# Approximation Properties of Generalizedλ-Bernstein–Stancu-Type Operators

Authors: Qing-Bo Cai; Gülten Torun; Ülkü Dinlemez Kantar
Journal: Journal of Mathematics
 (2021)

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## Abstract

The present study introduces generalizedλ-Bernstein–Stancu-type operators with shifted knots. A Korovkin-type approximation theorem is given, and the rate of convergence of these types of operators is obtained for Lipschitz-type functions. Then, a Voronovskaja-type theorem was given for the asymptotic behavior for these operators. Finally, numerical examples and their graphs were given to demonstrate the convergence of Gm,λα,βf,x to fx with respect to m values.


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## Body

## 1. Introduction

The Bernstein operators, which are positive linear operators, are of great importance for the theory of approximation. In [1], Bernstein operators were introduced by Bernstein to prove the Weierstrass approximation theorem. For g∈C0,1, the classical Bernstein operators are given as(1)Bmg,x=∑j=0mgjmqm,jx,x∈0,1,where the Bernstein basis functions qm,jx are defined as(2)qm,jx≔mjxj1−xm−j;j=0,1,…,m.There are many generalizations of theBmg,x operators. In [2], Gadjiev and Ghorbanalizadeh gave Bernstein–Stancu operators with shifted knots:(3)Sm,α,βg,x=m+β2mm∑j=0mgj+α1m+β1pm,jx,where x∈Am≔α2/m+β2,m+α2/m+β2 and pm,jx≔mjx−α2/m+β2jm+α2/m+β2−xm−j for j=0,1,…,m, and αiandβi are positive real numbers, and 0≤αi≤βi for i=1,2 . In the case of αi=βi=0 for i=1,2, operator (3) reduces to the classical Bernstein operator (1). In addition, the case of α2=β2=0 was handled and examined by Stancu in [3] and these operators were called the classical Bernstein–Stancu operators. A Durrmeyer variant of the Bernstein–Stancu operators in (3) was studied by Dinlemez Kantar and Ergelen in [4], and a Voronovskaja-type approximation theorem for these operators was given. Several studies were conducted on some approximation properties, and asymptotic-type results were given for these operators in [5–14].In [5], Cai et al. introduced the Bernstein operators with shape parameter λ∈−1,1 as follows:(4)Bmλg,x=∑j=0mgjmq˜m,jλ,x,m∈ℕ,where x∈0,1 and q˜m,jλ,x are Bézier basis functions with shape parameter λ∈−1,1 defined by(5)q˜m,0λ,x≔qm,0x−λ/m+1qm+1,1x,q˜m,jλ,x≔qm,jx+λm−2j+1m2−1qm+1,jx−m−2j−1m2−1qm+1,j+1x;j=1,2,…,m−1,q˜m,mλ,x≔qm,mx−λm+1qm+1,mx.Later in [6], Cai proposed Kantorovich-type λ-Bernstein operators, as well as their Bézier variant, and examined the approximation results. Kantorovich-type λ-Bernstein operators were also studied by Acu et al. in [7], and they considered the approximation properties and asymptotic-type results.Recently, in [8], Srivastava et al. constructed λ-Bernstein–Stancu operators defined by(6)Bm,α,βλg,x=∑j=0mgj+αm+βq˜m,jλ,x,m∈ℕ,where x∈0,1 and Bézier basis functions q˜m,jλ,x with shape parameter λ are defined in (5).In the present study, we introduce the following generalizedλ-Bernstein–Stancu operators with shifted knots for g∈C0,1:(7)Gm,λα,βg,x=m+β2mm∑j=0mgj+α1m+β1p˜m,jλ,x,where αi and βi are positive real numbers satisfying 0≤αi≤βi for i=1,2 and Bézier basis functions p˜m,jλ,x with shape parameter λ∈−1,1 are defined by(8)p˜m,0λ,x≔pm,0x−λm+1pm+1,1x,p˜m,jλ,x≔pm,jx+λm−2j+1m2−1pm+1,jx−m−2j−1m2−1pm+1,j+1x;j=1,2,…,m−1,p˜m,mλ,x≔pm,mx−λm+1pm+1,mx,where pm,jx≔mjx−α2/m+β2jm+α2/m+β2−xm−j for j=0,1,…,m, αiandβi are positive real numbers, 0≤αi≤βi for i=1,2, and x∈Am≔α2/m+β2,m+α2/m+β2. We give approximation properties and Voronovskaja-type approximation theorem for asymptotic behavior of operator (7). When αi=βi=0 for i=1,2 and λ=0, operator (7) reduces to the classical Bernstein operator (1). When λ=0, it reduces to the Bernstein–Stancu operator (3). When αi=βi=0 for i=1,2, it reduces to the Bernstein operator with shape parameter λ∈−1,1 (4). In the following section, several lemmas are given to prove the main results.


