Approximation Properties of λ-Bernstein-Kantorovich-Stancu Operators

Murat Bodur; Nesibe Manav; Fatma Tasdelen

doi:10.1515/ms-2022-0010

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Approximation Properties of λ-Bernstein-Kantorovich-Stancu Operators

Murat Bodur , Nesibe Manav and Fatma Tasdelen

Published/Copyright: February 16, 2022

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From the journal Mathematica Slovaca Volume 72 Issue 1

Abstract

The goal of this paper is to construct a new type of Bernstein operators depending on the shape parameter λ ∈ [–1,1]. For these new type operators a uniform convergence result is presented. Furthermore, order of approximation in the sense of local approximation is investigated and Voronovskaja type theorem is proved. Lastly, some graphical results are given to show the rate of convergence of constructed operators to a given function f.

Keywords: λ-Bernstein operators; λ-Kantorovich-Stancu operators; Voronovskaja type theorem; rate of convergence

MSC 2010: Primary 41A10; Secondary 41A25; 41A36

1 Introduction

In 1912, the Russian mathematician S. N. Bernstein gave one of the most known proof of the Weierstrass’s theorem [17], in [5], by defining the following operators; B_n : C[0,1] → C[0,1],

(1.1)Bn(f;x)=∑k=0nbn,k(x)f(kn),

where

bn,k(x)=(nk)xk(1−x)n−k,x∈[0,1].

In 1987, Lupaş [12] introduced the first q-analogue of Bernstein operators and investigated its approximating and shape-preserving properties. In 1996, Phillips [14] proposed another q-variant of the classical Bernstein operators, the so-called Phillips q-Bernstein operators. We have to remind that the q-variants of Bernstein polynomials provide one shape parameter for constructing free-form curves and surfaces, Phillips q-Bernstein operator was applied well in this area, as Khan and Lobiyal [11] indicated before.

Leonid V. Kantorovich, in 1930, introduced a generalization of Bernstein operators as K_n : L₁[0,1] → C[0,1]

(1.2)Kn(f;x)=(n+1)∑k=0nbn,k(x)∫kn+1k+1n+1f(t)dt.

K_n operators are obtained from the classical Bernstein operators (1.1), by replacing the approximating function value f(kn) with integral of f in a neighborhood of kn [10].

In 1968, D. D. Stancu [16] constructed positive linear operators Snα,β:C[0,1]→C[0,1]

(1.3)Snα,β(f;x)=∑k=0nbn,k(x)fk+αn+β.

Here f ∈ C[0,1], α and β real parameters such that 0 ≤ α ≤ β and n∈ℕ. Note that for α = β = 0 the operators turn to classical Bernstein operators [5]. Some recent papers mentioning different types of generalizations of Benstein operators and its applications are cited in [11] and [13].

In 2004, D. Barbosu [2] gave a Kantorovich-Stancu form as below:

(1.4)Knα,β(f;x)=(n+β+1)∑k=0nbn,k(x)∫k+αn+β+1k+α+1n+β+1f(t)dt.

Some interesting approximation properties of Stancu type operators were given by Barbosu in [3, 4].

Constructing a new kind of Bézier curve or a new type of Bernstein operators by using different basis is currently of high interest to researchers. In 2010, Zhangxiang Ye and his colleagues [18] proposed their study not only constructing a new kind of basis functions of Bézier curve with single shape parameter, but also providing a practical algorithm of curve modeling. They discussed some important properties of the basis functions and the corresponding curves. Thus, they explained why a curve or a surface with shape parameters is a good choice.

In 2018, inspiring by a paper in [18], Cai et al. [6] introduced a new generalization of Bernstein operators as follows

(1.5)Bn,λ(f;x)=∑k=0nb˜n,k(λ;x)f(kn),

where λ ∈ [−1,1] and b˜n,k for k = 0,1,…, are defined by

(1.6)b˜n,0(λ;x)=bn,0(x)−λn+1bn+1,1(x),b˜n,i(λ;x)=bn,i(x)+λn−2i+1n2−1bn+1,i(x)−n−2i−1n2−1bn+1,i+1(x),b˜n,n(λ;x)=bn,n(x)−λn+1bn+1,n(x).

It is easy to verify that ∑k=0nb˜n,k(λ;x)=1 and these operators become the classical Bernstein operators for λ = 0. They obtained some approximation properties of λ-Bernstein operators and established an asymptotic formula. Also, they gave some numerical examples to show the rate of convergence of B_n,λ(f; x) to f(x).

