# Simultaneous Approximation for Generalized Srivastava-Gupta Operators

Authors: Tuncer Acar; Lakshmi Narayan Mishra; Vishnu Narayan Mishra
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We introduce a new Stancu type generalization of Srivastava-Gupta operators to approximate integrable functions on the interval0,∞ and estimate the rate of convergence for functions having derivatives of bounded variation. Also we present simultenaous approximation by new operators in the end of the paper.


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## Body

## 1. Introduction

To approximate integrable functions on the interval0,∞, Srivastava and Gupta [1] introduced a general sequence of linear positive operators Gn,c as follows:(1)Gn,cf;x=n∑k=1∞‍pn,kx;c∫0∞‍pn+c,k-1t;cftdt+pn,0x;cf0,for a function f∈Hα0,∞, where Hα0,∞α≥0 is the class of locally integrable functions defined on 0,∞ and satisfying the growth condition (2)ft≤MtαM>0;α≥0;t⟶∞,(3)pn,kx;c=-xkk!ϕn,ckx,(4)ϕn,cx=e-nx,c=01+cx-n/c,c∈N:=1,2,3,….The general sequence of operatorsGn,c has many interesting properties in approximation theory, which is an interesting area of research in the present era, and several researchers have studied these operators; we can mention some important studies on these operators (see [1–3]). In [4], author introduced King and Stancu type generalization of Srivastava-Gupta operators and presented some direct results. Also, Verma and Agrawal [5] introduced a new generalization of Srivastava-Gupta operators and studied the rate of convergence for the functions having the derivatives of bounded variation (BV). The rate of convergence for the functions having the derivatives of (BV) is an active area of research and many researchers studied this direction. We refer the readers to [6–10] and references therein.Stancu [11, 12] introduced generalizations of Bernstein polynomials with one and two parameters (resp.), satisfying the condition 0≤α≤β, as(5)snαf,x=∑k=0nfknnk∏s=0k-1x+αs∏s=0n-k-11-x+αs∏s=0n-11+αskkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk0≤x≤1,snα,βf,x=∑k=0nfk+αn+βnkxk(1-x)n-kkkkkkkkkkkkkkkkkkkkkkkkkkkk0≤x≤1,for any f∈C0,1. Stancu type generalization of approximation operators present better approach depending on α,β. Therefore, this kind of generalizations and their approximation properties have been studied intensively. We refer the readers to [13–17] and references therein. Mishra et al. [18, 19], V. N. Mishra, and L. N. Mishra [20] have established very interesting results on approximation properties of various functional classes using different types of positive linear summability operators.The purpose of this paper is to introduce a new Stancu type generalization of the operators defined in [5] as(6)Gn,r,cα,βf;x=nΓn/c+rΓn/c-r+1Γn/c+1Γn/c∑k=0∞pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1t;cfnt+αn+βdt.By the definition of operators, it is clear that Gn,r,cα,βf;x is positive and linear. For α=β=0, Gn,r,c0,0f;x reduces to operators defined in [5]. In this study we obtain the rate of convergence for the functions having the derivatives of bounded variation. Also, in the end of the paper, we study the simultaneous approximation.