## 2. Some Preliminary Results

Lemma 1.
For generalizedλ-Bernstein–Stancu operators with shifted knots, we have the following equalities:(9)iGm,λα,β1,x=1,iiGm,λα,βs,x=m+β2m+β1x−α2m+β2+α1m+β1,+λ1m+β1m−1mm+β2−2x−α2m+β2+m+β2mmx−α2m+β2m+1,+α1−1m+α2m+β2−xm+1+α1m+β1m+β2mmx−α2m+β2m+α2m+β2−xm.iiiGm,λα,βs2,x=m+β2m+β12m−1m+β2mx−α2m+β22+1+2α1x−α2m+β2+α12m+β12+λ1m+β12m−1−mm+β2+2mx−α2m+β2−4mx−α2m+β22+m+β2mm2m+1x−α2m+β2m+1+m+α2m+β2−xm+1+α1m+β12m−12mm+β2−4x−α2m+β2+m+β2mm2x−α2m+β2m+1+α1−2m+α2m+β2−xm+1+α12m+β12m+β2mmx−α2m+β2m+α2m+β2−xm,ivGm,λα,βs3,x=m+β2m+β13m−1m−2m+β22m2x−α2m+β23+3m−11+α1m+β2mx−α2m+β22+1+3α11+α1x−α2m+β2+α13m+β13+λ3mm+β13x−α2m+β22−2x−α2m+β23+1m+β13m−1mm+β2−2x−α2m+β2−6mx−α2m+β22+m+β2mm3m2+3m+1x−α2m+β2m+1−m+α2m+β2−xm+1+α12m+β13m+β2mm−3x−α2m+β2m+1+α1x−α2m+β2m+α2m+β2−xm+α1m+β13m−13α1−1mm+β2+6mx−α2m+β2−12m−6α1x−α2m+β22+m+β2mmα12−3α1−1m+α2m+β2−xm+1+6m+1x−α2m+β2m+1+3α1m+β13m+1m+β2mmx−α2m+β2m+1+6α12mm+β13m2−1m+β2mmx−α2m+β2m+1,vGm,λα,βs4,x=m+β2m+β14m−1m−2m−3m+β23m3x−α2m+β24+2m−1m−22α1+3m+β22m2x−α2m+β23+m−16α1α1+2+7m+β2mx−α2m+β22+22α13+3α12+2α1+1x−α2m+β2+α14m+β14+λ2m+β14x−α2m+β2+3m+22x−α2m+β22+2mm−8x−α2m+β23−4mm−2x−α2m+β24+2m2+5m−23m+β2mmx−α2m+β2m+1+1m+β14m−1−mm+β2+2x−α2m+β2+44x−α2m+β22−m+β2mm45x−α2m+β2m+1−m+α2m+β2−xm+1+α1m+β1412α1x−α2m+β2+12mx−α2m+β22−24mx−α2m+β23+m+β2mm12m+2+α1x−α2m+β2m+1+α13x−α2m+β2m+α2m+β2−xm+α1m+β14m−122α12−3α1+2mm+β2−2x−α2m+β2−24mα1+1x−α2m+β22+m+β2mm22α12+9α1+14x−α2m+β2m+1+α13−4α12+6α1−4m+α2m+β2−xm+1.Proof.
If we use Bézier basis functions (8) in λ-Bernstein–Stancu operators (7), we obtain(10)Gm,λα,β1,x=m+β2mm∑j=0mp˜m,jλ,x=m+β2mm∑j=0mp˜m,jx=m+β2mmmm+β2m=1,Gm,λα,βs,x=m+β2mm∑j=0mj+α1m+β1p˜m,jλ,x=m+β2mmα1m+β1pm,0x−λm+1pm+1,1x+∑j=1m−1j+α1m+β1pm,jx+λm−2j+1m2−1pm+1,jx−m−2j−1m2−1pm+1,j+1x+m+α1m+β1pm,mx−λm+1pm+1,mx=m+β2mm∑j=0mj+α1m+β1pm,jx+λΥ1,mα,βx−Υ2,mα,β,where(11)Υ1,mα,βx=m+β2mm∑j=0mj+α1m+β1m−2j+1m2−1pm+1,jx,Υ2,mα,βx=m+β2mm∑j=1m−1j+α1m+β1m−2j−1m2−1pm+1,j+1x.