Very recently Acu et al. [1] presented a Kantorovich modification of λ-Bernstein operators. They considered the positive linear operators as

(1.7)Kn,λ(f;x)=(n+1)∑k=0nb˜n,k(λ;x)∫kn+1k+1n+1f(t)dt.

They proved a quantitative Voronovskaja type theorem by means of Ditzian-Totik modulus of smoothness. Also, a Grüss-Voronovskaja type theorem for λ-Kantorovich operators were given by them. They studied approximation properties of the Kantorovich variant of λ-Bernstein operators. Some articles can be given about related work [7, 8, 15]. According to this progress, we defined Kantorovich-Stancu type λ-Bernstein operators as below

(1.8)Sn,λα,β(f;x)=(n+β+1)∑k=0nb˜n,k(λ;x)∫k+αn+β+1k+α+1n+β+1f(t)dt,

where Bézier bases b˜n,k(λ;x) are defined in (1.6). Here, as we mentioned above α and β real parameters such that 0 ≤ α ≤ β and n∈ℕ. Besides, it should be taken into consideration that

if we chose α = β = 0, the sequence of operators defined in (1.8) reduce to operators in (1.7),
if we take λ = 0 in (1.8), then the Kantorovich-Stancu type λ-Bernstein operators tend to Kantorovich-Stancu type operators in (1.4),
For α = β = λ = 0 in (1.8), then the Kantorovich-Stancu type λ-Bernstein operators reduce to Kantorovich type operators in (1.2).

The present work is organized as follows: We demonstrate moments of (1.8) operators can be obtained using the concept of moment generating function. A uniform of these constructed operators (1.8) is mentioned. After, we investigate order of approximation in the sense of local approximation with using a classical approach, the second modulus of continuity, Peetre’s K-functional and Lipschitz type function. Then, a Voronovskaja type result is presented for given operators. Some graphical examples are also given in this paper in order to show the rate of convergence of (1.8) operators.

2 Preliminary results

In this part, we present the moments and the central moments of Kantorovich-Stancu type λ-Bernstein operators to use the proofs of the main theorems.

LEMMA 2.1

Lete_i (t) ≔ tⁱ, i = 0, 1, 2 .... Then the λ-Kantorovich-Stancu operatorsSn,λα,βsatisfy

Sn,λα,β(e0;x)=1,Sn,λα,β(e1;x)=nx+αn+β+1+12(n+β+1)+λ1−2x+xn+1−(1−x)n+1(n+β+1)(n−1),Sn,λα,β(e2;x)=n(n−1)x2(n+β+1)2+2(1−α)nx(n+β+1)2+α2+α+13(n+β+1)2 +λ2nx(1−2x)+2xn+1(n+1)−2x(n+β+1)2(n−1) +λ2α(1−2x+xn+1−(1−x)n+1)(n+β+1)2(n−1),Sn,λα,β(e3;x)=14(n+β+1)3{1+2nx(7−9x+9nx+4x2−6nx2+2n2x2) +4α(1+6nx−12nx2+12n2x2+12nxα+3α/2+α2)} +λ(n+3)(n2−1)(n+β+1)3(1−(1−x)n+1−xn+1) +λ2(n−1)(n+β+1)3{−1+(1−x)n+1−12x+6nx−30nx2 +6n2x2+12nx3−12n2x3+13xn+1+12nxn+1+6n2xn+1} +3λα2(n−1)(n+β+1)3{−4x+4nx−8nx2+4xn+1+4nxn+1 +2α−6(1−x)n+1α−4xα+2xn+1α},Sn,λα,β(e4;x)=15(n+β+1)4{5n4x4−30n3x4+40n3x3+55n2x4−120n2x3 +80nx3−75nx2+30nx+1}+α(n+β+1)4{4n3x3−12n2x3 +8nx3+18n2x2−18nx2+14nx+1+6n2x2α−6nx2α +4nxα2+12nxα+α3+2α2+2α} +2λ(n+3)(n2−1)(n+β+1)4(1−(1−x)n+1−xn+1+2α−2α(1−x)n+1−2αxn+1) +λ(n+β+1)4{x−nx2+6n2x2−44nx3−2n2x3−16nx4 −8n2x4−xn+1+29nxn+1+4n2xn+1−24nx3α+24nxn+1α} +λ(n−1)(n+β+1)4{−2+2(1−x)n+1−9x+nx(5−49x+13nx) +xn+1(11+44n−13n2))} +λα(n+β+1)4{−2+2(1−x)n+1−24x+12nx−60nx2+12n2x2+26xn+1 +48nxn+1−12n2xn+1−12xα+12nxα−24nx2α+12xn+1α+12nxn+1α +4α2−4(1−x)n+1α2−8xα2+4xn+1α2}.

