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Approximation of a kind of new type Bézier operators
Journal of Inequalities and Applications volume 2015, Article number: 412 (2015)
Abstract
In this paper, a kind of new 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 modulus of continuity. Then, by using the Ditzian-Totik modulus of smoothness, a direct theorem concerned with an approximation for these operators is also obtained.
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:
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:
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]).
Remark 1
For \(L_{n}(t^{j};x)\), \(j=0,1,2\), we have
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:
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.
Lemma 1
(see [2])
Let \(\alpha>0\). We have
Lemma 2
Let \(\alpha>0\). We have
Proof
By simple calculation, we obtain \(B_{n,k}(1)=1\), \(B_{n,k}(t)=\frac{k}{n}\), \(B_{n,k}(t^{2})=\frac{1}{n^{2}+1}(k^{2}+\frac{k}{n})\).
(i) Since \(\sum_{k=0}^{n}Q^{(\alpha)}_{n,k}(x)=1\), by (3) we can get \(L_{n,\alpha}(1;x)=1\).
(ii) By (3), we have
thus, by Lemma 1(i), we have \(\lim_{n\rightarrow\infty}L_{n,\alpha}(t;x)=x\) uniformly on \([0,1]\).
(iii) By (3), we have
thus, by Lemma 1, we have \(\lim_{n\rightarrow\infty}L_{n,\alpha}(t^{2};x)=x^{2}\) uniformly on \([0,1]\). □
Lemma 3
(see [11])
For \(x\in[0,1]\), \(k=0,1,\ldots,n\), we have
Lemma 4
(see [12])
For \(0<\alpha<1\), \(\beta>0\), we have
where the constant \(A_{s}\) only depends on s.
Lemma 5
For \(\alpha\geq1\), we have
Proof
(i) By (3), Lemma 3 and Remark 1, we obtain
since \(\max_{0\leq x\leq1 }x(1-x)=\frac{1}{4}\), and for any \(n\in N\), one can get \(\frac{n(n+1)}{n^{2}+1}\leq2\), so we have
(ii) In view of \(L_{n,\alpha}(1;x)=1\), by the Cauchy-Schwarz inequality, we have
thus, we get
□
Lemma 6
For \(0<\alpha<1\), we have
Here the constant \(M_{\alpha}\) only depends on α.
Proof
(i) By (3) and Lemma 3, we obtain
By Lemma 4, we have \(I_{1}\leq\frac{n(n+1)}{n^{2}+1}(n+1)^{-\alpha}(A_{\frac{2}{\alpha }})^{\alpha}\leq2(A_{\frac{2}{\alpha}})^{\alpha}n^{-\alpha}\), where the constant \(A_{\frac{2}{\alpha}}\) only depends on α.
Using the Hölder inequality, we have \(\sum_{k=0}^{n}P^{\alpha}_{n,k}(x)\leq(n+1)^{1-\alpha}[\sum_{k=0}^{n}P_{n,k}(x)]^{\alpha}\), and \(|\frac{k}{n}-2x\frac{k}{n}+x^{2}|\leq4\), so we have
Denote \(M_{\alpha}=2(A_{\frac{2}{\alpha}})^{\alpha}+4\), then we can get
(ii) Since
thus, we get
□
Lemma 7
For \(f\in C[0,1]\), \(x\in[0,1]\) and \(\alpha> 0\), we have
Proof
By (3) and Lemma 2(i), we have
□
3 Main results
First of all we give the following convergence theorem for the sequence \(\{L_{n,\alpha}(f;x)\}\).
Theorem 1
Let \(\alpha>0\). Then the sequence \(\{L_{n,\alpha}(f;x)\}\) converges to f uniformly on \([0,1]\) for any \(f\in C[0,1]\).
Proof
Since \(L_{n,\alpha}(f;x)\) is bounded and positive on \(C[0,1]\), and by Lemma 2, we have \(\lim_{n\rightarrow\infty}\|L_{n,\alpha}(e_{j};\cdot)-e_{j}\|=0\) for \(e_{j}(t)=t^{j}\), \(j=0,1,2\). So, according to the well-known Bohman-Korovkin theorem ([13], p.40, Theorem 1.9), we see that the sequence \(\{L_{n,\alpha}(f;x)\}\) converges to f uniformly on \([0,1]\) for any \(f\in C[0,1]\). □
Next we estimate the rates of convergence of the sequence \(\{ L_{n,\alpha}\}\) by means of the modulus of continuity.
