JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2015/936308 936308 Research Article Simultaneous Approximation for Generalized Srivastava-Gupta Operators Acar Tuncer 1 Mishra Lakshmi Narayan 2, 3 Mishra Vishnu Narayan 4 Banaś Józef 1 Department of Mathematics, Faculty of Science and Arts Kirikkale University Yahsihan, 71450 Kirikkale Turkey kku.edu.tr 2 Department of Mathematics National Institute of Technology Cachar District, Silchar, Assam 788 010 India nits.ac.in 3 L. 1627 Awadh Puri Colony Beniganj, Phase III, Opposite to Industrial Training Institute (I.T.I.) Ayodhya Main Road, Faizabad, Uttar Pradesh 224 001 India 4 Applied Mathematics and Humanities Department Sardar Vallabhbhai National Institute of Technology Ichchhanath Mahadev Dumas Road, Surat District, Surat, Gujarat 395007 India svnit.ac.in 2015 642015 2015 19 05 2014 09 10 2014 642015 2015 Copyright © 2015 Tuncer Acar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce a new Stancu type generalization of Srivastava-Gupta operators to approximate integrable functions on the interval 0, 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.

1. Introduction

To approximate integrable functions on the interval 0,, Srivastava and Gupta  introduced a general sequence of linear positive operators Gn,c as follows:(1)Gn,cf;x=nk=1pn,kx;c0pn+c,k-1t;cftdt+pn,0x;cf0,for a function fHα0,, where Hα0,α0 is the class of locally integrable functions defined on 0, and satisfying the growth condition (2)ftMtαM>0;α0;t,(3)pn,kx;c=-xkk!ϕn,ckx,(4)ϕn,cx=e-nx,c=01+cx-n/c,cN:=1,2,3,.

The general sequence of operators Gn,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 ). In , author introduced King and Stancu type generalization of Srivastava-Gupta operators and presented some direct results. Also, Verma and Agrawal  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  and references therein.

Stancu [11, 12] introduced generalizations of Bernstein polynomials with one and two parameters (resp.), satisfying the condition 0αβ, as(5)snαf,x=k=0nfknnks=0k-1x+αss=0n-k-11-x+αss=0n-11+αskkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk0x1,snα,βf,x=k=0nfk+αn+βnkxk(1-x)n-kkkkkkkkkkkkkkkkkkkkkkkkkkkk0x1,for any fC0,1. 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  and references therein. Mishra et al. [18, 19], V. N. Mishra, and L. N. Mishra  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  as(6)Gn,r,cα,βf;x=nΓn/c+rΓn/c-r+1Γn/c+1Γn/ck=0pn+rc,kx;c×0pn-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 . 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 the mth order moment be defined as(7)Un,r,mα,βx=Gn,cα,βt-xm;x=n-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;cnt+αn+β-xmdt,where n,mN0, 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+1cefm+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 of Un,r,mα,βx, taking the derivative of Un,r,mα,βx, we get (10)Un,r,mα,βx=-n-rcmk=0pn+rc,kx;c×0pn-r-1c,k+r-1t;cnt+αn+β-xm-1dt+n-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;cnt+αn+β-xmdt=-mUn,r,m-1α,βx+n-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;cnt+αn+β-xmdt.Hence, using the identity(11)x1+cxpn+rc,kx;c=k-n+rcxpn+rc,kx;cwe have(12)x1+cxUn,r,mα,βx+mUn,r,m-1α,βx=n-rck=0k-n+rcxpn+rc,kx;c×0pn-r-1c,k+r-1t;cnt+αn+β-xmdt=n-rck=0kpn+rc,kx;c×0pn-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-rck=0pn+rc,kx;ckkk×0k+r-1-n-r-1ctpn-r-1c,k+r-1kkkkkkkk×t;cnt+αn+β-xmdtkkk+n-rcn-r-1ck=0pn+rc,kx;ckkk×0pn-r-1c,k+r-1t;ctnt+αn+β-xmdtkkk-r-1n-rck=0pn+rc,kx;ckkk×0pn-r-1c,k+r-1t;cnt+αn+β-xmdtk=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-rck=0pn+rc,kx;ckki×0pn-r-1c,k+r-1t;cnt+αn+β-xm+1dtkk-αn+β-xn-rck=0pn+rc,kx;ckk×0pn-r-1c,k+r-1t;cnt+αn+β-xmdtk=0,I2=n-r-1cn+βn×n-rck=0pn+rc,kx;cki×0pn-r-1c,k+r-1t;cnt+αn+β-xm+1dtki-αn+β-xki×n-rck=0pn+rc,kx;ckkkkkkkkk×0pn-r-1c,k+r-1t;ckkkkkkkkkkkkk×nt+αn+β-xmdtk=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-1t;c=k+r-1-n-r-1ctpn-r-1c,k+r-1t;c,we have(16)I1=n-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;ctnt+αn+β-xmdt+cn-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;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-rck=0pn+rc,kx;ck×0pn-r-1c,k+r-1t;cnt+αn+β-xm+1dtk-αn+β-xn-rck=0pn+rc,kx;ck×0pn-r-1c,k+r-1t;cnt+αn+β-xmdtk=0.Now integrating by parts, we get(18)J1=-m+1n-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;cnt+αn+β-xmdt+mαn+β-xn-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;cnt+αn+β-xm-1dt=-m+1×nt+αn+β-xmn-rck=0pn+rc,kx;ck×0pn-r-1c,k+r-1t;cnt+αn+β-xmdt+mαn+β-x×n-rck=0pn+rc,kx;ck×0pn-r-1c,k+r-1t;cnt+αn+β-xm-1k=0dt=-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+1cα-n+βx+Un,r,m-1α,βx×cmα-n+βx2-mnα-n+βxn+βwhich is the desired result.

