Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.018338

ARTICLE

Note on a New Construction of Kantorovich Form q-Bernstein Operators Related to Shape Parameter λ

Qingbo Cai1 and Reşat Aslan2,*

1Fujian Provincial Key Laboratory of Data-Intensive Computing, Key Laboratory of Intelligent Computing and Information Processing, School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou, 362000, China
2Department of Mathematics, Faculty of Sciences and Arts, Harran University, Şanlıurfa, 63300, Turkey
*Corresponding Author: Reşat Aslan. Email: resat63@hotmail.com
Received: 16 July 2021; Accepted: 23 September 2021

1  Introduction

One of the uncomplicated and most elegant proof of the Weierstrass approximation theorem, the polynomials introduced by Bernstein [1], still guides many studies. Very recently, the quantum calculus shortly (q-calculus), which has numerous applications in various disciplines, has attracted the attention of many authors working on approximation theory. Firstly, Lupaş [2] investigated several approximation properties of the generalizations of q-Bernstein polynomials. Afterward, Phillips [3] obtained some convergence theorems and Voronovskaya type asymptotic formula for the most popular generalizations of the q-Bernstein polynomials. Based on the studies [2,3], many approximation properties of the well-known linear positive operators via q-calculus were investigated in details by several researchers. We refer readers for some notable results [4–13].

In 2007, a Kantorovich type Bernstein polynomials via q-calculus are constructed by Dalmanoğlu [14] as follows:

Bm,q(μ;y)=[m+1]q∑j=0mbm,j(y;q)q−j∫[j]q/[m+1]q[j+1]q/[m+1]qμ(t)dqt, y∈[0,1], bm,j(y;q):=[mj]qyj∏r=0m−j−1(1−qry), (1)

where Bm,q: C[0,1]→C[0,1] are defined for any m∈N and any function μ∈C[0,1].

She obtained several approximation properties of the constructed polynomials (1) and estimated the order of approximation in terms of the modulus of continuity. Now, before proceeding further, we recall several basic notations based on q-calculus see details in [15]. Let 0<q≤1, for all integer j > 0, the q-integer [j] q is defined by

The q-factorial [j] q! and q-binomial [jl]q are defined respectively, as below:

[j]q!:={[j]q[j-1]q..[1]q,j=1,2,..1,j=0

and

jlq:=[j]q![l]q![j-l]q!, (j≥l≥0).

The q-analogue of the integration on the interval [0, B] is defined as

∫0Bμ(t)dqt:=B(1-q)∑j=0∞μ(Bqj)qj,   q∈(0,1).

In 1962’s, French engineer Bézier handled the Bernstein basis functions, since they have a simple construction to utilization, to develop the shape design of surface and curve of cars. Moreover, these magnificent polynomials of Bernstein led to many application fields of Mathematics, such as computer graphics, computer-aided geometric design (CAGD), numerical solution of partial differential equations and, etc. Some applications in CAGD, one can refer to [16 –19].

In 2019, Cai et al. [20] proposed and studied some statistical approximation properties of a new generalization of (λ,q)-Bernstein polynomials via Bézier bases with shape parameter λ∈[−1,1] as follows:

B~m,q,λ(μ;y)=∑j=0mb~m,j(y;q,λ)μ([j]q[m]q), (2)

where

b˜ m,0(y;q,λ)=bm,0(y;q)−λ[m]q%+1bm+1,1(y;q),b˜ m,j(y;q,λ)=bm,j(y;q) +λ([m]q−2[j]q+1[m]q2−1bm+1,j(y;q)       −[m]q−2[j]q−1[m]q2−1bm+1,j+1%(y;q) ) (j=1,2,...,m−1),b˜ m,m(y;q,λ)=bm,m(y;q)−λ[m]q%+1bm+1,m(y;q), (3)

m≥2, y∈[0,1], 0<q≤1, λ∈[−1,1] and bm, j(y; q) are defined in (1).

In the year 2020, Mursaleen et al. [21] considered Chlodowsky kind (λ,q)-Bernstein-Stancu polynomials and derived Korovkin-type convergence, and Voronovskaya-type asymptotic theorems. For some recent relevant works we refer to [22–34].

Motivated by all write above-mentioned works, we will introduce and examine the following Kantorovich version of (λ,q)-Bernstein operators:

Rm,q,λ(μ;y)=[m+1]q∑j=0mb~m,j(y;q,λ)q−j∫[j]q/[m+1]q[j+1]q/[m+1]qμ(t)dqt. (4)

Remark 1.1. Let the operators Rm,q,λ(μ;y) be defined by (4). Then,

For λ=0, operators (4) reduce to the Kantorovich type q-Bernstein polynomials constructed by Dalmanoğlu [14].

For λ=0 and q = 1, operators (4) reduce to the classical Kantorovich polynomials [35].

For q = 1, operators (4) reduce to the Kantorovich type λ-Bernstein polynomials proposed by Cai [36].