Thanks to the linearity of Bernstein–Stancu operators (3), we obtain(12)m+β2mm∑j=0mj+α1m+β1pm,jx=m+β2m+β1x−α2m+β2+α1m+β1.
Now, we will computeΥ1,mα,βx and Υ2,mα,βx(13)Υ1,mα,βx=m+β2mm∑j=0mj+α1m+β1m−2j+1m2−1pm+1,jx=m+β2mm1m+β1m−1m+1x−α2m+β2∑j=0m−1pm,j+α1∑j=0mpm+1,jx−m+β2mm2m+1m+β1m2−1mx−α2m+β22∑j=0m−2pm−1,jx+1+α1x−α2m+β2∑j=0m−1pm,jx=1m+β1x−α2m+β2−m+β2mmx−α2m+β2m+1+2mm+β1m−1−x−α2m+β22+m+β2mmx−α2m+β2m+1+α1m+β1m−1−2x−α2m+β2−m+β2mmx−α2m+β2m+1+mm+β2,Υ2,mα,βx=m+β2mm∑j=1m−1j+α1m+β1m−2j−1m2−1pm+1,j+1x=m+β2mm1m+β1m+1m+1x−α2m+β2∑j=1m−1pm,jx−∑j=1m−1pm+1,j+1x+α1∑j=1m−1pm+1,j+1x−m+β2mm2m+β1m2−1mm+1x−α2m+β22∑j=0m−2pm−1,jx−m+1x−α2m+β2∑j=1m−1pm,jx+∑j=1m−1pm+1,j+1x+α1m+1x−α2m+β2∑j=1m−1pm,jx−α1∑j=1m−1pm+1,j+1x=1m+β1x−α2m+β2−m+β2mmx−α2m+β2m+1−1m+β1m+1mm+β2−m+β2mmm+α2m+β2−xm+1+x−α2m+β2m+1−2m+β1m2−1mm+β2−m+β2mmm+α2m+β2−xm+1+m+1x−α2m+β2m+α2m+β2−xm+x−α2m+β2m+1−α1m+β1m+β2mmx−α2m+β2m+α2m+β2−xm+α1m+β1m+1mm+β2−m+β2mmm+α2m+β2−xm+1+x−α2m+β2m+1+2m+β1m−1x−α2m+β2−m+β2mmx−α2m+β2m+α2m+β2−xm+x−α2m+β2m+1−2mm+β1m−1x−α2m+β22−m+β2mmx−α2m+β2m+1−2α1m+β1m−1x−α2m+β2−m+β2mmx−α2m+β2m+1+2α1m+β1m2−1mm+β2−m+β2mmm+α2m+β2−xm+1+x−α2m+β2m+1.
Using (12), Υ1,mα,βx, and Υ2,mα,βx in Gm,λα,βs,x, we obtain the following equation:(14)Gm,λα,βs,x=m+β2m+β1x−α2m+β2+α1m+β1+λ1m+β1m−1mm+β2−2x−α2m+β2+m+β2mmx−α2m+β2m+1+α1−1m+α2m+β2−xm+1+α1m+β1m+β2mmx−α2m+β2m+α2m+β2−xm.
Then, we have the following equality for the third moment by using the linearity ofGm,λα,βgs,x:(15)Gm,λα,βs2,x=m+β2mm∑j=0mj+α1m+β12p˜m,jλ,x=m+β2mmα1m+β12pm,0x−λm+1pm+1,1x+∑j=1m−1j+α1m+β12pm,jx+λm−2j+1m2−1pm+1,jx−m−2j−1m2−1pm+1,j+1x+m+α1m+β12pm,mx−λm+1pm+1,mx=m+β2mm∑j=0mj+α1m+β12pm,jx+λΥ3,mα,βx−Υ4,mα,βx,where(16)Υ3,mα,βx=m+β2mm∑j=0mj+α1m+β12m−2j+1m2−1pm+1,jx,Υ4,mα,βx=m+β2mm∑j=1m−1j+α1m+β12m−2j−1m2−1pm+1,j+1x.
