Abstract
In this paper, we define the \((p,q)\)-Bernstein polynomials of degree m of a quaternion variable. We obtain some approximation results, and also the Voronovskaja type result with quantitative upper estimates is proved.
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1 Introduction
The quaternion field is an extension of the class of complex numbers, i.e., \(\mathbb{C} \subset\mathbb{H}\). It is a non-commutative field defined by
where the complex units \(i,j,k \notin\mathbb{R}\) satisfy
For \(\omega= y_{1} + y_{2}i + y_{3}j + y_{4}k\), the norm is defined as \(\| w \| = \sqrt{y_{1}^{2} + y_{2}^{2} + y_{3}^{2} + y_{4}^{2}}\) on \(\mathbb {H}\).
We need to give some basic details of analyticity of a function of quaternion variable and some properties of \((p,q)\)-calculus for our purpose.
Definition 1.1
(see Gal [7], p. 296)
A function \(\mathcal{G}: \mathbb{D}_{\mathcal{R}} \longrightarrow\mathbb{H}\) is left Weierstrass analytic (or left \(\mathcal{W}\)-analytic) in \(\mathbb{D}_{\mathcal{R}}\) if \(\mathcal{G}(w) = \sum_{l=0}^{\infty}c_{l}w^{l}\) for all \(w \in\mathbb {D}_{\mathcal{R}} \), where \(\mathbb{D}_{\mathcal{R}}\) denotes the open ball, i.e., \(\mathbb{D}_{\mathcal{R}} = \{w \in\mathbb{H}:\| w \| < \mathcal{R}\}\) and \(c_{l} \in \mathbb{H}\) for all \(l=0,1,2,\ldots\) . Similarly, \(\mathcal{G}\) is called right \(\mathcal{W}\)-analytic in \(\mathbb{D}_{\mathcal{R}}\) if \(\mathcal{G}(w) = \sum_{l=0}^{\infty}w^{l}c_{l}\) for all \(w \in\mathbb {D}_{\mathcal{R}}\).
It is understood here that in any closed ball \(\overline{\mathbb{D}}_{r}=\{\omega\in\mathbb{H}:\| w\| \leq r\}, 0< r<\mathcal{R}\), the partial sums \(\sum_{l=0}^{n}c_{l}w^{l}\) and \(\sum_{l=0}^{n}w^{l}c_{l}\) converge to \(\mathcal{G}\) uniformly, with respect to the metric \(d(w,z)=\| w-z\| \).
Remark
The two concepts given in Definition 1.1 coincide with the Weierstrass type analyticity in the case of complex variable and that is equivalent to the holomorphy concept given by Cauchy. Also, it is better known that the only functions of the trivial form \(\mathcal {G}(w)=Cw+D\) are analytic in the Cauchy sense in the case of quaternion variable (Mejlihzon [17]), while the classic classes of left (or right) monogenic functions at each \(w = y_{1}+y_{2}i+y_{3}j+y_{4}k\in\mathbb{D}_{\mathcal {R}}\subset \mathbb{H}\) were introduced by Moisil [18] as the class of functions \(\mathcal{G}=\mathcal{G}_{1}+\mathcal{G}_{2}i+\mathcal {G}_{3}j+\mathcal{G}_{4}k\) satisfying \(T\mathcal{G}(w)=0\) (or \(\mathcal{G}T(w)=0\) respectively) at each \(w\in\mathbb{D}_{\mathcal{R}}\), where \(T=\frac{\partial }{\partial y_{1}}+\frac{\partial}{\partial y_{2}}i+\frac{\partial}{\partial y_{3}}j+\frac{\partial}{\partial y_{4}}k\) does not coincide with the class of left \(\mathcal{W}\)-analytic (or right \(\mathcal{W}\)-analytic, respectively) function defined in Definition 1.1. For more details concerning the properties of the left (or right) \(\mathcal{W}\)-analytic functions, see [11], [10].
