# Some Approximation Properties of Modified Jain-Beta Operators

Authors: Vishnu Narayan Mishra; Prashantkumar Patel
Journal: Journal of Calculus of Variations
 (2013)

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## Abstract

Generalization of Szász-Mirakyan operators has been considered by Jain, 1972. Using these generalized operators, we introduce new sequences of positive linear operators which are the integral modification of the Jain operators having weight functions of some Beta basis function. Approximation properties, the rate of convergence, weighted approximation theorem, and better approximation are investigated for these new operators. At the end, we generalize Jain-Beta operator with three parametersα, β, and γ and discuss Voronovskaja asymptotic formula.


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## Body

## 1. Introduction

For0<ϑ<∞,  |κ|<1, let
(1)wκ(i,ϑ)=ϑ(ϑ+iκ)i-1e-(ϑ+iκ)i!i=0,1,2,…;
then
(2)∑i=0∞‍wκ(i,ϑ)=1.
Equation (1) is a Poisson-type distribution which has been considered by Consul and Jain [1].In 1970, Jain [2] introduced and studied the following class of positive linear operators:
(3)Pnκ(f,x)=∑i=0∞‍wκ(i,nx)f(in),
where 0≤κ<1 and wκ(i,nx) has been defined in (1).The parameterκ may depend on the natural number n. It is easy to see that κ=0; (3) reduces to the well-known Szász-Mirakyan operators [3]. Different generalization of Szász-Mirakyan operator and its approximation properties is studied in [4, 5]. Kantorovich-type extension of Pnκ was given in [6]. Integral version of Jain operators using Beta basis function is introduced by Tarabie [7], which is as follows:
(4)Pnκ(f,x)=∑i=1∞‍1B(n+1,i)wκ(i,nx)×∫0∞‍ti-1(1+t)n+i+1f(t)dt+e-nxf(0).
In Gupta et al. [8] they considered integral modification of the Szász-Mirakyan operators by considering the weight function of Beta basis functions. Recently, Dubey and Jain [9] considered a parameter γ in the definition of [8]. Motivated by such types of operators we introduce new sequence of linear operators as follows:Forx∈[0,∞) and γ>0,
(5)Pn,γκ(f,x)=∑i=1∞‍wκ(i,nx)∫0∞‍bn,i,γ(t)f(t)dt+e-nxf(0),
where wκ(i,nx) is defined in (1) and
(6)bn,i,γ(t)=γΓ(n/γ+i+1)Γ(i)Γ(n/γ+1)(γt)i-1(1+γt)(n/γ)+i+1.
The operators defined by (5) are the integral modification of the Jain operators having weight function of some Beta basis function. As special case, γ=1 the operators (5) reduced to the operators recently studied in [7]. Also, if κ=0 and γ=1, then the operators (5) turn into the operators studied in [8].In the present paper, we introduce the operators (5) and estimate moments for these operators. Also, we study local approximation theorem, rate of convergence, weighted approximation theorem, and better approximation for the operators Pn,γκ. At the end, we propose Stancu-type generalization of (5) and discuss some local approximation properties and asymptotic formula for Stancu-type generalization of Jain-Beta operators.