## 2. Auxiliary Results

In order to prove our main results, we need the following lemmas.Lemma 1.
Let themth order moment be defined as(7)Un,r,mα,βx=Gn,cα,βt-xm;x=n-rc∑k=0∞‍pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xmdt,where n,m∈N∪0, and then, for n>m+r+1c, we have the following recurrence relation:(8)n-r+m+1cn+βUn,r,m+1x=nx1+cxUn,r,mα,βx′+mUn,r,m-1α,βx+Un,r,mα,βx×m+r+n+rcxn+α-n+βxkkkkkk×n-r+2m+1cefm+r+n+rcxn+α-n+βx+Un,r,m-1α,βx×cmα-n+βx2-mnα-n+βxn+β,Un,r,0α,βx=1,Un,r,1α,βx=α-n+βxn+β+nr+n+rcxn-r+1cn+β,Un,r,2α,βx=nx1+cxn-r+1cn-r+2cn+β2kkkkkk+αn+β-x+nr+n+rcxn-r+1cn+βkkkkkk×n1+r+n+rcxn-r+2cn+βkkkkkk+αn+β-x+nr+n+rcxn-r+1cn+β2kkkkkk×α-n+βxkkkkkk+α-n+βxcα-n+βx-nn-r+2cn+β2.Furthermore, Un,r,mα,βx is polynomial of degree m in x and (9)Un,r,mα,βx=On+β-m+1/2.Proof.
By definition ofUn,r,mα,βx, taking the derivative of Un,r,mα,βx, we get (10)Un,r,mα,βx′=-n-rcm∑k=0∞‍pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xm-1dt+n-rc∑k=0∞‍pn+rc,k′x;c×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xmdt=-mUn,r,m-1α,βx+n-rc∑k=0∞‍pn+rc,k′x;c×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xmdt.Hence, using the identity(11)x1+cxpn+rc,k′x;c=k-n+rcxpn+rc,kx;cwe have(12)x1+cxUn,r,mα,βx′+mUn,r,m-1α,βx=n-rc∑k=0∞‍k-n+rcxpn+rc,kx;c×∫0∞pn-r-1c,k+r-1t;cnt+αn+β-xmdt=n-rc∑k=0∞‍kpn+rc,kx;c×∫0∞pn-r-1c,k+r-1t;cnt+αn+β-xmdt-n+rcxUn,r,mα,βx=I-n+rcxUn,r,mα,βx.We can write I as(13)I=nt+αn+β-xmn-rc∑k=0∞‍pn+rc,kx;ckkk×∫0∞k+r-1-n-r-1ctpn-r-1c,k+r-1kkkkkkkk×t;cnt+αn+β-xmdtkkk+n-rcn-r-1c∑k=0∞‍pn+rc,kx;ckkk×∫0∞pn-r-1c,k+r-1t;ctnt+αn+β-xmdtkkk-r-1n-rc∑k=0∞‍pn+rc,kx;ckkk×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xmdt∑k=0∞‍=I1+I2-r-1Un,r,mα,βx.To estimate I2 using t=n+β/nnt+α/n+β-x-α/n+β-x, we have (14)I2=n-r-1cn+βn×n-rc∑k=0∞‍pn+rc,kx;ckki×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xm+1dtkk-αn+β-xn-rc∑k=0∞‍pn+rc,kx;ckk×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xmdt∑k=0∞‍,I2=n-r-1cn+βn×n-rc∑k=0∞‍pn+rc,kx;cki×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xm+1dtki-αn+β-xki×n-rc∑k=0∞‍pn+rc,kx;ckkkkkkkkk×∫0∞‍pn-r-1c,k+r-1t;ckkkkkkkkkkkkk×nt+αn+β-xmdt∑k=0∞‍=n-r-1cn+βn×Un,r,m+1α,βx-αn+β-xUn,r,mα,βx.Next