Many authors have worked on q-analogue of different operators, for instance, in [9, 23, 25, 26]. Recently in several areas of mathematical sciences the \((p,q)\)-calculus has many interesting applications (see [5, 12, 16]). For \(p=1\), the notion of \((p,q)\)-calculus is reduced to q-calculus and the transition from the q-case to the \((p,q)\)-case with an extra parameter p is fairly straightforward. One advantage of using the parameter p has been mentioned in approximation by \((p,q)\) Lorentz operators in compact disk [20]. For more \((p,q)\)-approximation, we refer to [1,2,3,4, 13,14,15] and [27].
Most recently, Mursaleen et al. introduced and studied approximation properties of the \((p,q)\)-analogues of many well-known operators, such as Bernstein operators [19], Lorentz operators on compact disk [20], Bleimann–Butzar–Hahn operators [21], divided-difference and Bernstein operators [22], and many more.
Now recall some notations and definitions on \((p,q)\)-calculus.
For \(0< q< p\) and any positive integer m, \((p,q)\) integers are defined as
The \((p,q)\)-binomial expansion is defined as
and for the integers \(0\leq k\leq m\), \((p,q)\)-binomial coefficients are defined by
where
The \((p,q)\)-analogue of Bernstein operators is defined as follows [22]:
The Euler identity is defined by
We introduce the following.
Definition 1.2
Let \(q>p\geq1\) and \(\mathcal{R}>1\). For a function \(\mathcal{G}:\mathbb{D}_{\mathcal{R}}\longrightarrow\mathbb{H}\), because of non-commutativity, we define three distinct \((p,q)\)-Bernstein polynomials of a quaternion variable:
by labeling them as the left \((p,q)\)-Bernstein polynomials, the right \((p,q)\)-Bernstein polynomials, and the middle \((p,q)\)-Bernstein polynomials, respectively.
2 Approximation results
Firstly, we show that for any continuous function \(\mathcal{G}\) these three kinds of \((p,q)\)-Bernstein polynomials do not converge. For example, if we take \(\mathcal{G} = iwi\), we get easily
We can obtain convergence result for the classes of functions in Definition 1.1. For this we need some auxiliary results for \((p,q)\)-operators in complex plane similar as done in [28] for q-operators.
Lemma 2.1
Let \(q\geq p\geq1\) be fixed. Then, for \(n\geq2\),
where \(c_{j}\geq0\ (j=1,2,\ldots,i)\) and \(c_{1}+c_{2}+\cdots+c_{i}=1\). Besides, if \(m\geq n\), then
Also, for any \(r\geq1\),
Proof
The proof is simple, one can prove this lemma with the help of Lemma 3 of [24]. So we skip it. □
Lemma 2.2
Let \(c_{1},c_{2},\ldots,c_{k}\in(0,1)\). Then
and
Proof
For the proof, see (Wang [28], Lemma 2). □
Lemma 2.3
Let \(q \geq p \geq1\) be fixed. If \(m \geq n \geq2\) and \(r \geq1\), then for any w, \(\|w\| \leq r\),
Proof
It follows from (2.1) and (2.2) that, for \(\|w\| \leq r\),
Hence the lemma is proved. □
Theorem 2.4
Let \(q>p\geq1\). Suppose that \(\mathcal{G}:\mathbb {D}_{\mathcal{R}}\longrightarrow\mathbb{H}\) such that \(\mathcal{G}(w)\in \mathbb{R}\) for all \(w\in{}[0,1]\). We have the following representation formula:
where \([\vartriangle^{k}\mathcal{G}(0)]_{p,q}=\sum_{k=0}^{n}(-1)^{k}p^{\frac{(n-k)(n-k-1)}{2}}q^{\frac{k(k-1)}{2}}\binom {n}{k}_{p,q}\mathcal{G}(n-k) \).