From the linearity of the Bernstein–Stancu operators (3)and Bernstein basis functions, we obtain(17)m+β2mm∑j=0mj+α1m+β12pm,jx=m+β2m+β12m−1m+β2x−α2/m+β22m+1+2α1x−α2m+β2+α12m+β12.
Next, we computeΥ3,mα,βx and Υ4,mα,βx(18)Υ3,mα,βx=m+β2mm∑j=0mj+α1m+β12m−2j+1m2−1pm+1,jx=m+β2mm1m+β12m−1mm+1x−α2m+β22∑j=0m−2pm−1,jx+m+11+2α1x−α2m+β2∑j=0m−1pm,jx+α12∑j=0mpm+1,jx−m+β2mm1m+β12m2−12m+1mm−1x−α2m+β23∑j=0m−3pm−2,jx+2m+1m3+2α1x−α2m+β22∑j=0m−2pm−1,jx+2m+11+2α1+α12x−α2m+β2∑j=0m−1pm,jx=1m+β12x−α2m+β2−2mx−α2m+β23+2m−1m+β2mmx−α2m+β2m+1+mm−5m+β12m−1x−α2m+β22−m+β2mmx−α2m+β2m+1+2α1m+β12x−α2m+β2+α1m+β12m−12m+1m+β2mmx−α2m+β2m+1+α1m+β2mmx−α2m+β2m+1−2x−α2m+β2+mm+β2−4mx−α2m+β22Υ4,mα,βx=m+β2mm∑j=1m−1j+α1m+β12m−2j−1m2−1pm+1,j+1x=m+β2mm1m+β12m+1mm+1x−α2m+β22∑j=0m−2pm−1,jx−m+11−2α1x−α2m+β2∑j=1m−1pm,jx+1−2α1+α12∑j=1m−1pm+1,j+1x−m+β2mm2m+1m+β12m2−1mm−1x−α2m+β23∑j=0m−3pm−2,jx+2α1mx−α2m+β22∑j=0m−2pm−1,jx+1−2α1+α12x−α2m+β2∑j=1m−1pm,jx+m+β2mm21−2α1+α12m+β12m2−1∑j=1m−1pm+1,j+1x=1m+β12−x−α2m+β2+mx−α2m+β22−2mx−α2m+β23+m+1m+β2mmx−α2m+β2m+1+1m+β12m+1mm+β2−m+β2mmx−α2m+β2m+1+m+α2m+β2−xm+1+2m+β12m−1−x−α2m+β2+m+β2mmx−α2m+β2m+1+2m+β12m2−1mm+β2−m+β2mmx−α2m+β2m+1+m+α2m+β2−xm+1+α1m+β122x−α2m+β2+m+β2mm2x−α2m+β2m+1−α1x−α2m+β2m+α2m+β2−xm+α1α1−2m+β12m+1mm+β2−m+β2mmx−α2m+β2m+1+m+α2m+β2−xm+1−2α1m+β12m−1α1−2x−α2m+β2+2mx−α2m+β22−α1m+β2mmx−α2m+β2m+1+2α1α1−2m+β12m2−1mm+β2−m+β2mmx−α2m+β2m+1+m+α2m+β2−xm+1.
Combining (17), Υ3,mα,βx, and Υ4,mα,βx, we obtain the result for Gm,λα,βs2,x.
Using the same technique in the above moments, we obtainGm,λα,βs3,x and Gm,λα,βs4,x.Corollary 1.