to estimate I1 using the equality(15)t1+ctpn-r-1c,k+r-1′t;c=k+r-1-n-r-1ctpn-r-1c,k+r-1t;c,we have(16)I1=n-rc∑k=0∞‍pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1′t;ctnt+αn+β-xmdt+cn-rc∑k=0∞‍pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1′t;ct2nt+αn+β-xmdt=J1+J2.Putting t=((n+β)/n)[(nt+α/n+β-x)-(α/n+β-x)], we get(17)J1=n+βn×n-rc∑k=0∞‍pn+rc,kx;ck×∫0∞pn-r-1c,k+r-1′t;cnt+αn+β-xm+1dtk-αn+β-xn-rc∑k=0∞‍pn+rc,kx;ck×∫0∞pn-r-1c,k+r-1′t;cnt+αn+β-xmdt∑k=0∞‍.Now integrating by parts, we get(18)J1=-m+1n-rc∑k=0∞‍pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xmdt+mαn+β-xn-rc∑k=0∞‍pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xm-1dt=-m+1×nt+αn+β-xmn-rc∑k=0∞‍pn+rc,kx;ck×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xmdt+mαn+β-x×n-rc∑k=0∞‍pn+rc,kx;ck×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xm-1∑k=0∞‍dt=-m+1Un,r,mα,βx+mαn+β-xUn,r,m-1α,βx.Proceeding in a similar manner, we obtain the estimate J2 as(19)J2=-cm+2n+βnUn,r,m+1x+2cm+1n+βnαn+β-xUn,r,mx-cmn+βnαn+β-x2Un,r,m-1α,βx.Combining the equations, we have(20)n-r+m+1cn+βUn,r,m+1α,βx=nx1+cxUn,r,mα,βx′+mUn,r,m-1α,βx+Un,r,mα,βx×m+r+n+rcxn+α-n+βxkkk×n-r+2m+1cα-n+βx+Un,r,m-1α,βx×cmα-n+βx2-mnα-n+βxn+βwhich is the desired result.
Moments form=0,1,2 can be easily obtained by using the above recurrence relation.Remark 2.
For sufficiently largen, C>2, and x∈0,∞, it can be seen from Lemma 1 that (21)Un,r,2α,βx≤Cσr,cα,βxn+β,where σr,cα,βx=x1+cx+xα+βx+r1+cx for the convenient notation.Remark 3.
By using Cauchy-Schwarz inequality, it follows from Remark2 that, for sufficiently large n, C>2, and x∈0,∞,(22)n-rc∑k=0∞‍pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1t;cnt+αn+β-xdt≤Un,r,2α,βx1/2≤Cσr,cα,βxn+β.Lemma 4.
Letx∈0,∞ and C>2; then, for sufficiently large n, we have(23)λn,rx,y=n-rc∑k=0∞‍pn+rc,kx;ckkkkkkkk×∫0y‍pn-r-1c,k+r-1t;cdtλn,rx,y≤Cx1+cxnx-y2,0≤y≤x,1-λn,rx,z=n-rc∑k=0∞‍pn+rc,kx;ckkkkkkkk×∫z∞‍pn-r-1c,k+r-1t;cdt1-λn,rx,z≤Cx1+cxnz-x2,x≤z≤∞.Proof.