Proof
In the expression of \(\mathcal{B}^{m}_{p,q}(\mathcal {G})(w)\), the real values \(\mathcal{G}(\frac{[l]_{p,q}}{p^{l-m}[m]_{p,q}})\) commute with the other terms. As we take the condition on \(\mathcal{G}\), so that \(w^{l}\prod_{s=0}^{m-1-l}(p^{s}-q^{s}w)=\prod_{s=0}^{m-1-l}(p^{s}-q^{s}w)w^{l}, \alpha w = w \alpha\), for all \(\alpha\in\mathbb{R}\), \(w \in\mathbb{D}\) and that we can interchange the order of terms in the product \(\prod_{s=0}^{m-1-l}(p^{s}-q^{s}w)\). By the same reason in the case of \((p,q)\)-Bernstein polynomials of real variable [22], we obtain that the coefficient of \(w^{m}\) in the expression of \(\mathcal{B}^{m}_{p,q}(\mathcal{G})(w)\) is
By using the Euler identity based on \((p,q)\)-analogue, we get the required result. □
Remark
Clearly, Theorem 2.4 holds also for middle \((p,q)\)-Bernstein operators \(\mathcal{B}^{m**}_{p,q}(\mathcal{G})(w)\) and right \((p,q)\)-Bernstein operators \(\mathcal{B}^{m*}_{p,q}(\mathcal{G})(w)\). In [7] and [8] upper estimates by q-Bernstein operators, \(q \geq1\), of quaternion variable were proved.
Theorem 2.5
(Gal [7])
Suppose that \(\mathcal{G}: \mathbb{D}_{\mathcal{R}} \longrightarrow\mathbb{H}\) is left \(\mathcal{W}\)-analytic in \(\mathbb{D}_{\mathcal{R}}\). Then, for all \(1 \leq r < \mathcal{R}, \|w\| \leq r\), and \(m \in\mathbb{N}\), we have
Theorem 2.6
(Gal [8])
Let \(1< q<\mathcal{R}\) and suppose that \(\mathcal{G}: \mathbb{D}_{\mathcal{R}} \longrightarrow\mathbb{H}\) is left \(\mathcal{W}\)-analytic in \(\mathbb{D}_{\mathcal{R}}\). Then, for all \(1 \leq r < \frac{\mathcal{R}}{q}, \|w\| \leq r\), and \(m \in\mathbb{N}\), we have
Remark
By the right Bernstein polynomials \(\mathcal {B}^{m*}_{q}(\mathcal{G})(w)\), a similar upper estimate in approximation can be obtained if \(\mathcal{G}\) is supposed to be right \(\mathcal{W}\)-analytic for \(q \geq1 \).
Theorem 2.7
Let \(\mathcal{R}>q>p>1\) and \(\mathcal{G}:\mathbb {D}_{\mathcal{R}}\longrightarrow\mathbb{H}\) be left \(\mathcal {W}\)-analytic in \(\mathbb{D}_{\mathcal{R}}\), i.e., \(\mathcal{G}(w)=\sum_{i=0}^{\infty }c_{i}w^{i}\) for all \(w\in\mathbb{D}_{\mathcal{R}}\), where \(c_{i}\in \mathbb{H}\) for all \(i=0,1,2,\ldots\) . Then, for all \(1< r<\frac {p\mathcal{R}}{q}\), \(\Vert w\Vert\leq r\), and \(m\in\mathbb{N}\), we have
Proof
Denoting \(e_{i}(w)=w^{i}\). Firstly we will show that
Here \(\mathcal{G}_{n}(w)=\sum_{i=0}^{n}c_{i}e_{i}(w), n \in\mathbb {N}\), is the partial sum of the expansion of \(\mathcal{G}\), due to the linearity of \(\mathcal{B}^{m}_{p,q}\), we get
it is enough to prove that \(\lim_{n\longrightarrow\infty}\mathcal{B} ^{m}_{p,q}(\mathcal{G}_{n})(w)=\mathcal{B}^{m}_{p,q}(\mathcal {G})(w)\) for all \(\|w\| \leq r\) and \(m \in\mathbb{N}\).