The present research is organized as follows: In Section 2, we compute several preliminary outcomes such as moments and central moments. In Section 3, we give a Korovkin-type convergence theorem and estimate the degree of convergence in terms of the ordinary modulus of continuity, the functions belong to Lipschitz-type class and Peetre’s K-functional, respectively. In the final section, we demonstrate the comparison of the convergence of operators (4) to a function with some illustrations and error estimation table using Maple software.

2  Preliminaries

Lemma 2.1. (See [20]) Let eu(t) = tu, u∈N∪{0}, y∈[0,1], 0<q≤1, λ∈[−1,1] and m > 1. Then, for the operators given by (2) we have

B~m,q,λ(e0;y)=1. (5)

B˜ m,q,λ(e1;y)=y+[m+1]qλy(1−ym)[m]q([m]q−1)       −2[m+1]qλy[m]q2−1(1−ym[m]q%+qy(1−ym−1))      +λq[m]q([m]q+1)(1−∏r=0m(1−qry)−ym+1−[m+1]qy(1−ym))      +λ[m]q2−1{2[m+1]qy2(1−ym−1%)−2[m+1]qy(1−ym)q[m]q      +2q[m]q(1−∏r=0m(1−qry)−ym+1)}. (6)

B~m,q,λ(e2;y)=y2+y(1−y)[m]q+[m+1]qλy[m]q([m]q−1)(qy(1−ym−1)+1−ym[m]q)−2[m+1]qλ[m]q([m]q2−1){y(1−ym)[m]q+q(q+2)y2(1−ym−1)+q3[m−1]qy3(1−ym−2)}−λq[m]q([m]q+1){[m+1]qy2(1−ym−1)−[m+1]qy(1−ym)q[m]q+1−∏r=0m(1−qry)−ym+1q[m]q}+2λ[m]q([m]q2−1)×{q[m−1]q[m+1]qy3(1−ym−2)−(1−q)[m+1]qy2(1−ym−1)q+[m+1]qy(1−ym)q2[m]q−1−∏r=0m(1−qry)−ym+1q2[m]q}. (7)

Lemma 2.2. Let y∈[0,1], 0<q≤1, λ∈[−1,1] and m > 1. Then, we have

Rm,q,λ(1;y)=1. (8)

Proof. Taking into consideration the definition of q-integral, it follows:

Hence, we get (8).

Lemma 2.3 Let y∈[0,1], 0<q≤1, λ∈[−1,1] and m > 1. Then, we have

Rm,q,λ(t;y)=y−qmy[m+1]q+λy(1−ym)[m]q−1−2[m]qλy[m]q2−1(1−ym[m]q+qy(1−ym−1))+λq[m+1]q([m]q+1)(1−∏r=0m(1−qry)−ym+1−[m+1]qy(1−ym))+λ[m]q[m+1]q([m]q2−1){2[m+1]qy2(1−ym−1)−2[m+1]qy(1−ym)q[m]q+2q[m]q(1−∏r=0m(1−qry)−ym+1)}+1(1+q)[m+1]q. (9)

Proof. By simple computing, one has

∫[j]q/[m+1]q[j+1]q/[m+1]qtdqt=qj(1+q)[m+1]q2((1+q)[j]q+1).

Then,

In view of (5) and (6), we obtain desired sequel (9).

Lemma 2.4. Let y∈[0,1], 0<q≤1, λ∈[−1,1] and m > 1. Then, we have

Rm,q,λ(t2;y)=y2−2qmy2[m+1]q+q2my2[m+1]q2+[m]qy[m+1]q2(2q+1+(q2+q+1)(1−y)q2+q+1)+λy[m+1]q([m]q−1)(q[m]qy(1−ym−1)+(q2+3q+2)(1−ym)q2+q+1)−2[m]qλ[m+1]q([m]q2−1){(q2+3q+2)y(1−ym)(q2+q+1)[m]q+q3y3[m−1]q(1−ym−2)+q(q3+3q2+5q+1)y2(1−ym−1)q2+q+1}−λq[m+1]q2([m]q2−1){[m]q[m+1]qy2(1−ym−1)−(3q2+2q+1)[m+1]qy(1−ym)q(q2+q+1)+3q2+2q+1q(q2+q+1)(1−∏r=0m(1−qry)−ym+1)}+2λ[m]q[m+1]q2([m]q2−1){q[m−1]q[m+1]qy3(1−ym−2)+(q3+q2+2q−1)[m+1]qy2(1−ym−1)q(q2+q+1)−(3q2+q−1)[m+1]qy(1−ym)q2(q2+q+1)+3q2+q−1q2(q2+q+1)(1−∏r=0m(1−qry)−ym+1)[m]q}+1(q2+q+1)[m+1]q2. (10)

Proof. As in previous Lemmas, again using the definition of q-integral, we get

∫[j]q/[m+1]q[j+1]q/[m+1]qt2dqt=qj([j+1]q2+[j+1]q[j]q+[j]q2)(q2+q+1)[m+1]q3.

By the fact that [j + 1]q = 1+q[j]q, one can has the following easily:

Then, by (5), (6) and (7), we arrive at (10) so the proof is done.