## 2. Basic Results

Lemma 1 (see [2]).
ForPnκ(tm,x),  m=0,1,2, one has
(7)Pnκ(1,x)=1,Pnκ(t,x)=x1-κ,Pnκ(t2,x)=x2(1-κ)2+xn(1-κ)3.Lemma 2.
The operatorsPn,γκ, n>γ defined by (5) satisfy the following relations:
(8)Pn,γκ(1,x)=1,Pn,γκ(t,x)=x1-κ,Pn,γκ(t2,x)=nx2(1-κ)2(n-γ)+x(1+(1-κ)2)(1-κ)3(n-γ).Proof.
By simple computation, we get(9)Pn,γκ(1,x)=∑i=1∞‍wκ(i,nx)+e-nx=Pnκ(1,x)=1,(10)Pn,γκ(t,x)=∑i=1∞‍wκ(i,nx)×∫0∞‍γΓ(n/γ+i+1)Γ(i)Γ(n/γ+1)·(γt)i-1(1+γt)(n/γ)+i+1·tdt=∑i=1∞‍wκ(i,nx)γΓ(n/γ+i+1)Γ(i)Γ(n/γ+1)·Γ(i+1)Γ(n/γ)γ2Γ(n/γ+i+1)=∑i=1∞‍wκ(i,nx)in=Pnκ(t,x)=x1-κ;(11)Pn,γκ(t2,x)=∑i=1∞‍wκ(i,nx)×∫0∞‍γΓ(n/γ+i+1)Γ(i)Γ(n/γ+1)·(γt)i-1(1+γt)(n/γ)+i+1·t2dt=∑i=1∞‍wκ(i,nx)γΓ(n/γ+i+1)Γ(i)Γ(n/γ+1)·Γ(i+2)Γ((n/γ)-1)γ3Γ(n/γ+i+1)=∑i=1∞‍wκ(i,nx)i2+in(n-γ)=∑i=1∞‍wκ(i,nx)(i2n(n-γ)+in(n-γ))=nn-γ∑i=1∞‍wκ(i,nx)i2n2+1n-γ∑i=1∞‍wκ(i,nx)in=nn-γ(x2(1-κ)2+xn(1-κ)3)+1(n-γ)x(1-κ)=nx2(1-κ)2(n-γ)+x(1+(1-κ)2)(1-κ)3(n-γ).Lemma 3.
Forx∈[0,∞), n>γ, and with φx=t-x, one has  (i)
Pn,γκ(φx,x)=κx/(1-κ),(ii)
Pn,γκ(φx2,x)=((κ2(n-γ)+γ)/((1-κ)2(n-γ)))x2 +((1+(1-κ)2)/((1-κ)3(n-γ)))x.Lemma 4.
Forx∈[0,∞), n>γ, one has
(12)Pn,γκ(φx2,x)≤n+2(1-κ)3(n-γ)(x2+x)=δκ,n,γ(x)(say).Proof.
Sincemax{x,x2}≤x+x2, (1-κ)2≤1, and (1-κ)-2≤(1-κ)-3, we have
(13)Pn,γκ(φx2,x)≤[κ2(n-γ)+γ+2(1-κ)3(n-γ)](x+x2)≤[n+2(1-κ)3(n-γ)](x+x2)(as|κ|<1),
which is required.


## 3. Some Local Approximation

LetBx2[0,∞)={f:foreveryx∈[0,∞),|f(x)|≤Mf(1+x2),  Mf being a constant depending on f}. By Cx2[0,∞), we denote the subspace of all continuous functions belonging to Bx2[0,∞). Also, Cx2*[0,∞) is subspace of all the function f∈Cx2[0,∞) for which limx→∞(f(x)/(1+x2)) is finite. The norm on Cx2*[0,∞) is ∥f∥x2=supx∈[0,∞)(|f(x)|/(1+x2)).If we look at Lemma2 and based on the famous Korovkin theorem [10], it is clear that {Pn,γκ}n>γ does not form an approximation process. To do this approximation process, we replace constant κ by a number κn∈[0,1). If