We give the proof for only first inequality, and the other is similar. Using Remark2 with α=β=0, for sufficiently large n and 0≤y≤x and nt+α/n+β≤t, we have(24)λn,rx,y=n-rc∑k=0∞‍pn+rc,kx;c×∫0y‍pn-r-1c,k+r-1t;cdt≤n-rc∑k=0∞‍pn+rc,kx;c×∫0y‍pn-r-1c,k+r-1t;ct-x2y-x2dt≤Cx1+cxnx-y2.Lemma 5.
Supposef is s times differentiable on 0,∞ such that f(s-1)(t)=O(tα), for some integer α>0 as t→∞. Then, for any r,s∈N0, and n>max⁡{α,r+s}, we have(25)DsGn,r,cα,βf;x=nn+βsGn,r+s,cα,βf;x(Dsf,x).Proof.
Using the identity(26)pn,k′x=npn+c,k-1x,c-pn+c,kx,c.One can observe that, even in case k=0, the above identity is true with the condition pn+c,negativex,c=0. Thus, applying (26), we have(27)DGn,r,cα,βf;x=nΓn/c+rΓn/c-r+1Γn/c+1Γn/c∑k=0∞‍Dpn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1t;cfnt+αn+βdt=nΓn/c+rΓn/c-r+1Γn/c+1Γn/c×∑k=0∞‍n+rcpn+r+1c,k-1x,c-pn+r+1c,kx,c×∫0∞‍pn-r-1c,k+r-1t;cfnt+αn+βdt=nn+rcΓn/c+rΓn/c-r+1Γn/c+1Γn/c×∑k=0∞‍pn+r+1c,kx,c×∫0∞‍pn-r-1c,k+rt;c-pn-r-1c,k+r-1t;ckkkkk×fnt+αn+βdt=-nn+rcΓn/c+rΓn/c-r+1n-rcΓn/c+1Γn/c×∑k=0∞‍pn+r+1c,kx,c×∫0∞‍Dpn-rc,k+rt;cfnt+αn+βdt=n2Γn/c+r+1Γn/c-rn+βΓn/c+1Γn/c×∑k=0∞‍pn+r+1c,kx,c×∫0∞‍pn-rc,k+rt;cDfnt+αn+βdt=nn+βGn,r+1,cα,βDf;x,which means that the identity is satisfied for s=1. Let us suppose that the result holds for s=m; that is,(28)DmGn,r,cα,βf;x=nn+βmGn,r+m,cα,βf;xDmf,x=nn+βm×nΓn/c+r+mΓn/c-r-m+1Γn/c+1Γn/c×∑k=0∞‍pn+r+mc,kx;c×∫0∞‍pn-r+m-1c,k+r+m-1t;cDmfnt+αn+βdt.Also, from (26) we can write(29)Dm+1Gn,r,cα,βf;x=nn+βm×nΓn/c+r+mΓn/c-r-m+1Γn/c+1Γn/c×∑k=0∞‍Dpn+r+mc,kx;c×∫0∞‍pn-r+m-1c,k+r+m-1t;cDmfnt+αn+βdt=nn+βm×nΓn/c+r+mΓn/c-r-m+1Γn/c+1Γn/c×∑k=0∞‍n+r+mc×pn+r+m+1c,k-1x,c-pn+r+m+1c,kx,c×∫0∞‍pn-r+m-1c,k+r+m-1t;cDmfnt+αn+βdt=nn+βm×cnΓn/c+r+m+1Γn/c-r-m+1Γn/c+1Γn/c×∑k=0∞‍pn+r+m+1c,kx;c×∫0∞‍pn-r+m-1c,k+r+mt;ckk-pn-r+m-1c,k+r+m-1t;cDmfnt+αn+βdt=-nn+βm×cnΓn/c+r+m+1Γn/c-r-m+1Γn/c+1Γn/c×∑k=0∞‍pn+r+m+1c,kx;c×∫0∞‍Dpn-r+mc,k+r+mt;cn-r+m-1cDmfnt+αn+βdtand, integrating by parts the last integral, we have(30)Dm+1Gn,r,cα,βf;x=nn+βm+1×nΓn/c+r+m+1Γn/c-r-mΓn/c+1Γn/c×∑k=0∞‍pn+r+m+1c,kx;c×∫0∞‍pn-r+mc,k+r+mt;cDm+1fnt+αn+βdt.Hence we have(31)Dm+1Gn,r,cα,βf;x=nn+βm+1×Gn,r+m+1,cα,βf;x(Dm+1f,x),in which the result is true for s=m+1, and hence by mathematical induction the proof of the lemma is completed.