LEMMA 2.2

By consideringLemma 2.1, λ-Kantorovich-Stancu operators possess

(2.1)Sn,λα,β(t−x;x)=α−(β+1)xn+β+1+12(n+β+1)+λ1−2x+xn+1−(1−x)n+1(n+β+1)(n−1)=ς(α,β,n,λ)(x),

(2.2)Sn,λα,β((t−x)2;x)=(β+1)2−n(n+β+1)2x2−(β+1)(2α+1)−n(n+β+1)2x+α2+α+13(n+β+1)2 +λ(n+β+1)2(n−1){2nx(1−2x)+2xn+1(n+1)−2x +2(α−(n+β+1)x)(1−2x+xn+1−(1−x)n+1)}=σ(α,β,n,λ)(x).

Besides, the consecutive limits hold

limn→∞nSn,λα,β(t−x;x)=2(α−x(β+1))+12

and

limn→∞nSn,λα,β((t−x)2;x)=x(1−x).

3 Convergence results

In this main chapter, we search the rate of convergence by means of the modulus of continuity, Peetre’s K-functional and elements of Lipschitz class. Here, C[0, 1] is the class of real valued functions defined on [0, 1] which is uniformly continuous with the norm ‖f‖=supx∈[0,1]|f(x)|.

THEOREM 3.1

For f ∈ C[0, 1], λ ∈ [–1, 1], Kantorovich-Stancu type λ-Bernstein operatorsSn,λα,βconverge uniformly to f on [0, 1].

Proof. By the Bohmann-Korovkin theorem, it is sufficient to show that

limn→∞‖Sn,λα,β(ti−xi)‖=0, i=0,1,2.

It can be clearly obtained these three conditions using Lemma 2.1.

Here, we will give some definitions to use in this section.

DEFINITION 1

For δ > 0 and f ∈ C[0, 1], the modulus of continuity ω(f; δ) of the function f is defined by

ω(f;δ)=supx,y∈[0,1]|x−y|≤δ|f(x)−f(y)|.

DEFINITION 2

The second modulus of continuity of f ∈ C[0,1] is defined by

ω2(f;δ):=sup0<t≤δ‖f(⋅+2t)−2f(⋅+t)+f(⋅)‖C[0,1].

DEFINITION 3 ( 9)

Peetre’s K-functional of the function f ∈ C[0, 1] is defined by

K2(f;δ):=infg∈W2[0,1]{‖f−g‖C[0,1]+δ‖g″‖C[0,1]},

where

W2[0,1]:={g∈C[0,1]:g′,g″∈C[0,1]}.

The following inequality

K2(f;δ)≤M{ω2(f;δ)}

is valid for all δ > 0. The positive constant M is independent of f and δ.

In the following theorem, the modulus of continuity is used to estimate the order of approximation to the function f.

THEOREM 3.2

Let f ∈ C[0, 1], then

|Sn,λα,β(f;x)−f(x)|≤2ω(f;δ),

whereδ:=δ(α,β,n,λ)=σ(α,β,n,λ)(x)..

Proof. Keeping in mind the following property of modulus of continuity

|f(t)−f(x)|≤ω(f;δ)((t−x)2δ2+1).

We obtain

|Sn,λα,β(f;x)−f(x)|≤Sn,λα,β(|f(t)−f(x)|;x)≤ω(f;δ)(1+1δ2Sn,λα,β((t−x)2;x)).

Lastly, choosing δ=σ(α,β,n,λ)(x), so

|Sn,λα,β(f;x)−f(x)|≤2ω(f;δ).

Hence, the proof is completed.

THEOREM 3.3

If g ∈ C¹[0, 1], then

|Sn,λα,β(g;x)−g(x)|≤ς(α,β,n,λ)(x)|g′(x)|+2σ(α,β,n;λ)(x)ω(g′,σ(α,β,n;λ)(x)).