(14)limn→∞κn=0,
then Lemma 2 gives
(15)limn→∞Pn,γκn(tj,x)=xj,j=0,1,2,
uniformly on any compact interval K⊂[0,∞). Based on Korovkin’s criteria we state the following.Theorem 5.
LetPn,γκn with n>γ be defined in (5), where limn→∞κn=0. For any compact K⊂[0,∞) and for each f∈Cx2*[0,∞) one has
(16)limn→∞Pn,γκn(f,x)=f(x),uniformlyinx∈K.Now, we establish a direct local approximation theorem for the modified operatorsPn,γκ in ordinary approximation. Let the space CB[0,∞) of all continuous and bounded functions be endowed with the norm ∥f∥=sup{|f(x)|:x∈[0,∞)}. Further let us consider the following K-functional:
(17)K2(f,δ)=infg∈W2{∥f-g∥+δ∥g′′∥},
where δ>0 and W2={g∈CB[0,∞):g′,g′′∈CB[0,∞)}. By the methods given in [11], there exists an absolute constant C>0 such that
(18)K2(f,δ)≤Cω2(f,δ),
where
(19)ω2(f,δ)=sup0<h≤δsupx∈[0,∞)|f(x+2h)-2f(x+h)+f(x)|
is the second order modulus of smoothness of f∈CB[0,∞).Theorem 6.
Forf∈CB[0,∞) and n>γ, one has
(20)|Pn,γκ(f,x)-f(x)|≤ω(f,κx1-κ)+Cω2(f,δκ,n,γ(x)+κx1-κ),
where C is a positive constant.Proof.
We introduce the auxiliary operators as follows:(21)P^n,γκ(f,x)=Pn,γκ(f,x)-f(x+κx1-κ)+f(x).
Let g∈W∞2 and x,t∈[0,∞). By Taylor’s expansion we have
(22)g(t)=g(x)+(t-x)g′(x)+∫xt‍(t-u)g′′(u)du.
Applying P^n,γκ, we get
(23)P^n,γκ(g,x)-g(x)=g′(x)P^n,γκ((t-x),x)+P^n,γκ(∫xt‍(t-u)g′′(u)du,x).
Applying Lemma 2, we get
(24)|P^n,γκ(g,x)-g(x)|≤P^n,γκ(|∫xt‍(t-u)g′′(u)du|,x)≤Pn,γκ((t-u)2,x)∥g′′∥+|∫xx+κx/(1-κ)‍(x+κx1-κ-u)g′′(u)du|≤[δκ,n,γ(x)+κx1-κ]∥g′′∥.
Since
(25)|Pn,γκ(f,x)|≤∑i=1∞‍wκ(i,nx)×∫0∞‍bn,i,γ(t)|f(t)|dt+e-nx|f(0)|≤∥f∥,(26)|Pn,γκ(f,x)-f(x)|≤|P^n,γκ(f-g,x)-(f-g)(x)|+|P^n,γκ(g,x)-g(x)|+|(x+κx1-κ)-f(x)|≤2∥f-g∥+[δκ,n,γ(x)+κx1-κ]∥g′′∥+ω(f,κx1-κ).
Taking infimum overall g∈W2, we get
(27)|Pn,γκ(f,x)-f(x)|≤K(f,δκ,n,γ(x)+κx1-κ)+ω(f,κx1-κ).
In view of (18)
(28)|Pn,γκ(f,x)-f(x)|≤Cω2(f,δκ,n,γ(x)+κx1-κ)+ω(f,κx1-κ),
which proves the theorem.