## 3. Main Results

Throughout the paper byDBq(0,∞) we denote the class of absolutely continuous functions f on 0,∞ (where q is a some positive integer) satisfying the conditions:(i)
ft≤C1tq and C1>0,(ii)
the functionf has the first derivative on the interval (0,∞) which coincide almost everywhere with a function which is of bounded variation on every finite subinterval of (0,∞). It can be observed that for all functions f∈DBq0,∞ we can have the representation(32)fx=fc+∫cx‍ψtdt,x≥c≥0.Theorem 6.
Letf∈DBq(0,∞), q>0, and x∈(0,∞). Then, for C>2 and sufficiently large n, we have(33)Γn/c2Γn/c+rΓn/c-rGn,r,cα,βf;x-fx≤C1+cxn∑k=1[n]‍⋁x-x/kx+(x/k)fx′x+xn⋁x-(x/n)x+(x/n)fx′x+C1+cxnxf2x-fx-xf′x++fx+On-q+f′x+Cx1+cxn+Cσr,cα,βxn+βf′x+-f′x-2+f′x++f′x-2×α-βxn-cr+1+2nrcx+nxc+nrn-r+1cn+β,where C is a constant which may be different on each occurrence.Proof.
Using the mean value theorem, we have(34)Γn/c2Γn/c+rΓn/c-rGn,r,cα,βf;x-fx=n-rc∑k=0∞‍pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1t;cfnt+αn+β-fxdt=∫0∞∫xnt+α/n+β‍n-rc∑k=0∞‍pn+rc,kx;c‍∑k=0∞‍×pn-r-1c,k+r-1t;cf′udu∫xnt+α/n+β‍dt.Also, using the identity(35)f′u=f′x++f′x-2+f′x(u)+f′x+-f′x-2sgnu-x+f′x-f′x++f′x-2χx(u),where (36)χxu=1,u=x;0,u≠x,we have(37)n-rc∑k=0∞‍pn+rc,kx;c×∫0∞‍∫xt‍f′x-f′x++f′x-2χxudu×pn-r-1c,k+r-1t;cdt=0.Thus, using the above identities, we can write(38)Γn/c2Γn/c+rΓn/c-rGn,r,cα,βf;x-fx≤∫0∞‍∫xtn-rc‍∑k=0∞‍pn+rc,kx;ci×pn-r-1c,k+r-1t;ci×f′x++f′x-2+f′xudu∑k=0∞‍dt+∫0∞‍∑k=0∞‍f′x+-f′x-2sgnu-x∫xt‍n-rc∑k=0∞‍pn+rc,kx;ckkkkkk×pn-r-1c,k+r-1t;ckkkkkk×f′x+-f′x-2sgnu-xdu∑k=0∞‍dt.Also, it can be verified that (39)∫0∞∫xt‍n-rc∑k=0∞‍pn+rc,kx;c‍kkk×pn-r-1c,k+r-1t;ckkk∑k=0∞‍×f′x+-f′x-2sgnu-xdudt≤f′x+-f′x-2Un,r,2x1/2,(40)∫0∞‍∫xt‍n-rc∑k=0∞‍pn+rc,kx;ckkk×pn-r-1c,k+r-1t;ckkk×f′x++f′x-2dudt≤f′x++f′x-2Un,r,1x.Combining (38)–(40), we get(41)Γn/c2Γn/c+rΓn/c-rGn,r,cα,βf;x-fx≤∫x∞‍∫xt‍fx′udun-rckkkk×∑k=0∞‍pn+rc,kx;cpn-r-1c,k+r-1t;cdtk+∫0x‍∫xt‍fx′udun-rckkkkk×∑k=0∞‍pn+rc,kx;cpn-r-1c,k+r-1t;cdt∫x∞‍∫xt‍fx′udu+f′x+-f′x-2Un,r,2x1/2+f′x++f′x-2Un,r,1x=An,rα,βf,x+Bn,rα,βf,x+f′x+-f′x-2×Un,r,2x1/2+f′x++f′x-2Un,r,1x.Applying Remark 2 and Lemma 1 in above equation, we have(42)Γn/c2Γn/c+rΓn/c-rGn,r,cα,βf;x-fx≤An,rα,βf,x+Bn,rα,βf,x+Cσr,cα,βxn+βf′x+-f′x-2+f′x++f′x-2×α-βxn-cr+1+2nrcx+nxc+nrn-r+1cn+β.In order to complete the proof of the theorem, it suffices to estimate the terms An,rα,βf,x and Bn,rα,βf,x. Applying Remark 2 with α=β=0, we get (43)An,rα,βf,x=∫x∞‍∫xt‍fx′udun-rckki×∑k=0∞‍pn+rc,kx;cpn-r-1c,k+r-1t;cdt∫x∞‍∫xt‍fx′udu≤n-rc∑k=0∞‍pn+rc,kx;cii×∫2x∞‍ft-fxpn-r-1c,k+r-1t;cdt∑k=0∞‍+f′x+ii×n-rc∑k=0∞‍pn+rc,kx;cii∑k=0∞‍×∫x2x‍pn-r-1c,k+r-1t;ct-xdtii+∫x2x‍fx′udu1-λn,rx,2xii+∫x2x‍fx′t·1-λn,rx,tdt≤n-rc∑k=0∞‍pn+rc,kx;cii×∫2x∞‍pn-r-1c,k+r-1t;cC1t2qdtii+fxx2n-rc∑k=0∞‍pn+rc,kx;cii×∫0∞‍pn-r-1c,k+r-1t;ct-x2dtii+f′x+n-rc∑k=0∞‍pn+rc,kx;cii×∫2x∞‍pn-r-1c,k+r-1t;ct-xdtii+Cx1+cxnx2f2x-x-xf′x+ii+C1+cxn∑k=1[n]‍⋁xx+x/kfx′xii+xn⋁xx+(x/n)fx′x.