Theorem 2
Let \(f\in C[0,1]\), \(x\in[0,1]\). Then
-
(i)
when \(\alpha\geq1\), we have \(\|L_{n,\alpha}(f;\cdot)-f\|\leq(1+\sqrt{\frac{\alpha}{2}})\omega(f,\frac {1}{\sqrt{n}})\);
-
(ii)
when \(0<\alpha<1\), we have \(\|L_{n,\alpha}(f;\cdot)-f\|\leq(1+\sqrt{M_{\alpha}})\omega(f,n^{-\frac {\alpha}{2}})\).
Here the constant \(M_{\alpha}\) only depends on α.
Proof
(i) When \(\alpha\geq1\), by Lemma 2(i), we have
so, by Lemma 5(ii), we obtain \(|L_{n,\alpha}(f;x)-f(x)|\leq(1+\sqrt{\frac{\alpha}{2}})\omega(f,\frac {1}{\sqrt{n}})\). The desired result follows immediately.
(ii) When \(0<\alpha<1\), by Lemma 2(i), we have
so, by Lemma 6(ii), we obtain \(|L_{n,\alpha}(f;x)-f(x)|\leq(1+\sqrt{M_{\alpha}})\omega(f,n^{-\frac {\alpha}{2}})\). The desired result follows immediately. □
Theorem 3
\(Let f\in C^{1}[0,1]\), \(x\in[0,1]\). Then
-
(i)
when \(\alpha\geq1\), we have
$$ \bigl\vert L_{n,\alpha}(f;x)-f(x)\bigr\vert \leq\bigl\Vert f'\bigr\Vert \sqrt{\frac{\alpha}{2n}} +\omega \biggl(f',\frac{1}{\sqrt{n}}\biggr) \biggl(1+\sqrt{ \frac{\alpha}{2}}\biggr)\sqrt{\frac {\alpha}{2n}}; $$ -
(ii)
when \(0<\alpha<1\), we have
$$ \bigl\vert L_{n,\alpha}(f;x)-f(x)\bigr\vert \leq\bigl\Vert f'\bigr\Vert \sqrt{M_{\alpha}n^{-\alpha}} +\omega \bigl(f',n^{-\frac{\alpha}{2}}\bigr) (1+\sqrt{M_{\alpha}}) \sqrt{M_{\alpha }n^{-\alpha}}, $$where the constant \(M_{\alpha}\) only depends on α.
Proof
Let \(f\in C^{1}[0,1]\). For any \(t, x\in[0,1]\), \(\delta>0\), we have
hence, by the Cauchy-Schwarz inequality, we have
So we get
-
(i)
When \(\alpha\geq1\), taking \(\delta=\frac{1}{\sqrt{n}}\) in (5), by Lemma 5 and inequality (5), we obtain the desired result.
-
(ii)
When \(0<\alpha<1\), taking \(\delta=n^{-\frac{\alpha}{2}}\) in (5), by Lemma 6 and inequality (5), we obtain the desired result.
□
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
be the Ditzian-Totik modulus of first order, and let
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])
for some absolute constant \(C>0\).
Now we state our next main result.
Theorem 4
Let \(f\in C[0,1]\), \(\alpha\geq1\), \(\varphi(x)=\sqrt{x(1-x)}\), \(x\in [0,1]\), \(0\leq\lambda\leq1\). Then there exists an absolute constant \(C>0\) such that
Proof
Let \(g\in W_{\lambda}\), by Lemma 2(i) and Lemma 7, we have
Since \(g(t)=\int_{x}^{t}g'(u)\, du+g(x)\), \(L_{n,\alpha}(1;x)=1\), so, we have
By the Hölder inequality, we get
also, in view of \(1\leq\sqrt{u}+\sqrt{1-u}<2\), \(0\leq u\leq1\), we have
thus, by (10) and (11), we obtain
also, by (9) and (12), we have
In view of (4) and Lemma 2(i), by the Cauchy-Schwarz inequality, we have
so, by (13) and (14), we obtain
thus, by (8) and (15), we have
Then, in view of (16), (6), and (7), we obtain
where C is a positive constant, in different places the value of C may be different. □
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.
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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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Ren, MY., Zeng, XM. Approximation of a kind of new type Bézier operators. J Inequal Appl 2015, 412 (2015). https://doi.org/10.1186/s13660-015-0940-9
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DOI: https://doi.org/10.1186/s13660-015-0940-9