## 4. Rate of Convergence and Weighted Approximation

For any positivea, by
(29)ωa(f,δ)=sup|t-x|≤δsupx,t∈[0,a]|f(t)-f(x)|,
we denote the usual modulus of continuity of f on the closed interval [0,a]. We know that, for a function f∈Cx2[0,∞), the modulus of the continuity ωa(f,δ) tends to zero.Now we give a rate of convergence theorem for the operatorPn,γκ.Theorem 7.
Letf∈Cx2[0,∞) and ωa+1(f,δ) be its modulus of continuity on the finite interval [0,a+1]⊂[0,∞), where a>0. Then, for n>γ,
(30)∥Pn,γκ(f,·)-f∥C[0,a]≤6Mf(1+a2)δκ,n,γ(x)+ωa+1(f,δκ,n,γ(x)).Proof.
Forx∈[0,a] and t>a+1, since t-x>1, we have
(31)|f(t)-f(x)|≤Mf(2+x2+t2)≤Mf(2+3x2+2(t-x)2)≤6Mf(1+a2)(t-x)2.
For x∈[0,a] and t≤a+1, we have
(32)|f(x)-f(t)|≤ωa+1(f,|t-x|)≤(1+|t-x|δ)ωa+1(f,δ),
with δ>0.
From (31) and (32) we can write
(33)|f(t)-f(x)|≤6Mf(1+a2)(t-x)2+(1+|t-x|δ)ωa+1(f,δ),
for x∈[0,a] and t≥0. Thus,
(34)|Pn,γκ(f,x)-f(x)|≤Pn,γκ(|f(t)-f(x)|,x)≤6Mf(1+a2)Pn,γκ((t-x)2,x)+ωa+1(f,δ)(1+1δPn,γκ((t-x)2,x)1/2).
Hence, by Schwarz’s inequality and Lemma 3, for x∈[0,a],
(35)|Pn,γκ(f,x)-f(x)|≤6Mf(1+a2)δκ,n,γ(x)+ωa+1(f,δ)(1+1δδκ,n,γ(x)).
By taking δ=δκ,n,γ(x), we get
(36)∥Pn,γκ(f,·)-f∥C[0,a]≤6Mf(1+a2)δκ,n,γ(x)+ωa+1(f,δκ,n,γ(x)),
which proves the theorem.Now we will discuss the weighted approximation theorem, where the approximation formula holds true on the interval[0,∞).Theorem 8.
Iff∈Cx2*[0,∞), limn→∞κn=0, and n>γ, then,
(37)limn→∞∥Pn,γκn(f,·)-f∥x2=0.Proof.
Using the theorem in [12] we see that it is sufficient to verify the following three conditions:
(38)limn→∞∥Pn,γκn(tr,x)-xr∥x2=0,r=0,1,2.
Since Pn,γκn(1,x)=1, the first condition of (38) is fulfilled for r=0. By Lemma 2 we have
(39)∥Pn,γκn(t,x)-x∥x2=supx∈[0,∞)|Pn,γκn(t,x)-x|1+x2≤|x1-κn-x|supx∈[0,∞)x1+x2≤κnx1-κn,
and the second condition of (38) holds for r=1 as n→∞ with κn→0.
Similarly, we can write, forn>γ,
(40)∥Pn,γκn(t2,x)-x2∥x2=supx∈[0,∞)|Pn,γκn(t2,x)-x2|1+x2≤|n(1-κn)2(n-γ)-1|supx∈[0,∞)x21+x2+|(1+(1-κn)2)(1-κ)3(n-γ)|supx∈[0,∞)x1+x2≤|κn2(n-γ)+γ(1-κn)2(n-γ)|+|(1+(1-κn)2)(1-κn)3(n-γ)|,
which implies that
(41)limn→∞∥Pn,γκn(t2,x)-x2∥x2=0withκn⟶0.
Thus, the proof is completed.


