write \(a_{k}(f)=(n+s_{n}+1)\int_{I_{k}} f(t)\,dt\), where \(I_{k}=[\frac {k}{n+s_{n}+1},\frac{k+1}{n+s_{n}+1}]\). Noting that \(J_{n, n+1}(x)=0\), we have
Then
Next, we estimate the four parts in (3.5):
For \(x\in E_{n}^{c}\), \(\delta_{n}(x)\le \frac{2}{\sqrt{n}}\), and since \(\int_{0}^{1}p_{n-1, k}(x) \,dx=\frac{1}{n}\), we get
Since \(J^{\alpha-1}_{n,k}(x)-J^{\alpha-1}_{n,k+1}(x)\le 1\) and \(J'_{n,k+1}(x)=np_{n-1, k}(x)\), it is easy to see that
To estimate \(\int_{E_{n}}\delta_{n}(x)J_{2}\,dx\), we recall that by [3], p.129, (9.4.15),
with \(p'_{n,k}(x)=\frac{n}{\varphi^{2}(x)} (\frac{k}{n}-x )p_{n,k}(x)\), \(x\in (0, 1)\). By the Hölder inequality we have
To estimate \(\int_{E_{n}}\delta_{n}(x)J_{1}\,dx\), we will consider two cases, \(\alpha\ge2\) and \(1<\alpha<2\) (\(J_{1}=0\) when \(\alpha=1\)).
For \(\alpha\ge2\), we have \(J^{\alpha-1}_{n,k}(x)- J^{\alpha-1}_{n,k+1}(x)\le(\alpha-1)p_{n,k}(x)\), and we need a result of [4], p.375,
Since \(J'_{n,k}(x)=np_{n-1, k-1}(x) \ge0\), we have
For \(1<\alpha<2\), applying \(u(a)-u(b)=u'(\xi)(a-b)\) (\(a<\xi<b\)), we get that there exists \(\xi_{k}(x)\), \(J_{n,k+1}(x)<\xi_{k}(x)<J_{n,k}(x)\) such that
Hence, we have
Since \(p_{n,k}(x)=\frac{n(1-x)}{n-k}p_{n-1,k}(x)\) for \(k< n\), from (3.9) we can deduce that
and
In order to estimate \(\frac{\sqrt{n}}{n-k}L_{1}\), choose \(l\in \mathbb{N}\) such that \(l(\alpha-1)>1\). Then, for \(k< n\), we have
Therefore, for \(k\le n-1\), we get
By (3.11)-(3.14), for \(1<\alpha<2\), we obtain
Estimates (3.5)-(3.8) and (3.15) imply (3.1).
Now we prove (3.2). For \(f\in W_{p}\), noting that \(J'_{n,0}(x)=0\), we have
Hence,
First, we estimate \(\int_{0}^{1}\delta_{n}(x)Q_{3}\,dx\). For \(1\le k\le n-2\) and \(0< u<\frac{2}{n+s_{n}+1}\), we have \(\frac{k}{n}(1-\frac{k}{n})\le C(\frac{k}{n+s_{n}+1}+u)(1-\frac{k}{n+s_{n}+1}-u)\) and (similarly to [1], p.155)
Therefore,
Noting that \(\varphi^{-1}(\frac{k}{n})<\sqrt{n}\) for \(0< k< n-1\), we get
Hence, we have
Since \(\sqrt{n}\delta_{n}(u)\ge1\), for \(Q_{1}\), we write
Therefore, we have
Similarly, we have
This is (3.2). The proof of Lemma 3.1 is complete. □
Theorem 3.2
Let \(f\in L_{p}[0, 1]\) (\(1\le p\le\infty\)), \(\varphi(x)=\sqrt{x(1-x)}\), and \(0<\beta<1 \). Then
implies \(\omega_{\varphi}(f, t)_{p}=O (t^{\beta} )\).
Proof
By Lemma 3.1, for appropriate g, we have
which by the Berens-Lorentz lemma implies that
From relation (1.2) and (3.20) we see that the proof of Theorem 3.2 is complete. □
References
Ditzian, Z, Totik, V: Moduli of Smoothness. Springer, New York (1987)
Chang, G: Generalized Bernstein-Bézier polynomial. J. Comput. Math. 1(4), 322-327 (1983)
Liu, Z: Approximation of continuous by the generalized Bernstein-Bézier polynomials. Approx. Theory Appl. 4(2), 105-130 (1986)
Zeng, X, Piriou, A: On the rate of convergence of two Bernstein-Bézier type operators for bounded variation functions. J. Approx. Theory 95, 369-387 (1998)
Guo, S, Qi, Q, Liu, G: The central approximation theorem for Baskakov-Bézier operators. J. Approx. Theory 147, 112-124 (2007)
Gupta, V: Simultaneous approximation for Bézier variant of Szász-Mirakyan-Durrmeyer operators. J. Math. Anal. Appl. 328, 101-105 (2007)
Liu, G, Yang, X: On the approximation for generalized Szász-Durrmeyer type operators in the space \(L_{p}[0,+\infty)\). J. Inequal. Appl. 2014, 447 (2014)
Liu, G: Pointwise approximation for Bernstein-Sikkema-Bézier operator. Math. Pract. Theory 43(1), 199-204 (2013) (in Chinese)
Milovanović, GV, Rassias, MT: Analytic Number Theory, Approximation Theory and Special Functions. Springer, New York (2014)
Cao, J: A generalization of the Bernstein polynomials. J. Math. Anal. Appl. 209, 140-146 (1997)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (grant 11371119), the Key Foundation of Education Department of Hebei Province (grant ZD2016023), and by Natural Science Foundation of Education Department of Hebei Province (grant Z2014031).
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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.
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Liu, G., Yang, X. Approximation for a generalization of Bernstein operators. J Inequal Appl 2016, 204 (2016). https://doi.org/10.1186/s13660-016-1147-4
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DOI: https://doi.org/10.1186/s13660-016-1147-4