In this paper, we introduce a new family of generalized Bernstein operators based on q integers, called \((\alpha,q)\)-Bernstein operators, denoted by \(T_{n,q,\alpha}(f)\). We investigate a Kovovkin-type approximation theorem, and obtain the rate of convergence of \(T_{n,q,\alpha}(f)\) to any continuous functions f. The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to \(f(x)\). We also obtain the monotonicity with n and q of \(T_{n,q,\alpha}(f)\).
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1 Introduction
A generalization of Bernstein polynomials based on q-integers was proposed by Lupaş in 1987 in [1]. However, the Lupaş q-Bernstein operators are rational functions rather than polynomials. In 1997, Phillips [2] proposed the Phillips q-Bernstein polynomials, and for decades thereafter the application of q integers in positive linear operators became a hot topic in approximation theory, such as generalized q-Bernstein polynomials [3‐6], Durrmeyer-type q-Bernstein operators [7‐9], Kantorovich-type q-Bernstein operators [10‐13], etc. As we know, q integers play important roles not only in approximation theory, but also in CAGD. Based on the Phillips q-Bernstein polynomials [2], which are generalizations of Bernstein polynomials, generalized Bézier curves and surfaces were introduced in [14‐16]. In [14], Oruç and Phillips constructed q-Bézier curves using the basis functions of Phillips q-Bernstein polynomials. Dişibüyük and Oruç [15, 16] defined the q generalization of rational Bernstein–Bézier curves and tensor product q-Bernstein–Bézier surfaces. Moreover, Simeonov et al. [17] introduced a new variant of the blossom, the q blossom, which is specifically adapted to developing identities and algorithms for q-Bernstein bases and q-Bézier curves. In 2014, Han et al. [18] proposed a generalization of q-analog Bézier curves with one shape parameter, and established degree evaluation and de Casteljau algorithms and some other properties. In 2016, Han et al. [19] introduced a new generalization of weighted rational Bernstein–Bézier curves based on q integers, and investigated the generalized rational Bézier curve from a geometric point of view, obtaining degree evaluation and de Casteljau algorithms, etc.
Recently, Chen et al. [20] introduced a new family of α-Bernstein operators, and investigated some approximation properties, such as the rate of convergence, Voronovskaja-type asymptotic formulas, etc. They also obtained the monotonic and convex properties. For \(f(x)\in [0,1]\), \(n\in \mathbb{N}\), and any fixed real α, the α-Bernstein operators they introduced are defined as
where \(f_{i}=f ( \frac{i}{n} ) \). For \(i=0,1,\ldots,n\), the α-Bernstein polynomial \(p_{n,i}^{\alpha }(x)\) of degree n is defined by \(p_{1,0}^{(\alpha )}(x)=1-x\), \(p_{1,1}^{(\alpha )}(x)=x\) and
(2)
where \(n\geq 2\).
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Motivated by above research, in this paper we propose the q analogue of α-Bernstein operators, called \((\alpha , q)\)-Bernstein operators, which are defined as
Here, we mention some definitions based on q integers, the details of which can be found in [21, 22]. For any fixed real number \(0< q\leq 1\) and each non-negative integer k, we denote q-integers by \([k]_{q}\), where
Also, q-factorial and q-binomial coefficients are defined as follows:
The q-analog of \((1+x)^{n}\) is defined by \((1+x)_{q}^{n}:=\prod_{s=0}^{n-1} ( 1+q^{s}x ) \). The q derivative and q derivative of the product are defined as \(D_{q}f(x):=\frac{d_{q}f(x)}{d _{q}x}=\frac{f(qx)-f(x)}{(q-1)x} \) and \(D_{q}(f(x)g(x)):=f(qx)D_{q}g(x)+g(x)D _{q}f(x)\), respectively. We also have \(D_{q}x^{n}=[n]_{q}x^{n-1}\) and \(D_{q}(1-x)_{q}^{n}=-[n]_{q}(1-qx)_{q}^{n-1}\).
The rest of this paper is organized as follows. In the next section, we give some basic properties of the operators \(T_{n,q,\alpha }(f)\), such as the moments and central moments for proving the convergence theorems, the forward difference form of \(T_{n,q,\alpha }(f)\) for proving shape-preserving properties, etc. In Sect. 3, we obtain the convergence property and the rate of convergence theorem. In Sect. 4, we investigate some shape-preserving properties, such as monotonicity- and convexity-preserving properties with respect to \(f(x)\), and also we study the monotonicity with n and q of \(T_{n,q,\alpha }(f)\).