By Theorem 2.4 we have
for all \(m,n \in\mathbb{N}\) and \(\|w\| \leq\mathcal{R}\), it follows
which by \(\lim_{n\longrightarrow\infty}\|(\mathcal{G}_{n}-\mathcal {G})\|_{r}=0\) implies the desired conclusion.
Here \(\|(\mathcal{G}_{n}-\mathcal{G})\|_{r}=\max{\|(\mathcal {G}_{n}(w)-\mathcal{G}(w)\|; \|w\|\leq r}\).
Consequently, we obtain
It remains to estimate \(\|\mathcal{B}^{m}_{p,q}(e_{i})(w)-e_{i}(w)\|\), firstly for all \(0 \leq i \leq m\) and secondly for \(i \geq m+1\), where
Set
by relationship given in [22], we can write
where \([0,\frac{p^{m-1}[1]_{p,q}}{[m]_{p,q}},\ldots,\frac{p^{m-k}[k]_{p,q}}{[m]_{p,q}};e_{i} ]\) denotes the divided difference of \(e_{i}(w)=w^{i}\).
Recall that the divided difference of a function \(\mathcal{F}\) on the knots \(y_{0},y_{1},\ldots,y_{j}\) is given by
therefore it follows
However, by the relationship given in [22], we get the formula
which combined with the above relationship (2.8) implies
Since each \(e_{i}\) is convex of any order and \(\mathcal{B}^{m}_{p,q}(e_{i})(1)=e_{i}(1)=1\) for all i, it follows that all \(\mathcal{A}^{m,k,i}_{p,q}\geq0\) and \(\sum_{k=0}^{m}\mathcal{A}^{m,k,i}_{p,q} = 1\) for all i and m.
Also, note that \(\mathcal{A}^{m,i,i}_{p,q}= (1-\frac {p^{m-1}[1]_{p,q}}{[m]_{p,q}} ) (1-\frac{p^{m-2}[2]_{p,q}}{[m]_{p,q}} )\ldots (1-\frac{p^{m-i+1}[i-1]_{p,q}}{[m]_{p,q}} )\) for all \(i\geq1\) and that \(\mathcal{A}^{m,0,0}_{p,q} = 1\).
In the estimation of \(\|\mathcal{B}^{m}_{p,q}(e_{i})(w)-e_{i}(w)\|\), we distinguish two cases: (1) \(0\leq i \leq m\); (2) \(i >m\).
Case 1. We have
Since \(\|e_{k}(w)\| \leq r^{k}\) for all \(\|w\| \leq r\) and \(k \geq 0\), by [24] we immediately get
for all \(\|z\|< r\).
Case 2. Here we have
From both of the above cases we conclude
where \(\|w\| \leq r, m \in\mathbb{N}\), which proves the desired result. □
Remark
Our results generalize the results of Gal [8] (see also [9]), which can be obtained by taking \(p=1\) in our results. Taking an extra parameter p gives more flexibility to study a general class of positive linear operators. By taking \(q>p=1\) in Theorem 2.7, we get the estimate of Theorem 2.6.
3 Voronovskaja type result
Theorem 3.1
Suppose that \(1 \leq p \leq q < \mathcal{R}\) and \(\mathcal{G} : \mathbb{D}_{\mathcal{R}} \longrightarrow\mathbb{H}\) is left \(\mathcal{W}\)-analytic in \(\mathbb{D}_{\mathcal{R}}\), i.e., \(\mathcal{G}(w) = \sum_{k=0}^{\infty}c_{k}w^{k}\), for all \(w \in\mathbb {D}_{\mathcal{R}} \), where \(c_{k} \in\mathbb{H}\) for all \(k=0,1,2,\ldots\) . Also, denote \(\mathcal{S}^{i}_{p,q}= [1]_{p,q} + [2]_{p,q} +\cdots+ [i-1]_{p,q}\), \(i \geq 2 \).