Using Lemma1, we obtain the following inequalities of central moments Gm,λα,βs−xn,x for n=1,2 and for fixed x∈0,1 and λ∈−1,1:(19)Gm,λα,βs−x,x=β2−β1m+β1x+α1−α2m+β1+λ−2+m+β2/mmx−α2/m+β2m+α1m−2α1+1m+β2/mmm+α2/m+β2−xmm+β1m−1x+m+2α2m+β1m−1m+β2−α2m+β2/mmx−α2/m+β2mm+β1m−1m+β2+α1−α1α2−1m+2α1α2−α2m+β2/mmm+α2/m+β2−xmm+β1m−1m+β2≤β2+β1m+β1x+α1+α2m+β1+λ2+m+β2/mmx−α2/m+β2m+α1m+2α1+1m+β2/mmm+α2/m+β2−xmm+β1m−1x+m+2α2m+β1m−1m+β2+α2m+β2/mmx−α2/m+β2mm+β1m−1m+β2+α1+α1α2+1m+2α1α2+α2m+β2/mmm+α2/m+β2−xmm+β1m−1m+β2≔Ψm,λα,βxGm,λα,βs−x2,x=−m2+β22−2β2β1−2β2+β12m−β22mm+β12x2+m2+β2−2β2α2+2α2+2β2α1+2α2β1−2β1α1m+2β2α2mm+β12x+α22+α12−α2−2α1α2m−α22mm+β12+λ4β1m+β12m−1−2m+β2/mmx−α2/m+β2m−2α1m−4α1+2m+β2/mmm+α2/m+β2−xmm+β1m−1x2+2β2−2β1+4α2−4α1m−4α1β2−4α2β1m+β12m−1m+β2+2m2+2α1+2α2+2β2+1mm+β12m−1m+β2+β2+2α1β2+2α2β1m+β12m−1m+β2m+β2mmx−α2m+β2m+α12+2α1α2−2α1+2m2m+β12m−1m+β2+α12β2−2α12+2α1α2β1−4α1α2−2α1β1+2β1+2α1+2α2−1m−2α12β2+2α1β2−β2+2α2β1m+β12m−1m+β2+−4α1α2m+β12m−1m+β2m+β2mmm+α2m+β2−xmx+2α1−2α2−1m2m+β12m+β2m−1+2α1β2−2α2β2+4α1α2−4α22−β2m+4α1α2β2m+β12m+β2m−1−2α2m+2α1α2+α2m+β2/mmx−α2/m+β2mm+β12m+β2m−1+α12−α12α2−2α1+1m+2α12α2−2α1α2+α2m+β2/mmm+α2/m+β2−xmm+β12m+β2m−1≤m2+β22+2β2β1+2β2+β22m+β22mm+β12x2+m2+β2+2β2α2+2α2+2β2α1+2α2β1+2β1α1m+2β2α2mm+β12x+α22+α12+α2+2α1α2m+α22mm+β12+λ4β1m+β12m−1+2m+β2/mmx−α2/m+β2mm+β1m−1+2α1m+4α1+2m+β2/mmm+α2/m+β2−xmm+β1m−1x2+2β2+2β1+4α2+4α1m+4α1β2+4α2β1m+β12m−1m+β2+2m2+2α1+2α2+2β2+1m+β2+2α1β2+2α2β1m+β2/mmx−α2/m+β2mm+β12m−1m+β2+α12+2α1α2+2α1+2m2+α12β2+2α12+2α1α2β1+4α1α2+2α1β1+2β1+2α1+2α2+1mm+β12m−1m+β2+2α12β2+2α1β2+β2+2α2β1+4α1α2m+β12m−1m+β2m+β2mmm+α2m+β2−xmx+2α1+2α2+1m2m+β12m+β2m−1+2α1β2+2α2β2+4α1α2+4α22+β2m+4α1α2β2m+β12m+β2m−1+2α2m+2α1α2+α2m+β2/mmx−α2/m+β2mm+β12m+β2m−1+α12+α12α2+2α1+1m+2α12α2+2α1α2+α2m+β2/mmm+α2/m+β2−xmm+β12m+β2m−1≔Φm,λα,βx.
Using Lemma1 and the linearity of Gm,λα,βgs,x, we have the following Corollary 2.Corollary 2.
We obtain the following equalities:ilimmm⟶∞Gm,λα,βs−x,x=β2−β1x+α1−α2iilimmm⟶∞Gm,λα,βs−x2,x=x−x2iiilimm2m⟶∞Gm,λα,βs−x4,x=3x4−6x3+24α22−24α2α1−6α12+3x2