## 5. Better Error Approximation

In this section, we modified operator (5), in such way that the linear functions are preserved. The technique, which replaced x by appropriate function, was studied for many operators, for example, Bernstein, Szász, Szász-Beta operators, and so on [13–20].We start by defining(42)rκ(x)=(1-κ)x.
We note that rκ(x)∈[0,∞), for any 0≤κ<1. By replacing x by rκ(x) we give the following modification of the operators Pn,γκn:
(43)Pn,γ*κ(f,x)=∑i=1∞‍wκ(i,nrκ(x))×∫0∞‍bn,i,γ(t)f(t)dt+e-nrκ(x)f(0),
where
(44)wκ(i,nrκ(x))=nrκ(x)(nrκ(x)+iκ)i-1e-(nrκ(x)+iκ)i!
and x∈[0,∞), n>γ; the term bn,i,γ(t) is given in (5).Lemma 9.
Forx∈[0,∞) and n>γ, one has  (i)
Pn,γ*κ(1,x)=1,(ii)
Pn,γ*κ(t,x)=x,(iii)
Pn,γ*κ(t2,x)=nx2/(n-γ)+((2-2κ+κ2)x)/((1-κ)2(n-γ)).Lemma 10.
Forx∈[0,∞), n>γ, and with φx=t-x, one has  (i)
Pn,γ*κ(φx,x)=0,(ii)
Pn,γ*κ(φx2,x)=x2γ/(n-γ)+((2-2κ+κ2)x)/((1-κ)2(n-γ)).Lemma 11.
Forx∈[0,∞), n>γ, one has
(45)Pn,γ*κ(φx2,x)≤2+γ(1-κ)2(n-γ)(x+x2)=τκ,n,γ(x)(say).Proof.
Sincemax{x,x2}≤x+x2 and (1-κ)2≤1, we have
(46)Pn,γ*κ(φx2,x)≤(2-2κ+κ2)+γ(1-κ)2(1-κ)2(n-γ)(x+x2)=((1-κ)2+1)+γ(1-κ)2(1-κ)2(n-γ)(x+x2)≤2+γ(1-κ)2(n-γ)(x+x2),
which is required.Theorem 12.
Letf∈CB[0,∞). Then for x∈[0,∞) and n>γ, one has
(47)|Pn,γ*κ(f,x)-f(x)|≤Mω2(f,τκ,n,γ(x)).Proof.
Letg∈W∞2 and x∈[0,∞). Using Taylor’s expansion
(48)g(t)=g(x)+g′(x)(t-x)+∫xt‍(t-u)g′′(u)du,t∈[0,∞)
and Lemma 10, we have
(49)Pn,γ*κ(g,x)-g(x)=Pn,γ*κ(∫xt‍(t-u)g′′(u)du).
Also, |∫xt‍(t-u)g′′(u)du|≤(t-x)2∥g′′∥. Thus,
(50)|Pn,γ*κ(g,x)-g(x)|≤Pn,γ*κ((t-x)2,x)∥g′′(x)∥=(2-2κ+κ2)x+x2γ(1-κ)2(1-κ)2(n-γ)∥g′′∥.
Since |Pn,γ*κ(f,x)|≤∥f∥,
(51)|Pn,γ*κ(f,x)-f(x)|≤|Pn,γ*κ(f-g,x)-(f-g)(x)|+|Pn,γ*κ(g,x)-g(x)|≤2∥f-g∥+(2-2κ+κ2)x+x2γ(1-κ)2(1-κ)2(n-γ)∥g′′∥≤2∥f-g∥+τκ,n,γ(x)∥g′′∥.
Finally taking the infimum on right side over all g∈W∞2, we get
(52)|Pn,γ*κ(f,x)-f(x)|≤K2(f,τk,n,γ).
In view of (18), we obtain
(53)|Pn,γ*κ(f,x)-f(x)|≤Cω2(f,τk,n,γ),
which proves the theorem.Remark 13.
We claim that the error estimation in Theorem12 is better than that of (20), provided f∈C[0,∞) and x>0. Indeed, in order to get this better estimation we must show that τκ,n,γ(x)≤δκ,n,γ(x)+κx/(1-κ). One can obtain that
(54)δκ,n,γ(x)+κx1-κ≤(n+2+κ(1-κ)2(n-γ)(1-κ)3(n-γ))(x+x2)≤(2n+2-γ(1-κ)3(n-γ))(x+x2).
Also,
(55)τκ,n,γ(x)≤δκ,n,γ(x)+κx1-κ⟺2+γ(1-κ)2(n-γ)(x+x2)≤2n+2-γ(1-κ)3(n-γ)(x+x2)⟺(2+γ)≤2n+2-γ(1-κ)⟺2-2κ+γ-γκ≤(2n+2-γ)⟺2(n-γ)+(2+γ)κ≥0,
which holds true, with n>γ>0 and κ>0. Thus, τκ,n,γ(x)≤δκ,n,γ(x)+κx/(1-κ).