(i) If \(q > p \geq1\), \(1 < r< \frac{p\mathcal{R}}{q^{2}}, \|w\| \leq r\), and \(m \in\mathbb{N}\), then the following upper estimate
holds, where \(\mathcal{C}^{r}_{p,q}(\mathcal{G}) = \max \{ \frac{p^{m-n+1}}{(q-p)(q-1)},\frac{p^{m-n+1}}{(q-p)^{2}(q-1)} \}\sum_{i=2}^{\infty}\|c_{i}\|.(i-1)^{2}(q^{2}r)^{i}\).
(ii) If \(q \geq p \geq1\), then for any \(1 < r < \frac{p\mathcal {R}}{q}\), we have
uniformly in \(\overline{\mathbb{D}}_{r}\), where \(E_{p,q}(\mathcal{G})(w)=\sum_{i=2}^{\infty}c_{i}\mathcal{S}^{i}_{p,q}[w^{i-1}-w^{i}], w \in \mathbb{H}\).
Proof
First we recall some important relationship for our proof. Let \(1 < r <\frac{p\mathcal{R}}{q}\).
From Theorem 2.7, we get
where
here we know \(\mathcal{A}^{m,k,i}_{p,q} \geq0\) for all \(0 \leq k \leq m, i \geq 0\) and \(\sum_{k=0}^{i}\mathcal{A}^{m,k,i}_{p,q}=1\) for all \(0 \leq i \leq m\)
First, we need to prove that \(E_{p,q}(\mathcal{G})(w)\) is left \(\mathcal{W}\)-analytic in \(\overline{\mathbb{D}}_{r}\), where
for \(1 < r < \frac{p\mathcal{R}}{q}\), using the inequality
By (3.1), \(\mathcal{S}^{i}_{p,q} \leq\frac{q^{i}}{(q-p)(q-1)}\) for \(q > p > 1\), it immediately follows
for all \(w \in\overline{\mathbb{D}}_{r}\). These show that, for \(q \geq p \geq1\), the function \(E_{p,q}(\mathcal{G})(w)\) is well-defined and left \(\mathcal{W}\)-analytic in \(\overline{\mathbb{D}}_{r}\).
By (2.7) we obtain
where
and
We have to estimate the expression
To estimate \(\|L^{i,m}_{p,q}(w)\|\), we discuss two cases: 1) \(3 \leq i \leq m \); 2) \(i \geq m+1 \).
Case (1). We obtain
Taking into account (3.1), (3.2) and following Lemma 2.3, we arrive at
for all \(w \in\overline{\mathbb{D}}_{r}\) and \(m \in\mathbb{N}\).
In the case of complex variable, the estimate in (3.4) remains exactly the same, with \(\|. \|\) replaced by \(|. |\), because all the calculations and estimates are made with real numbers as calculated in [6].
Case (2). Here we get
By (3.2), for all \(w \in\overline{\mathbb{D}}_{r}\), it follows
and similarly
Also, by (3.1) we get \(\mathcal{S}^{i}_{p,q} \leq\frac {q^{i}}{(q-p)(q-1)}\), for all \(w \in\overline{\mathbb{D}}_{r}\) it follows
and similarly
By (3.5) we obtain
for all \(w \in\overline{\mathbb{D}}_{r}\), where
Since \([i-1]^{2}_{p,q} \leq[i]^{2}_{p,q} \leq\frac{q^{2i}}{(q-p)^{2}(q-1)^{2}}\) for \(w \in\overline{\mathbb{D}}_{r}\) with \(1 < r < \frac{p\mathcal{R}}{q}\), we obtain by (3.4)
We immediately obtain the upper estimate in (i) by collecting (3.6) and (3.7).
(ii) For the case \(1< r<\frac{p\mathcal{R}}{q^{2}}\) and \(q>p\geq1\), we can get the desired conclusion by multiplying (i) with \([m]_{p,q}\) and passing the limit \(m\longrightarrow\infty\). But (ii) holds under a more general condition \(1< r<\frac{p\mathcal{R}}{q}\).
But by Theorem 2.7,
while for \(i>m\), using (3.2), we have
for all \(w\in\overline{\mathbb{D}}_{r}\).