## 3. Convergence Properties ofGm,λα,β

For the asymptotic behavior ofGm,λα,βg,x operators, we give the following Korovkin-type approximation theorem.Theorem 1.
Ifg∈C0,1 and λ∈−1,1, then Gm,λα,βg,x operators converge uniformly to g on 0,1, where C0,1 is a Banach space of all continuous functions on 0,1 with norm g≔supx∈0.1gx.Proof.
Using the equalitiesi,ii, and iii of Lemma 1, we obtain(20)limm⟶∞Gm,λα,βsi,x−xi=0,i=0,1,2.
Therefore, the proof is completed using Korovkin theorem.
Forε>0, the Peetre K-functional is defined as(21)K2g,ε≔infh∈C20,1g−h+εh″,where C20,1≔h∈C0,1:h′,h″∈C0,1. For g∈C0,1, the second-order modulus of continuity is defined as(22)φ2g,ε≔sup0<l≤εsupx,x+l,x+2l∈0,1gx+2l−2gx+l+gx.
By [9, Theorem 2.4], there exists an absolute constant C>0 such that(23)K2g,ε≤Cφ2g,ε.
Also, the usual modulus of continuity ofg∈C0,1 is defined as φg,ε≔sup0<l≤ε,supx,x+l∈0,1gx+l−gx.Theorem 2.
We obtain the following inequality forg∈C0,1 and λ∈−1,1:(24)Gm,λα,βg,x−gx≤4Cφ2g,Ψm,λα,β2x+Φm,λα,βx8+φg,Ψm,λα,βx,where Ψm,λα,βx and Φm,λα,βx are defined in Corollary 1.Proof.
Let us define the following operators:(25)G˜m,λα,βg,x=Gm,λα,βg,x−gm+β2m+β1x+α1−α2m+β1+λ1m+β1m−1mm+β2−2x−α2m+β2+m+β2mmx−α2m+β2m+1+α1−1m+α2m+β2−xm+1+α1m+β1m+β2mmx−α2m+β2m+α2m+β2−xm+gx.
From the linearity ofG˜m,λα,βg,x and the equalities i and ii of Lemma 1, we obtain(26)G˜m,λα,βs−x,x=0.
Using Taylor’s expansion forh∈C20,1, we write(27)hs=hx+h′xs−x+∫xss−uh″udu.
Applying generalizedλ-Bernstein–Stancu operators to both sides of (27) and using (26), we yield(28)G˜m,λα,βh,x=hx+G˜m,λα,β∫xss−uh″udu,xG˜m,λα,βh,x−hx≤∫xGm,λα,βs,xGm,λα,βs,x−uh″udu+Gm,λα,β∫xss−uh″udu,x≤h″2Gm,λα,βs−x,x2+h″2Gm,λα,βs−x2,x≤h″2Ψm,λα,β2x+Φm,λα,βx.
And, we find(29)Gm,λα,βg,x−gx≤G˜m,λα,βg−h,x−g−hx+G˜m,λα,βg,x−gx+gGm,λα,βs,x−gx≤4g−h+Ψm,λα,β2x+Φm,λα,βxh″2+φg,Ψm,λα,βx.
Therefore, if we take infimum on the right side of (29), overall h∈C20,1, we obtain(30)Gm,λα,βg,x−gx≤4K2g,Ψm,λα,β2x+Φm,λα,βx8+φg,Ψm,λα,βx.
Using inequality (23), we have(31)Gm,λα,βg,x−gx≤4Cφ2g,Ψm,λα,β2x+Φm,λα,βx8+φg,Ψm,λα,βx.
Thus, Theorem2 is proved.Remark 1.
Sincelimm⟶∞Ψm,λα,βx=0 and limm⟶∞Φm,λα,βx=0 for ∀x∈0,1, these limits give us a rate of pointwise convergence of the operators Gm,λα,βg,x to gx.
The space of the Lipschitz-type functions is defined as(32)Lipκξ=g∈C0,1:gy−gx≤κy−xξ;x,y∈ℝ,where κ>0 and 0<ξ≤1 [10].
In the following theorem, we obtain the rate of convergence of generalizedλ-Bernstein–Stancu operators Gm,λα,βg,x for functions in Lipκξ.Theorem 3.
Ifg∈Lipκξ, x∈0,1, and λ∈−1,1, then we have(33)Gm,λα,βg,x−gx≤κΦm,λα,βxξ/2,where Φm,λα,βx is defined in Corollary 1.Proof.
Becauseg∈Lipκξ and Gm,λα,βg,x are linear positive operators, we obtain the inequality by using Hölder’s inequality:(34)Gm,λα,βg,x−gx≤Gm,λα,βgs−gx,x=m+β2mm∑j=0mp˜m,jλ,xgj+α1m+β1−gx≤m+β2mmκ∑j=0mp˜m,jλ,xj+α1m+β1−xξ≤κ∑j=0mm+β2mmp˜m,jλ,xj+α1m+β1−x2ξ/2m+β2mmp˜m,jλ,x2−ξ/2≤κ∑j=0mm+β2mmp˜m,jλ,xj+α1m+β1−x2ξ/2∑j=0mm+β2mmp˜m,jλ,x2−ξ/2≤κΦm,λα,βxξ/2.
Hence, we proved Theorem3.
Finally, we give the main result of the article in Theorem4.Theorem 4.
Ifg∈C0,1, then for every x∈0,1 and λ∈−1,1, we obtain(35)limm⟶∞mGm,λα,βg,x−gx=g′xβ2−β1x+α1−α2+g″x2x−x2,where g″x exists.Proof.
Using Taylor’s formula, we can write the following equation for a fixedx∈0,1:(36)gs=gx+g′xs−x+g″x2s−x2+rs,xs−x2,where rs,x is Peano form of the remainder, rs,x∈C0,1, and lims⟶xrs,x=0. If we apply Gm,λα,βg,x to (36), then we have(37)Gm,λα,βg,x−gx=g′xGm,λα,βs−x,x+g″x2Gm,λα,βs−x2,x,+Gm,λα,βrs,xs−x2,x,and we obtain(38)limm⟶∞mGm,λα,βg,x−gx=g′xlimm⟶∞mGm,λα,βs−x,x+g″x2limm⟶∞mGm,λα,βs−x2,x+limm⟶∞mGm,λα,βrs,xs−x2,x.
Using Cauchy–Schwarz inequality, we obtain(39)Gm,λα,βrs,xs−x2,x≤Gm,λα,βr2s,x,x1/2Gm,λα,βs−x4,x1/2.
Becauselims⟶xrs,x=0 and Gm,λα,βs−x4,x are finite operators, we have(40)limmm⟶∞Gm,λα,βrs,xs−x2,x=0.In the end, by using the equalities i and ii of Corollary 2 and (40) in (38), we yield(41)limm⟶∞mGm,λα,βg,x−gx=g′xβ2−β1x+α1−α2+g″x2x−x2.
Thus, we proved Theorem4.