2 Auxiliary results
For proving the main results, we require the following lemmas.
where\(\triangle_{q}^{0}g_{i}=g_{i}\), which is defined in (6).
Proof
We can obtain (8) easily by [2]. Next, in order to prove (9), we use induction on r. It is clear that (9) holds for \(r=0\). Let us assume that (9) holds for some \(r=k\geq 0\). For \(r=k+1\), we have
This shows that (9) holds when k is replaced by \(k+1\), and this completes the proof of Lemma 2.4. □
Since \(f [ \frac{[j]_{q}}{[n]_{q}},\frac{[j+1]_{q}}{[n]_{q}},\ldots, \frac{[j+k]_{q}}{[n]_{q}} ] =\frac{[n]_{q}^{k}\triangle_{q}^{k}f _{j}}{q^{\frac{k(2j+k-1)}{2}}[k]_{q}!}=\frac{f^{(k)}(\xi )}{k!}\), where \(\xi \in ( \frac{[j]_{q}}{[n]_{q}},\frac{[j+k]_{q}}{[n]_{q}} ) \), the q differences of the monomial \(x^{k}\) of order greater than k are zero. We see from Lemma 2.4 that, for all \(n\geq k\), \(T_{n,q,\alpha } ( t^{k};x ) \) is a polynomial of degree k. Actually, the \((\alpha , q)\)-Bernstein operators are degree-reducing on polynomials; that is, if f is a polynomial of degree m, and then \(T_{n,q,\alpha }(f)\) is a polynomial of degree \(\leq \min\{m,n\}\). In particular, we have the following results.
Lemma 2.5
Letting\(f(t)=t^{k}\), \(n-1\geq k\geq 2\), we have
For \(f(t)=t^{2}\), we have \(\triangle_{q}^{0}f_{0}=f_{0}=0\), \(\triangle_{q}^{1}f_{0}=f_{1}-f_{0}=\frac{1}{[n]_{q}^{2}}\), \(\triangle_{q}^{1}f_{1}=f_{2}-f_{1}=\frac{2q+q^{2}}{[n]_{q}^{2}}\), \(\triangle_{q}^{2}f_{0}=\triangle_{q}^{1}f_{1}-q\triangle_{q}^{1}f _{0}=f_{2}-[2]_{q}f_{1}+qf_{0}=\frac{q[2]_{q}}{[n]_{q}^{2}}\), and \(\triangle_{q}^{2}f_{1}=f_{3}-[2]_{q}f_{2}+qf_{1}=\frac{q^{3}+q^{4}}{[n]_{q} ^{2}}\). By (9), we have \(\triangle_{q}^{0}g_{0}=0\), and
Then (11) is proved by (10). This completes the proof of Lemma 2.6. □
3 Convergence properties
We now state the well-known Bohman–Korovkin theorem, followed by a proof based on that given by Cheney [23].
Theorem 3.1
Let\(\{L_{n}\}\)denote a sequence of monotone linear operators that map a function\(f\in C[a,b]\)to a function\(L_{n}f\in C[a,b]\), and let\(L_{n}f\rightarrow f\)uniformly on\([a,b]\)for\(f=1, t\)and\(t^{2}\). Then\(L_{n}f\rightarrow f\)uniformly on\([a,b]\)for all\(f\in C[a,b]\).
Theorem 3.1 leads to the following theorem on the convergence of \((\alpha , q)\)-Bernstein operators.
Theorem 3.2
Let\(q:=\{q_{n}\}\)denote a sequence such that\(q_{n}\in (0,1)\)and\(\lim_{n\rightarrow \infty }q_{n}=1\). Then, for any\(f\in C[0,1]\)and\(\alpha \in [0,1]\), \(T_{n,q,\alpha }(f;x)\)converges uniformly to\(f(x)\)on\([0,1]\).