For estimating the integral(44)n-rc∑k=0∞‍pn+rc,kx;c∫2x∞‍pn-r-1c,k+r-1t;cC1t2qdt,we proceed as follows: since t≥2x implies that t≤2t-x so by Schwarz inequality and Lemma 1,(45)n-rc∑k=0∞‍pn+rc,kx;c∫2x∞‍pn-r-1c,k+r-1t;cC1t2qdt≤C12qn-rc∑k=0∞‍pn+rc,kx;c×∫0∞‍pn-r-1c,k+r-1t;cC1t-x2qdt≤C12qUn,r,2qx=On-qasn⟶∞.By using Hölder’s inequality and Remark 2 (α=β=0), we get the estimate as follows: (46)f′x+n-rc∑k=0∞‍pn+rc,kx;c×∫2x∞‍pn-r-1c,k+r-1t;ct-xdt≤f′x+×n-rc∑k=0∞‍pn+rc,kx;ckkk×∫0∞‍pn-r-1c,k+r-1t;ct-x2dt∑k=0∞‍1/2≤f′x+Cx(1+cx)n.Collecting the estimates from (43)–(46), we obtain(47)An,rα,βf,x≤On-q+f′x+×Cx(1+cx)n+C(1+cx)nx×f2x-fx-xf′x++fx+C1+cxn∑k=1n⋁xx+x/kfx′x+xn⋁xx+(x/n)fx′x.On the other hand, to estimate Bn,rα,βf,x by applying Lemma 4 with y=x-x/n and integration by parts, we have(48)Bn,rα,βf,x=∫0x‍∫xt‍fx′udtλn,rx,t≤∫0y‍+∫yx‍fx′tλn,rx,tdt≤Cx1+cxn∫0y‍⋁txf′x1x-t2dt+∫yx‍⋁txf′xdt=Cx1+cxn∫1n‍⋁x-x/uxf′xdu+xn⋁x-(x/n)xf′x≤Cx1+cxn∑k=1[n]‍⋁x-(x/k)xf′x+xn⋁x-(x/n)xf′x,where u=x/x-t.
Combining (41), (47), and (48), we get the desired result.Corollary 7.
Letfs∈DBq(0,∞), q>0, and x∈(0,∞). Then, for C>2 and n sufficiently large, one has(49)Γn/c2Γn/c+rΓn/c-rn+βnsk×DsGn,r,cα,βf;x-fsxΓn/c2Γn/c+rΓn/c-r≤C(1+cx)n∑k=1[n]‍⋁x-(x/k)x+(x/k)Ds+1fx+xn⋁x-(x/n)x+(x/n)Ds+1fx+C(1+cx)nx×f2x-x-xDs+1fx++fx+On-q+Ds+1fx+Cx1+cxn+Cσr,cα,βxn+βDs+1fx+-Ds+1fx-2+Ds+1fx++Ds+1fx-2×α-βxn-cr+1+2nrcx+nxc+nrn-r+1cn+β,where ⋁abfx denotes the total variation of fx on a,b and the auxiliary function Ds+1fx is defined by(50)Ds+1fxt=Ds+1ft-Ds+1fx-,0≤t≤x0,t=xDs+1ft-Ds+1fx+,x<t<∞.


## 4. Conclusion

The results of our lemmas and theorems are more general rather than the results of any other previously proved lemmas and theorems, which will enrich the literature of applications of quantum calculus in operator theory and convergence estimates in the theory of approximations by positive linear operators. The researchers and professionals working or intend to work in areas of mathematical analysis and its applications will find this research paper to be quite useful. Consequently, the results so established may be found useful in several interesting situations appearing in the literature on mathematical analysis, pure and applied mathematics, and mathematical physics. Some interesting applications of the positive approximation linear operators can be seen in [21–24].



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