Abstract
In this paper, we discuss the approximation properties of the complex weighted Kantorovich type operators. Quantitative estimates of the convergence, the Voronovskaja type theorem, and saturation of convergence for complex weighted Kantorovich polynomials attached to analytic functions in compact disks will be given. In particular, we show that for functions analytic in \(\{ z\in C:\vert z\vert < R \} \), the rate of approximation by the weighted complex Kantorovich type operators is \(1/n\).
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1 Introduction
The first constructive (and simple) proof of Weierstrass approximation theorem was given by Bernstein [1]. He gave an alternative proof to the Weierstrass approximation theorem and introduced the following polynomial:
where \(p_{n,k}(x)=\binom{n}{k}x^{k} ( 1-x ) ^{n-k}\), \(x\in [ 0,1 ] \).
For any \(f\in L^{p}([0,1])\), \(1\leq p\leq\infty\), Ditzian and Totik (see [2]) introduced the Kantorovich-Bernstein operator as follows:
where \(p_{n,k}(x)=\binom{n}{k}x^{k} ( 1-x ) ^{n-k}\), \(k=0,1,2,\ldots, n\); \(x\in [ 0,1 ]\).
Let
be the classical Jacobi weights.
Let
and equip \(L_{w}^{p}\) with norm
In [2], Ditzian and Totik studied the case of weighted approximation properties of \(K_{n}(f;x)\) in \(L_{w}^{p}\) under the restrictions \(-\frac{1}{p}<\alpha,\beta<1-\frac{1}{p}\) on the weighted parameters. In [3], Della Vecchia et al. removed the restrictions on α, β and introduced a weighted generalization of the \(K_{n}(f;x)\) as follows:
where
and \(f\in L_{w}^{p}\). \(K_{n}^{\ast}(f;x)\) allows a wider class of functions than the operator \(K_{n}(f;x)\). Because, Della Vecchia et al. dropped the restrictions \(\alpha,\beta<1-\frac{1}{p}\) and obtained (see [3]) direct and converse theorems and a Voronovskaya-type relation for the weighted Kantorovich operator (3). Also Della Vecchia et al. solved the saturation problem of the weighted Kantorovich operator (3) (see [4, Theorem 2.1]).
In [5], Yu introduced the following modified operator of \(K_{n} ( f;x ) \):
and direct and converse theorems and a Voronovskaya-type relation were obtained with modification of the classical Kantorovich-Bernstein operators (1) with Jacobi weights \(w(x)=x^{\alpha}(1-x)^{\beta}\), where \(-\frac {1}{p}<\alpha,\beta\).
The goal of the present note is to introduce the complex weighted Kantorovich type operator
where \(\frac{-1}{p}<\alpha,\beta\) and \(1\leq p\leq\infty\), \(B(\cdot ,\cdot)\) is a beta function, and we study the convergence properties of \(K_{n} ( f;z ) \). Notice that the approximation properties of complex generalized Kantorovich type operators are studied in [6].
In the recent books of Gal [7, 8] (see references therein), a systematic study of the overconvergence phenomenon in a complex approximation was made for the following important classes of Bernstein-type operators: Bernstein, Bernstein-Faber, Bernstein-Butzer, Bernstein-Stancu, Bernstein-Kantorovich, Favard-Szasz-Mirakjan, Baskakov, Bernstein-Durrmeyer, and Balazs-Szabados.
Let \(\mathbb{D}_{R}\) be a disc \(\mathbb{D}_{R}:= \{ z\in\mathbb{C}:\vert z\vert < R \} \) in the complex plane ℂ. Denote by \(H ( \mathbb{D}_{R} ) \) the space of all analytic functions on \(\mathbb{D}_{R}\). For \(f\in H ( \mathbb{D}_{R} ) \) we assume that \(f ( z ) =\sum_{m=0}^{\infty}a_{m}z^{m}\).
We start with the following quantitative estimates of the convergence for complex weighted Kantorovich type operators attached to an analytic function in a disk of radius \(R>1\) and center 0.
Theorem 1
Let \(f\in H ( \mathbb{D}_{R} ) \). If \(1\leq r< R\), then for all \(\vert z\vert \leq r\) we have
where \(n\in\mathbb{N}\).
The next theorem gives a Voronovskaja type result in compact disks, for complex weighted Kantorovich type operators attached to an analytic function in \(\mathbb{D}_{R}\), where \(R>1\), and with center 0.
Theorem 2
Let \(f\in H ( \mathbb{D}_{R} ) \). If \(1\leq r< R\) then, for all \(\vert z\vert \leq r\), we have
where \(n\in\mathbb{N}\).
As an application of Theorem 2 we present the order of approximation for complex weighted Kantorovich type operators.
Theorem 3
Let \(f\in H ( \mathbb{D}_{R} ) \). If \(1\leq r< R\) and if f is not a constant function, then the estimate
holds, where the constant \(C_{r} ( f ) \) depends on f, α, β, and r but it is independent of n.
2 Auxiliary results
Lemma 4
For all \(n\in\mathbb{N}\), \(m\in\mathbb{N}\cup \{ 0 \} \), \(z\in\mathbb{C}\) we have
where \(e_{m} ( z ) =z^{m}\).