Proof
From Lemma 2.1, we see that \(T_{n,q,\alpha }(f;x)=f(x)\) for \(f(t)=1\) and \(f(t)=t\). Since \(\lim_{n\rightarrow \infty }q_{n}=1\), we see from (10) that \(T_{n,q,\alpha }(f;x)\) converges uniformly to \(f(x)\) for \(f(t)=t^{2}\) as \(n\rightarrow \infty \). It also follows that \(T_{n,q,\alpha }\) is a monotone operator by Lemma 2.3; the proof is then completed by applying the Bohman–Korovkin theorem 3.1. □
As we know, the space \(C{[0,1]}\) of all continuous functions on \([0,1]\) is a Banach space with sup-norm \(\Vert f \Vert :=\sup_{x\in [0,1]} \vert f(x) \vert \). Letting \(f\in C{[0,1]}\), the Peetre K functional is defined by \(K_{2}(f;\delta ):=\inf_{g\in C^{2}{[0,1]}} \{ \Vert f-g \Vert +\delta \Vert g'' \Vert \}\), where \(\delta >0\) and \(C^{2}{[0,1]}:=\{g \in C{[0,1]}: g', g''\in C{[0,1]}\}\). By [24], there exists an absolute constant \(C>0\), such that
where \(\omega_{2}(f;\delta ):=\sup_{0< h\leq \delta } \sup_{x,x+h,x+2h\in [0,1]} \vert f(x+2h)-2f(x+h)+f(x) \vert \) is the second-order modulus of smoothness of \(f\in C{[0,1]}\).
Theorem 3.3
For\(f\in C{[0,1]}\), \(\alpha \in [0,1]\), \(q\in (0,1)\), we have
where C is a positive constant. Theorem 3.3 is proved. □
Remark 3.4
Letting \(q:=\{q_{n}\}\) denote a sequence such that \(q_{n}\in (0,1)\) and \(\lim_{n\rightarrow \infty }q_{n}=1\), we know that, under the conditions of theorem 3.3, the convergence rate of the operators \(T_{n,q, \alpha }(f)\) to f is \(1/\sqrt{[n]_{q}}\) as \(n\rightarrow \infty \). This convergence rate can be improved depending on the choice of q, at least as fast as \(1/\sqrt{n}\).
Example 3.5
Letting \(f(x) = 1 - \cos(4e^{x})\), the graphs of \(f(x)\) and \(T_{n,q,0.9}(f;x)\) with different values of n and q are shown in Fig. 1. Figure 2 shows the graphs of \(f(x)\) and \(T_{10,0.9,\alpha }(f;x)\) with \(\alpha =0.6\) and \(\alpha =0.9\).
Figure 1
Convergence of \(T_{n,q,\alpha }(f;x)\) to \(f(x)\) for fixed \(\alpha =0.9\)
Figure 2
Convergence of \(T_{n,q,\alpha }(f;x)\) to \(f(x)\) for fixed \(q=0.9\)
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4 Shape-preserving properties
The \((\alpha , q)\)-Bernstein operators \(T_{n,q,\alpha }(f;x)\) have a monotonicity-preserving property.
Theorem 4.1
Let\(f\in C{[0,1]}\). Iffis a monotonically increasing or monotonically decreasing function on\([0,1]\), so are all its\((\alpha , q)\)-Bernstein operators for fixed\(q\in (0,1)\)and\(\alpha \in [0,1]\).
where \(f_{i}=\frac{[i]_{q}}{[n+1]_{q}}\), \(g_{i}= ( 1- \frac{q^{n-i}[i]_{q}}{[n]_{q}} ) f_{i}+ \frac{q^{n-i}[i]_{q}}{[n]_{q}}f_{i+1}\). Then the q derivative of \(T_{n+1,q,\alpha }(f;x)\) is
and we denote the first and second parts of the right-hand side of the last equation by \(\Lambda_{1}\) and \(\Lambda_{2}\), respectively. We then have
Therefore, by using (15) and (16), the derivative of \(( \alpha , q)\)-Bernstein operators \(T_{n,q,\alpha }(f;x)\) may be expressed in the form
Since if f is monotonically increasing on \([0,1]\), the forward differences \(\triangle_{q}^{1}f_{i}\) and \(\triangle_{q}^{1}f_{i+1}\) are non-negative, and so is \(D_{q} [ T_{n,q,\alpha }(f;x) ] \). Hence, \((\alpha , q)\)-Bernstein operators \(T_{n,q,\alpha }(f;x)\) are monotonically increasing on \([0,1]\) for fixed \(q\in (0,1)\) and \(\alpha \in [0,1]\). On the contrary, if f is monotonically decreasing on \([0,1]\), then operators \(T_{n,q,\alpha }(f;x)\) are monotonically decreasing on \([0,1]\) for fixed \(q\in (0,1)\) and \(\alpha \in [0,1]\). Theorem 4.1 is proved. □
The \((\alpha , q)\)-Bernstein operators \(T_{n,q,\alpha }(f;x)\) have a convexity-preserving property
Theorem 4.2
Let\(f\in C{[0,1]}\). Iffis convex on\([0,1]\), so are all of its\(( \alpha , q)\)-Bernstein operators\(T_{n,q,\alpha }(f;x)\)for fixed\(q\in (0,1)\)and\(\alpha \in [0,1]\).