Moments for m=0,1,2 can be easily obtained by using the above recurrence relation.

Remark 2.

For sufficiently large n, C>2, and x0,, it can be seen from Lemma 1 that (21)Un,r,2α,βxCσr,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 Remark 2 that, for sufficiently large n, C>2, and x0,,(22)n-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;cnt+αn+β-xdtUn,r,2α,βx1/2Cσr,cα,βxn+β.

Lemma 4.

Let x0, and C>2; then, for sufficiently large n, we have(23)λn,rx,y=n-rck=0pn+rc,kx;ckkkkkkkk×0ypn-r-1c,k+r-1t;cdtλn,rx,yCx1+cxnx-y2,0yx,1-λn,rx,z=n-rck=0pn+rc,kx;ckkkkkkkk×zpn-r-1c,k+r-1t;cdt1-λn,rx,zCx1+cxnz-x2,xz.

Proof.

We give the proof for only first inequality, and the other is similar. Using Remark 2 with α=β=0, for sufficiently large n and 0yx and nt+α/n+βt, we have(24)λn,rx,y=n-rck=0pn+rc,kx;c×0ypn-r-1c,k+r-1t;cdtn-rck=0pn+rc,kx;c×0ypn-r-1c,k+r-1t;ct-x2y-x2dtCx1+cxnx-y2.

Lemma 5.