Proof
The recurrence formula can be derived by direct computation:
□
Lemma 5
We have
Lemma 6
For all \(z\in\mathbb{D}_{r}\), \(r\geq1\) we have
where \(e_{m} ( z ) =z^{m}\).
Proof
Indeed, using the inequality \(\vert B_{n} ( e_{j};z ) \vert \leq r^{j}\) (see [7]) we get
□
Lemma 7
For all \(z\in\mathbb{C}\), \(z\in\mathbb{N\cup\{}0\mathbb{\}}\) we have
Proof
We know that (see [7])
Taking the derivative of (10) and using the above formula we have
It follows that
Here we used the identity
□
Define
Lemma 8
Let \(n,m\in\mathbb{N}\), we have the following recurrence formula:
Proof
It is immediate that \(E_{n,m} ( z ) \) is a polynomial of degree less than or equal to m and that \(E_{n,0} ( z ) =E_{n,1} ( z ) =0\).
Using (11) we get
which is the desired recurrence formula. □
3 Proofs of the main results
Proof of Theorem 1
By use of the above recurrence we obtain the following relationship:
For \(\vert z\vert \leq r\) we can easily estimate the sum in the above formula as follows:
It is well known that, by a linear transformation, the Bernstein inequality in the closed unit disk becomes
for all \(\vert z\vert \leq r\), where \(P_{m} ( z ) \) is a complex polynomial of degree ≤m. From the above recurrence formula (12) we get
By writing the last inequality for \(m=1,2,\ldots\) , we can easily obtain, step by step, the following:
Since \(K_{n,q} ( f;z ) \) is analytic in \(\mathbb{D}_{R}\) (see [7, p.6]), we can write
which together with (13) immediately implies for all \(\vert z\vert \leq r\)
□
Proof of Theorem 2
A simple calculation and the use of the recurrence formula (10) lead us to the following relationship:
Firstly we estimate \(I_{3}\), \(I_{8}\). It is clear that
Secondly using the known inequality
to estimate \(I_{5}\), \(I_{6}\), \(I_{9}\), we have
Finally we estimate \(I_{4}\), \(I_{7}\). We use [7, Theorem 1.5.1]:
Using (13), (15), (16), and (17) in (14) finally we have (\(m\geq3\))
As a consequence, we get
The result follows from the fact that
Note that since \(f^{ ( 3 ) }=\sum_{m=4}^{\infty}a_{m}m ( m-1 ) ( m-2 ) z^{m-3}\) and the series is absolutely convergent for all \(\vert z\vert < R\), the finiteness of the involved constants in the statement easily follows. □
Proof of Theorem 3
For all \(z\in\mathbb{D}_{R}\) and \(n\in\mathbb{N}\) we get
We apply
to get
Because by hypothesis f is not a constant in \(\mathbb{D}_{R}\), it follows
Indeed, assuming the contrary it follows that \(\frac{(\alpha+1)-z ( \alpha+\beta+2 ) }{ ( \alpha+\beta+2 ) }mz^{m-1}+\frac{z(1-z)}{2}m(m-1)z^{m-2}=0\), for all \(z\in\overline{\mathbb {D}}_{R}\), that is,
for all \(z\in\overline{\mathbb{D}}_{R}\backslash \{ 0 \} \). Thus \(a_{m}=0\), \(m=1,2,3,\ldots \) . Thus, f is constant, which is in contradiction with the hypothesis.
Now, by Theorem 2 we have
Consequently, there exists \(n_{1}\) (depending only on f and r) such that, for all \(n\geq n_{1}\), we have
which implies
For \(1\leq n\leq n_{1}-1\) we have
which finally implies that
for all n, with \(C_{r} ( f ) =\min \{ \frac{1}{2}M_{r,1} ( f ) ,\ldots,\frac{1}{2}M_{r,n_{1}-1} ( f ) ,\frac{1}{2}\Vert \frac{(\alpha+1)-z ( \alpha+\beta+2 ) }{ ( \alpha+\beta+2 ) }mz^{m-1}+\frac{z(1-z)}{2}m(m-1)z^{m-2}\Vert _{r} \} \). □
References
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Ditzian, Z, Totik, V: Moduli of Smoothness. Springer, Berlin (1987)
Della Vecchia, B, Mastroianni, G, Szabados, J: A weighted generalization of the classical Kantorovich operator. Rend. Circ. Mat. Palermo Suppl. 82, 1-27 (2010)
Della Vecchia, B, Mastroianni, G, Szabados, J: A weighted generalization of the classical Kantorovich operator. II. Saturation. Mediterr. J. Math. 10(1), 1-15 (2013)
Yu, DS: Weighted approximation by modified Kantorovich-Bernstein operators. Acta Math. Hung. 141(1-2), 132-149 (2013)
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Mahmudov, N.I., Kara, M. Approximation properties of weighted Kantorovich type operators in compact disks. J Inequal Appl 2015, 46 (2015). https://doi.org/10.1186/s13660-015-0571-1
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DOI: https://doi.org/10.1186/s13660-015-0571-1