Proof. Let g ∈ C¹[0, 1]. For any x, t ∈ [0, 1], we have

g(t)−g(x)=g′(x)(t−x)+∫xt(g′(y)−g′(x))dy,

so, applying the operators at above equality, we get

(3.1)Sn,λα,β(g(t)−g(x);x)=g′(x)Sn,λα,β(t−x;x)+Sn,λα,β(∫xt(g′(y)−g′(x))dy;x).

Taking into account the following property of modulus of continuity

|g(y)−g(x)|≤ω(g;δ)(|y−x|δ+1),δ>0,

we can write

(3.2)|∫xt|g′(y)−g′(x)|dy|≤ω(g′;δ)[(t−x)2δ+|t−x|].

By using (3.2) in (3.1), we get

|Sn,λα,β(g;x)−g(x)|≤|g′(x)|⋅|Sn,λα,β(t−x;x)|+ω(g′;δ){1δSn,λα,β((t−x)2;x)+Sn,λα,β(|t−x|;x)}.

Using Cauchy-Schwarz inequality then we reach

|Sn,λα,β(g;x)−g(x)|≤|g′(x)||Sn,λα,β(t−x;x)| +ω(g′;δ){1δSn,λα,β((t−x)2;x)+1}Sn,λα,β((t−x)2;x)≤|g′(x)|ς(α,β,n,λ)(x)+ω(g′,δ){1δσ(α,β,n;λ)(x)+1}σ(α,β,n;λ)(x).

Choosing δ=σ(α,β,n,λ)(x), we find the desired inequality.

Now, we interest in the rate of convergence by the known method K-functional.

THEOREM 3.4

If f ∈ C[0, 1], then

|Sn,λα,β(f;x)−f(x)|≤Mω2(f,12σ(α,β,n,λ)(x)+ς2(α,β,n,λ)(x))+ω(f,ς(α,β,n,λ)(x)),

where M is a positive constant.

Proof. Denoting

εn,λα,β(x)=nx+αn+β+1+12(n+β+1)+λ1−2x+xn+1−(1−x)n+1(n+β+1)(n−1)

and

(3.3)S˜n,λα,β(f;x)=Sn,λα,β(f;x)+f(x)−f(εn,k(x)).

It follows immediately

S˜n,λα,β(e0;x)=Sn,λα,β(e0;x)=1S˜n,λα,β(e1;x)=Sn,λα,β(e1;x)+x−εn,λ(x)=x.

Applying S˜n,λα,β to Taylor’s formula, we get

S˜n,λα,β(g;x)=g(x)+S˜n,λα,β(∫xt(t−y)g″(y)df;x)

Therefore,

S˜n,λα,β(g;x)=g(x)+Sn,λα,β(∫xt(t−y)g″(y)dy;x)−∫xεn,k(x)(εn,k(x)−y)g″(y)dy.

This implies that

In view of (3.3), we obtain

(3.4)|S˜n,λα,β(f;x)|≤|Sn,λα,β(f;x)|+|f(x)|+|f(εn,λα,β(x))|≤3‖f‖.

Now, for f ∈ C[0, 1] and g ∈ W²[0, 1], using (3.3) and (3.4), we get

|Sn,λα,β(f;x)−f(x)|=|S˜n,λα,β(f;x)−f(x)+f(εn,λα,β(x))−f(x)|≤|S˜n,λα,β(f−g;x)|+|S˜n,λα,β(g;x)−g(x)|+|g(x)−f(x)|+|f(εn,λα,β(x))−f(x)|≤4‖f−g‖+[σ(α,β,n,λ)(x)+ς2(α,β,n,λ)(x)]‖g″‖ +ω(f,ς(α,β,n,λ)(x)).

Taking the infimum on the right side over all g ∈W²[0, 1], we have

|Sn,λα,β(f;x)−f(x)|≤4K2(f,14(σ(α,β,n,λ)(x)+ς2(α,β,n,λ)(x))) +ω(f,ς(α,β,n,λ)(x)).

Conclusively, when using the relation between K-functional and the second modulus of continuity as shown in Definition 3, we reach the desired result.

We investigate the rate of convergence with the help of the functions of Lipschitz class. For this, we have to give below definition primarily.

Definition 4

Let f be a real valued continuous function defined on [0, ∞). Then the following statement holds for f in order γ (0 < γ ≤ 1) on [0, ∞),

|f(x)−f(y)|≤M|x−y|γ

for all x, y ∈[0, ∞) and M > 0. These set of Lipschitz continuous functions of order γ with Lipschitz constant M is denoted by Lip_M(γ).

