1 Introduction
In view of the Bézier basis function, which was introduced by Bézier [
1], in 1983, Chang [
2] defined the generalized Bernstein-Bézier polynomials for any
\(\alpha>0\), and a function
f defined on
\([0,1]\) as follows:
$$ B_{n,\alpha}(f;x)=\sum_{k=0}^{n}f \biggl(\frac{k}{n}\biggr)\bigl[J^{\alpha}_{n,k}(x)-J^{\alpha}_{n,k+1}(x) \bigr], $$
(1)
where
\(J_{n,n+1}(x)=0\), and
\(J_{n,k}(x)=\sum_{i=k}^{n}P_{n,i}(x)\),
\(k=0,1,\ldots,n\),
\(P_{n,i}(x)=\bigl ({\scriptsize\begin{matrix}{} n\cr i\end{matrix}} \bigr )x^{i}(1-x)^{n-i}\).
\(J_{n,k}(x)\) is the Bézier basis function of degree
n.
Obviously, when \(\alpha=1\), \(B_{n,\alpha}(f;x)\) become the well-known Bernstein polynomials \(B_{n}(f;x)\), and for any \(x\in[0,1]\), we have \(1=J_{n,0}(x)>J_{n,1}(x)>\cdots>J_{n,n}(x)=x^{n}\), \(J_{n,k}(x)-J_{n,k+1}(x)=P_{n,k}(x)\).
During the last ten years, the Bézier basis function was extensively used for constructing various generalizations of many classical approximation processes. Some Bézier type operators, which are based on the Bézier basis function, have been introduced and studied (
e.g., see [
3‐
9]).
In 2012, Ren [
10] introduced Bernstein type operators as follows:
$$ L_{n}(f;x)=f(0)P_{n,0}(x)+\sum _{k=1}^{n-1}P_{n,k}(x)B_{n,k}(f)+f(1)P_{n,n}(x), $$
(2)
where
\(f\in C[0,1]\),
\(x\in[0,1]\),
\(P_{n,k}(x)=\bigl ( {\scriptsize\begin{matrix}{}n\cr k\end{matrix}} \bigr )x^{k}(1-x)^{n-k}\),
\(k=0,1,\ldots,n\), and
\(B_{n,k}(f)= \frac{1}{B(nk,n(n-k))}\int_{0}^{1}t^{nk-1}(1-t)^{n(n-k)-1}f(t)\, dt\),
\(k=1,\ldots,n-1\),
\(B(\cdot,\cdot)\) is the beta function.
The moments of the operators
\(L_{n}(f;x)\) were obtained as follows (see [
10]).
In the present paper, we will study the Bézier variant of the Bernstein type operators
\(L_{n}(f;x)\), which have been given by (
2). We introduce a new type of Bézier operators as follows:
$$ L_{n,\alpha}(f;x)=f(0)Q^{(\alpha)}_{n,0}(x)+\sum _{k=1}^{n-1}Q^{(\alpha)}_{n,k}(x)B_{n,k}(f)+f(1)Q^{(\alpha)}_{n,n}(x), $$
(3)
where
\(f\in C[0,1]\),
\(x\in[0,1]\),
\(\alpha>0\),
\(Q^{(\alpha)}_{n,k}(x)=J^{\alpha}_{n,k}(x)-J^{\alpha}_{n,k+1}(x)\),
\(J_{n,n+1}(x)=0\),
\(J_{n,k}(x)=\sum_{i=k}^{n}P_{n,i}(x)\),
\(k=0,1,\ldots,n\),
\(P_{n,i}(x)=\bigl ( {\scriptsize\begin{matrix}{}n\cr i\end{matrix}} \bigr )x^{i}(1-x)^{n-i}\), and
\(B_{n,k}(f)=\frac{1}{B(nk,n(n-k))}\int_{0}^{1}t^{nk-1}(1-t)^{n(n-k)-1}f(t)\, dt\),
\(k=1,\ldots,n-1\),
\(B(\cdot,\cdot)\) is the beta function.
It is clear that \(L_{n,\alpha}(f;x)\) are linear and positive on \(C[0,1]\). When \(\alpha=1\), \(L_{n,\alpha}(f;x)\) become the operators \(L_{n}(f;x)\).
The goal of this paper is to study the approximation properties of these operators with the help of the Korovkin type approximation theorem. We also estimate the rates of convergence of these operators by using a modulus of continuity. Then we obtain the direct theorem concerned with an approximation for these operators by means of the Ditzian-Totik modulus of smoothness.
In the paper, for \(f\in C[0, 1]\), we denote \(\|f\|=\max\{{|f (x)| : x\in[0, 1]}\}\). \(\omega(f,\delta)\) (\(\delta>0\)) denotes the usual modulus of continuity of \(f\in C[0,1]\).
2 Some lemmas
Now, we give some lemmas, which are necessary to prove our results.
3 Main results
First of all we give the following convergence theorem for the sequence \(\{L_{n,\alpha}(f;x)\}\).
Next we estimate the rates of convergence of the sequence \(\{ L_{n,\alpha}\}\) by means of the modulus of continuity.
Finally we study the direct theorem concerned with an approximation for the sequence \(\{L_{n,\alpha}\}\) by means of the Ditzian-Totik modulus of smoothness. For the next theorem we shall use some notations.
For
\(f\in C[0,1]\),
\(\varphi(x)=\sqrt{x(1-x)}\),
\(0\leq\lambda\leq 1\),
\(x\in [0,1]\), let
$$\omega_{\varphi^{\lambda}}(f,t)=\sup_{0< h\leq t}\sup_{x\pm \frac{h\varphi^{\lambda}(x)}{2}\in[0,1]} \biggl\vert f\biggl(x+\frac{h\varphi^{\lambda }(x)}{2}\biggr)-f\biggl(x-\frac{h\varphi^{\lambda}(x)}{2} \biggr)\biggr\vert $$
be the Ditzian-Totik modulus of first order, and let
$$ K_{\varphi^{\lambda}}(f,t)=\inf_{g\in W_{\lambda}}\bigl\{ \| f-g \|+t\bigl\Vert \varphi^{\lambda}g'\bigr\Vert \bigr\} $$
(6)
be the corresponding
K-functional, where
\(W_{\lambda}=\{f|f\in \mathit{AC}_{\mathrm{loc}}[0,1], \|\varphi^{\lambda}f'\|<\infty, \|f'\|<\infty\}\).
It is well known that (see [
14])
$$ K_{\varphi^{\lambda}}(f,t)\leq C\omega_{\varphi^{\lambda}}(f,t), $$
(7)
for some absolute constant
\(C>0\).
Now we state our next main result.
4 Conclusions
In this paper, a new kind of type Bézier operators is introduced. The Korovkin type approximation theorem of these operators is investigated. The rates of convergence of these operators are studied by means of the modulus of continuity. Then, by using the Ditzian-Totik modulus of smoothness, a direct theorem concerned with an approximation for these operators is obtained. Further, we can also study the inverse theorem and an equivalent theorem concerned with an approximation for these operators.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 61572020), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant No. JA12324), and the Natural Science Foundation of Fujian Province of China (Grant Nos. 2014J01021 and 2013J01017).
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.