## 6. Stancu Approach

In 1968, Stancu introduced Bernstein-Stancu operator, which is a linear positive operator with two parametersα and β satisfying the condition 0≤α≤β. Inspired by the Stancu-type generalization of Bernstein operator and the recent important work on several other operators are discuss in [21–27], we propose following modification of the operator Pn,γκ as
(56)P~n,γκ,α,β(f,x)=∑i=1∞‍wκ(i,nx)∫0∞‍bn,i,γ(t)f(nt+αn+β)dt+e-nxf(αn+β),
where wκ(i,nx) and bn,i,γ(t) are defined in (5).Lemma 14.
ForP~n,γκ,α,β(ts,x),  s=0,1,2, the following inequalities holds:  (i)
P~n,γκ,α,β(1,x)=1,(ii)
P~n,γκ,α,β(t,x)=(nx+α(1-κ))/((n+β)(1-κ)),(iii)
P~n,γκ,α,β(t2,x) = n3x2/((n+β)2(n-γ)(1-κ)2) + (nx(2n+2 (-n+2α(n-γ))κ +(n-4α(n-γ))κ2 +2α(n-γ)κ3)) / ((n+β)2(n-γ)(1-κ)3) + α2/(n+β)2.The proof of the above lemma can be obtained by using linearity of operators and Lemma2.Lemma 15.
If one denotes central moments byΦn,m,γκ,α,β(x)=P~n,γκ,α,β((t-x)m,x),  m=1,2, then one has
(57)Φn,1,γκ,α,β(x)=x(nκ+βκ-β)(n+β)(1-κ)+α(n+β),Φn,2,γκ,α,β(x)=(((n+β)2(n-γ)(1-κ)2)-1(n3κ2+n2(-2βκ+2βκ2+γ(1-κ2))+nβ(β(1-κ)2+2γκ(1-κ))-β2γ(1-κ)2)×((n+β)2(n-γ)(1-κ)2)-1(n3κ2+n2(-2βκ+2βκ2+γ(1-κ2)))x2+(((n+β)2(n-γ)(1-κ)3)-1(n2((1-κ)2+1+2α(2κ3-5κ2+4κ-1)))×((n+β)2(n-γ)(1-κ)3)-1+((1-κ)3(-2β+4γ)κ3nα((-2β+4γ)κ32β-2γ+(-6β+8γ)κ+(6β-10γ)κ2+(-2β+4γ)κ3)+2αβγ(1-κ)3n2((1-κ)2+1+2α(2κ3-5κ2+4κ-1)))×((n+β)2(n-γ)(1-κ)3)-1(n2((1-κ)2+1+2α(2κ3-5κ2+4κ-1))))x+α2(n+β)2.Theorem 16.
LetP~n,γκn,α,β with n>γ be defined in (56), where limn→∞κn=0. For any compact A⊂[0,∞) and for each f∈Cx2*[0,∞), one has
(58)limn→∞P~n,γκn,α,β(f,x)=f(x),uniformlyinx∈A.The proof is based on Korovkin’s criterion and Lemma14.Theorem 17.
Letf∈CB[0,∞) and n>γ, one has
(59)|P~n,γκ,α,β(f,x)-f(x)|≤ω(f,Φn,1,γκ,α,β(x))+Bω2(f,Φn,1,γκ,α,β(x)+Φn,2,γκ,α,β(x)),
for every 0≤α≤β and x∈[0,∞), where B is a positive constant.The proof of Theorem17 is just similar to Theorem 6.Now, we establish the Voronovskaja-type asymptotic formula for the operatorsP~n,γκ,α,β(f,x).Theorem 18.
Letf be bounded and integrable on [0,∞), first and second derivatives of f exist at a fixed point x∈[0,∞), and κn∈(0,1) such that κn→0 as n→∞; then
(60)limn→∞n(P~n,γκn,α,β(f,x)-f(x))=(α-βx)f′(x)+12(γx2+2(1-α)x)f′′(x).Proof.
Letf, f′, f′′∈Cx2*[0,∞) and x∈[0,∞) be fixed. By Taylor’s expansion we can write
(61)f(t)=f(x)+f′(x)(t-x)+12f′′(x)(t-x)2+r(x,t)(t-x)2,
where r(t,x) is Peano form of the remainder, r(·,x)∈Cx2*[0,∞), and limt→xr(t,x)=0.
ApplyingP~n,γκn,α,β to the previous, we obtain
(62)n[P~n,γκn,α,β(f,x)-f(x)]=nf′(x)P~n,γκn,α,β(t-x,x)+12nf′′(x)P~n,γκn,α,β((t-x)2,x)+nP~n,γκn,α,β(r(t,x)(t-x)2,x).
By Cauchy-Schwarz’s inequality, we have
(63)P~n,γκn,α,β(r(t,x)(t-x)2,x)≤P~n,γκn,α,β(r(t,x)2,x)P~n,γκn,α,β((t-x)4,x).
Observe that r2(x,x)=0 and r2(·,x)∈Cx2*[0,∞). Then it follows that
(64)limn→∞nP~n,γκn,α,β(r(t,x)2,x)=r2(x,x)=0,
uniformly with respect to x∈[0,A].
Now, from (63) and (64), we obtain limn→∞nP~n,γκn,α,β(r(t,x)(t-x)2,x)=0.
Usingκn→0 as n→∞, we obtain
(65)limn→∞n(Φn,1,γκn,α,β(x))=α-βx,limn→∞n(Φn,2,γκn,α,β(x))=γx2+2(1-α)x.
Using above limits, we have
(66)limn→∞n[P~n,γκn,α,β(f,x)-f(x)]=limn→∞n[12f′(x)P~n,γκn,α,β(t-x,x)+12f′′(x)P~n,γκn,α,β((t-x)2,x)]+limn→∞n[P~n,γκn,α,β(r(t,x)(t-x)2,x)]=(α-βx)f′(x)+12(γx2+2(1-α)x)f′′(x),
which proves the theorem.Remark 19.
In particular, ifα=β=0 and γ=1, then the operators P~n,γκn,α,β(f,x), κn∈(0,1) such that κn→0 as n→∞, reduce to the Jain-Beta operators recently introduced by Tarabie [7]. We obtain the following conclusion of the above asymptotic formula for the Jain-Beta operator in the ordinary approximation as follows:
(67)limn→∞n[P~n,1κn,0,0(f,x)-f(x)]=12(x2+2x)f′′(x).



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*Source: 489249-2013-12-26.xml*