Also, since \(\mathcal{S}_{p,q}^{i}\leq(i-1)[i-1]_{p,q}\), it is immediate that
Therefore, we easily obtain
valid for all \(w\in\overline{\mathbb{D}}_{r}\).
For all \(w\in\overline{\mathbb{D}}_{r}\) and \(m>m_{0}\), we have
Now, since \(\frac{4}{[m]_{p,q}^{t}}\longrightarrow0\) as \(m\longrightarrow \infty\) and \(\sum_{i=2}^{m_{0}}\Vert c_{i}\Vert .i^{4}q^{(1+t)^{i}}.r^{i}<\infty\), for given \(\epsilon>0\), there exists an index \(m_{1}\) such that \(\frac{4}{[m]_{p,q}^{t}}.\sum_{i=2}^{m_{0}}\Vert c_{i}\Vert.i^{4}q^{(1+t)^{i}}.r^{i}<\epsilon\) for all \(m>m_{1}\).
Finally, for all \(m>\max\{m_{0},m_{1}\}\) and \(w\in\overline{\mathbb {D}}_{r} \), we get
which shows that
The theorem is proved. □
Corollary 3.2
Let \(1 < p < q < \mathcal{R}\) and \(\mathcal{G} : \mathbb {D}_{\mathcal{R}} \longrightarrow\mathbb{H}\) be right \(\mathcal{W}\)-analytic in \(\mathbb{D}_{\mathcal{R}}\), i.e., \(\mathcal{G}(w) = \sum_{k=0}^{\infty }w^{k}c_{k}\), for all \(w \in\mathbb{D}_{\mathcal{R}}\), where \(c_{k} \in\mathbb{H}\) for all \(k=0,1,2,\ldots\) . Then, for all \(1 \leq r < \frac{p\mathcal{R}}{q}, \| w\| \leq r\), and \(n \in\mathbb{N}\), we have
Remarks
(i) However, it is easy to observe that the middle \((p,q)\)-Bernstein type polynomials \(\mathcal{B}^{m**}_{p,q}(\mathcal{G})(w)\) cannot be obtained as an estimate of the form in Theorem 2.7, because when \(\mathcal{G}\) is right \(\mathcal{W}\)-analytic or left \(\mathcal {W}\)-analytic, it cannot be written for the middle \((p,q)\)-Bernstein type polynomials \(\mathcal{B}^{m**}_{p,q}(\mathcal{G})(w)=\sum_{k=0}^{\infty}c_{k}\mathcal{B}^{m**}_{p,q}(e_{k})(w)\).
(ii) Since the choice of \(p>1\) assures that \(\frac{p\mathcal {R}}{q} > \frac{\mathcal{R}}{q}\), this implies that the approximation estimated by \((p,q)\)-Bernstein operators in Theorem 2.7 holds in larger disks than those in the case when \(p=1\).
For the case \(p=q=1\), the ordinary Bernstein operators for quaternion variable have order of approximation \(\frac{1}{m}\) (see Gal [6]), which is weaker than the case of \(q>p=1\), i.e., \(\frac{1}{q^{m}}\) (see [8]).
However, \(\frac{p^{m}}{[m]_{p,q}}=(q-p)\frac{p^{m}}{q^{m}-p^{m}}\) implies that the order of approximation of \((p,q)\)-Bernstein operators for quaternion variable is \((\frac{p}{q} )^{m}\), which is also weaker than \(\frac{1}{q^{m}}\).
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Authors are thankful to the learned referees for their valuable comments which improved the presentation of the paper.
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This research project was supported by a grant from the “Research Center of the Female Scientific and Medical Colleges”, Deanship of Scientific Research, King Saud University.
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Bin Jebreen, H., Mursaleen, M. & Naaz, A. Approximation by quaternion \((p,q)\)-Bernstein polynomials and Voronovskaja type result on compact disk. Adv Differ Equ 2018, 448 (2018). https://doi.org/10.1186/s13662-018-1906-2
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DOI: https://doi.org/10.1186/s13662-018-1906-2