where \(f_{i}=\frac{[i]_{q}}{[n+2]_{q}}\), \(g_{i}= ( 1- \frac{q^{n-i+1}[i]_{q}}{[n+1]_{q}} ) f_{i}+ \frac{q^{n-i+1}[i]_{q}}{[n+1]_{q}}f_{i+1}\). The q-derivative of \(T_{n+2,q,\alpha }(f;x)\) can easily obtained by the proof theorem 4.1, which may be expressed as
Then we have
By some easy computations, we obtain
where \(\triangle_{q}^{2}g_{i}= ( 1- \frac{q^{n-i+1}[i]_{q}}{[n+1]_{q}} ) \triangle_{q}^{2}f_{i}+\frac{q ^{n-i-1}[i+2]_{q}}{[n+1]_{q}}\triangle_{q}^{2}f_{i+1}\). By the connection between the second-order q differences and convexity, we know that \(\triangle_{q}^{2}f_{i}\) and \(\triangle_{q}^{2}f_{i+1}\) are all non-negative since f is convex on \([0,1]\). Hence, we obtain \(D_{q}^{2} [ T_{n+2,q,\alpha }(f;x) ] \geq 0\), and then the convexity-preserving property of \(T_{n,q,\alpha }(f;x)\). Theorem 4.2 is proved. □
Next, if \(f(x)\) is convex, the \((\alpha , q)\)-Bernstein operators \(T_{n,q,\alpha }(f;x)\), for n fixed, are monotonic in q.
where \(\frac{[i]_{q}}{[n]_{q}}\leq x<\frac{[i+1]_{q}}{[n]_{q}}\), \(f _{i}=f ( \frac{[i]_{q}}{[n]_{q}} ) \), \(i=0,\ldots,n-1\). Then it is straightforward to check that \(g_{i}=g ( \frac{[i]_{q}}{[n-1]_{q}} ) \). Since f is convex on \([0,1]\), the intrinsic linear polynomial function \(g(x)\) must be convex on \([0,1]\) as well. Therefore, by the classical results of q-Bernstein operators (see [3]), we note that
We have \(B_{n-1}^{q_{2}}(g;x)\leq B_{n-1}^{q_{1}}(g;x)\) and \(B_{n}^{q_{2}}(f;x)\leq B_{n}^{q_{1}}(f;x)\), and the desired result is obvious. Theorem 4.3 is proved. □
Finally, if \(f(x)\) is convex, we give the monotonicity of \((\alpha , q)\)-Bernstein operators \(T_{n,q,\alpha }(f; x)\) with n.
Theorem 4.4
If\(f(x)\)is convex on\([0,1]\), for fixed\(q\in (0,1)\)and\(\alpha \in [0,1]\), we have
Letting the convex function \(f(x) = 1 - \sin(\pi x)\), \(x\in [0,1]\), the graphs of \(f(x)\) and \(T_{n,0.9,0.9}(f;x)\) with different values of \(n=10, 15, 20, 30\) are shown in Fig. 3. Figure 4 shows the graphs of \(f(x) = 1 - \sin(\pi x)\) and \(T_{10,q,0.9}(f;x)\) with \(q=0.6, 0.7, 0.8, 0.9\).
Figure 3
Monotonicity of \(T_{n,q,\alpha }(f;x)\) in the parameter n
Figure 4
Monotonicity of \(T_{n,q,\alpha }(f;x)\) in the parameter q
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5 Conclusion
In this paper, we proposed a new family of generalized Bernstein operators, named \((\alpha , q)\)-Bernstein operators, and denoted by \(T_{n,q,\alpha }(f)\). We study the rate of convergence of these operators, investigate their monotonicity-, convexity-preserving properties with respect to \(f(x)\), and also obtain their monotonicity with n and q of \(T_{n,q,\alpha }(f)\).
Acknowledgements
We thank Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing of Fujian Province University.
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All data generated or analyzed during this study are included in this published article.
Competing interests
The authors declare that there have no competing interests.
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