## 4. Numerical Examples

In this section, we show the theoretical results demonstrated in the previous sections by the following example.Example 1.
Let the trigonometric functionfx=1−cos4ex for x∈Am≔α2/m+β2,m+α2/m+β2, 0≤αi≤βii=1,2, and m=10,20,50,100. First, let us choose α1=α2=1 and β1=β2=2.
Forλ=−1, the graphs of Gm,λα,βf,x and fx are shown in Figure 1.
Forλ=1, the graphs of Gm,λα,βf,x and fx are shown in Figure 2.
Now, let us chooseα1=α2=10 and β1=β2=11. The graph of fx and the graph of Gm,λα,βf,x with λ=−1 and λ=1 are shown in Figures 3 and 4, respectively.
As a result, Figure5 reveals that the curve of G20,1α,βf,x with α1=α2=10 and β1=β2=11 studied in the article approaches the curve of the function fx much better than the curves of the operators B201f,x and B20,10,111f,x defined in [5, 8], respectively.Figure 1
Convergence ofGm,−11,2f,x to fx with respect to m values.Figure 2
Convergence ofGm,11,2f,x to fx with respect to m values.Figure 3
Convergence ofGm,−110,11f,x to fx with respect to m values.Figure 4
Convergence ofGm,110,11f,x to fx with respect to m values.Figure 5
Convergences ofGm,110,11f,x, B20,10,111f,x, and B201f,x to fx.



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