Corollary 2.1. Let y∈[0,1], 0<q≤1, λ∈[−1,1] and m > 1. As a consequence of Lemmas 2.2, 2.3 and 2.4, we arrive the following central moments:

Remark 2.1. It can be seen that for a fixed q∈(0,1), m→∞⁡lim[m]q=11−q. In order to provide the convergence results, we get the sequence q: = (qm) such that 0<qm<1,qm→1,1[m]qm→0 as m→∞.

3  Convergence Results of R\ttextit{m, q, }λ(μ;\ttextit{y})

In this section, we first introduce a Korovkin-type approximation theorem for Rm,q,λ(μ;y). It is a known fact that, the space C[0, 1] denotes the real-valued continuous function on [0, 1] and equipped with the norm for the function μ as ‖μ‖C[0,1]=y∈[0,1]⁡sup|μ(y)|.

Theorem 3.1. Assume that q:={qm} satisfies the conditions as in Remark. Then, for all μ∈C[0,1], λ∈[−1,1], y∈[0,1] and m > 1, we have

m→∞⁡lim‖Rm,qm,λ(μ;.)−μ‖=0. (11)

Proof. Consider the sequence of functions es(y) = ys, where s∈{0,1,2} and y∈[0,1]. According to the Bohman-Korovkin theorem [37], we have to show that

limm→∞‖Rm,qm,λ(es;.)−es‖=0, for s=0,1,2. (12)

The proof of (12) follows easily, using (8), (9) and (10). Hence, we get the required sequel.

Now, we will estimate the order of convergence in terms of the usual modulus of continuity, for the function belong to the Lipschitz type class and the Peetre’s K-functional. By first, we symbolize the usual modulus of continuity of μ∈C[0,1] as follows: ω(μ;η):= 0< a≤ηsup supy∈[0,1]|μ(y+a)-μ(y)|. Since η>0, ω(μ;η) has some useful properties see details by [38]. Also, we present an element of Lipschitz continuous function with LipL(ζ), where L > 0 and 0<ζ≤1. If the expression below: |μ(t)-μ(y)|≤L|t-y|ζ,(t,y∈ℝ), holds, then one can say μ is belong to LipL(ζ).

Also, the Peetre’s K-functional is defined byK2(μ,η)= infν∈C2[0,1]{∥μ-ν∥+η∥ν′′∥}, where η>0 and C2[0,1]={ν∈C[0,1]: ν′,ν′′∈C[0,1]}.

Taking into account [39], there exists an absolute constant C > 0 such that

K2(μ;η)≤Cω2(μ;η),η>0 (13)

where ω2(μ;η)= 0< z≤ηsup supy∈[0,1]|μ(y+2z)-2μ(y+z)+μ(y)|, is the second order modulus of smoothness of μ∈C[0,1].

Theorem 3.2. Assume that q:={qm} satisfies the conditions as in Remark 2.1. Then, for all μ∈C[0,1], λ∈[−1,1], y∈[0,1] and m > 1, we obtain |Rm,qm,λ(μ;y)-μ(y)|≤2ω(μ;βm(y,qm)), where βm(y,qm)=Rm,qm,λ((t−y)2;y) same as in Corollary 2.1.

Proof. Taking |μ(t)−μ(y)|≤(1+|t−y|δ)ω(μ;δ) into account and after operating Rm,qm,λ(.;y), we get |Rm,qm,λ(μ;y)-μ(y)|≤(1+1δRm,qm,λ(|t-y|;y))ω(μ;δ). Utilizing the Cauchy-Bunyakovsky-Schwarz inequality, it follows:

Choosing δ=βm(y,qm), which completes the proof.

Theorem 3.3. Assume that q:={qm} satisfies the conditions as in Remark 2.1. Then, for all μ∈LipL(ζ), 0<ζ≤1, y∈[0,1], λ∈[−1,1] and m > 1, we obtain |Rm,qm,λ(μ;y)-μ(y)|≤L(βm(y,qm))ζ2.

Proof. Let μ∈LipL(ζ). Using the linearity and monotonicity properties of the operators (4), therefore

Using the Hölder’s inequality and with p1=2ζ and p2=22−ζ, one has 1p1+1p2=1. Hence, we may write

Hence, Theorem 3.3 is proved.

Theorem 3.4. For all μ∈C[0,1], y∈[0,1] and λ∈[−1,1], the following inequality holds true |Rm,q,λ(μ;y)-μ(y)|≤Cω2(μ;12βm(y,q)+(αm(y,q))2)+ω(μ;αm(y,q)), where C > 0 is a positive constant, αm(y,q) and βm(y,q) are same as in Corollary 2.1.

Proof. Firstly, we define the following auxiliary operators:

R^m,q,λ(μ;y)=Rm,q,λ(μ;y)−μ(Rm,q,λ(t;y))+μ(y). (14)

In view of (8) and (9), we find R^m,q,λ(t-y;y)=0.

Using Taylor’s expansion, one has

ξ(t)=ξ(y)+(t−y)ξ′(y)+∫ty(t−u)ξ′′(u)du,(ξ∈C2[0,1]). (15)

After operating R^m,q,λ(.;y) to (15), yields