THEOREM 3.5

Let f ∈ Lip_M (γ), x ∈ [0, 1] and λ ∈ [–1, 1], then we have

|Sn,λα,β(f;x)−f(x)|≤M[σ(α,β,n,λ)(x)]γ2

where σ(α, β, n, λ)(x) is defined in (2.2).

Proof. For the positive linear operators Sn,λα,β(f;x) and f ∈ Lip_M(γ), while ∑k=0nb˜n,k(λ;x)=1, and keeping in mind well-known Hölder’s inequality, it can be written directly

Hence, the proof is completed.

4 Voronovskaja type theorem

Now, we prove the Voronovskaja type theorem for the operators Sn,λα,β.

THEOREM 4.1

Let f be a bounded function on [0,1], f^′, f^″ derivatives exist for any x ∈ (0,1) and λ ∈ [−1, 1]. Then, we have

limn→∞ 2n(Sn,λα,β(f;x)−f(x))=(2(α−x(β+1))+1)f′(x)+x(1−x)f″(x).

Proof. The Taylor’s expansion

f(t)=f(x)+f′(x)(t−x)+12f″(x)(t−x)2+h(t,x)(t−x)2,

where h(.) ∈ C [0, 1] and limt→x h(t;x)=0. Taking into account the linearity of the operators Sn,λα,β and applying it to both sides of the above Taylor’s expansion with a simple calculations, we obtain

n(Sn,λα,βf(x)−f(x))=nSn,λα,β(t−x;x)f′(x)+nSn,λα,β((t−x)2;x)f″(x)2+nSn,λα,β(h(t,x)(t−x)2;x).

Therefore, by using Lemma 2.2, we get

(4.1)limn→∞ n(Sn,λα,β(f;x)−f(x))=(2(α−x(β+1))+1)2f′(x)+x(1−x)2f″(x)+limn→∞ nSn,λα,β(h(t,x)(t−x)2;x).

Hence, by the Cauchy-Schwarz inequality, we have

(4.2)n|Sn,λα,β(h(t,x)(t−x)2;x)|≤Sn,λα,β(h2(t,x);x)n2Sn,λα,β((t−x)4;x).

Since limn→∞ n2Sn,λα,β((t−x)4;x) is bounded by Lemma 2.1 and h²(x, x) = 0, by considering (4.2) in (4.1), we obtain

limn→∞ nSn,λα,β(h(t−x)(t−x)2;x)=0

which completes the proof.

5 Graphical results

Finally, in this section, we provide some graphical examples that show the convergence of Kantorovich-Stancu type λ-Bernstein operators. With these given graphicals examples, it is possible to understand better how our operators converge to given functions.

Example 1

The first illustration demonstrates the convergence of the operators Sn,λα,β depending on n. Here, taken λ = 0.4, α = 0.2, β = 0.3 values and approximating function of f(x)=1−sin(72πx), we see the operators respectively for n = 20, n = 50, n = 80 values in Figure 1.

Figure 1

Example 2

Another example is illustrated in Figure 2 to show impact of shape parameters λ. For α = 0.2, β = 0.3, n = 6 be fixed and λ = —1, λ = 0, λ = 1 approximant Sn,λα,β convergence to f(x)=1−sin(72πx).

Now, we would like to take an attention to influence of different real parameters α (also can be shown different β) which is sense of Stancu.

Figure 2

Example 3

For λ = 0.3, n = 20, β = 2 and α = 0.5, α = 1, α = 1.5 the convergence of Kantorovich-Stancu type λ-Bernstein operators to f(x)=1−sin(72πx) are illustrated in Figure 3.

Figure 3

Example 4

The purpose of the last example as represented in Figure 4 is making a comparison between Kantorovich-Stancu type λ-Bernstein operators and classical Bernstein Kantorovich operators about approximating to f(x)=1−sin(72πx). Here, λ = 1, n = 20, α = 0.2, β = 0.3.

Figure 4

(Communicated by Gregor Dolinar)

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Received: 2020-03-26

Accepted: 2021-02-13

Published Online: 2022-02-16

Published in Print: 2022-02-16

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https://doi.org/10.1515/ms-2022-0010

Keywords for this article

λ-Bernstein operators; λ-Kantorovich-Stancu operators; Voronovskaja type theorem; rate of convergence

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