Suppose f 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,sN0, 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,kx=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/ck=0Dpn+rc,kx;c×0pn-r-1c,k+r-1t;cfnt+αn+βdt=nΓn/c+rΓn/c-r+1Γn/c+1Γn/c×k=0n+rcpn+r+1c,k-1x,c-pn+r+1c,kx,c×0pn-r-1c,k+r-1t;cfnt+αn+βdt=nn+rcΓn/c+rΓn/c-r+1Γn/c+1Γn/c×k=0pn+r+1c,kx,c×0pn-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=0pn+r+1c,kx,c×0Dpn-rc,k+rt;cfnt+αn+βdt=n2Γn/c+r+1Γn/c-rn+βΓn/c+1Γn/c×k=0pn+r+1c,kx,c×0pn-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=0pn+r+mc,kx;c×0pn-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=0Dpn+r+mc,kx;c×0pn-r+m-1c,k+r+m-1t;cDmfnt+αn+βdt=nn+βm×nΓn/c+r+mΓn/c-r-m+1Γn/c+1Γn/c×k=0n+r+mc×pn+r+m+1c,k-1x,c-pn+r+m+1c,kx,c×0pn-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=0pn+r+m+1c,kx;c×0pn-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=0pn+r+m+1c,kx;c×0Dpn-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=0pn+r+m+1c,kx;c×0pn-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 by DBq(0,) we denote the class of absolutely continuous functions f on 0, (where q is a some positive integer) satisfying the conditions:

ftC1tq and C1>0,

the function f 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 fDBq0, we can have the representation(32)fx=fc+cxψtdt,xc0.

Theorem 6.

Let fDBq(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-fxC1+cxnk=1[n]x-x/kx+(x/k)fxx+xnx-(x/n)x+(x/n)fxx+C1+cxnxf2x-fx-xfx++fx+On-q+fx+Cx1+cxn+Cσr,cα,βxn+βfx+-fx-2+fx++fx-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-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;cfnt+αn+β-fxdt=0xnt+α/n+βn-rck=0pn+rc,kx;ck=0×pn-r-1c,k+r-1t;cfuduxnt+α/n+βdt.Also, using the identity(35)fu=fx++fx-2+fx(u)+fx+-fx-2sgnu-x+fx-fx++fx-2χx(u),where (36)χxu=1,u=x;0,ux,we have(37)n-rck=0pn+rc,kx;c×0xtfx-fx++fx-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-fx0xtn-rck=0pn+rc,kx;ci×pn-r-1c,k+r-1t;ci×fx++fx-2+fxuduk=0dt+0k=0fx+-fx-2sgnu-xxtn-rck=0pn+rc,kx;ckkkkkk×pn-r-1c,k+r-1t;ckkkkkk×fx+-fx-2sgnu-xduk=0dt.Also, it can be verified that (39)0xtn-rck=0pn+rc,kx;ckkk×pn-r-1c,k+r-1t;ckkkk=0×fx+-fx-2sgnu-xdudtfx+-fx-2Un,r,2x1/2,(40)0xtn-rck=0pn+rc,kx;ckkk×pn-r-1c,k+r-1t;ckkk×fx++fx-2dudtfx++fx-2Un,r,1x.Combining (38)–(40), we get(41)Γn/c2Γn/c+rΓn/c-rGn,r,cα,βf;x-fxxxtfxudun-rckkkk×k=0pn+rc,kx;cpn-r-1c,k+r-1t;cdtk+0xxtfxudun-rckkkkk×k=0pn+rc,kx;cpn-r-1c,k+r-1t;cdtxxtfxudu+fx+-fx-2Un,r,2x1/2+fx++fx-2Un,r,1x=An,rα,βf,x+Bn,rα,βf,x+fx+-fx-2×Un,r,2x1/2+fx++fx-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-fxAn,rα,βf,x+Bn,rα,βf,x+Cσr,cα,βxn+βfx+-fx-2+fx++fx-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=xxtfxudun-rckki×k=0pn+rc,kx;cpn-r-1c,k+r-1t;cdtxxtfxudun-rck=0pn+rc,kx;cii×2xft-fxpn-r-1c,k+r-1t;cdtk=0+fx+ii×n-rck=0pn+rc,kx;ciik=0×x2xpn-r-1c,k+r-1t;ct-xdtii+x2xfxudu1-λn,rx,2xii+x2xfxt·1-λn,rx,tdtn-rck=0pn+rc,kx;cii×2xpn-r-1c,k+r-1t;cC1t2qdtii+fxx2n-rck=0pn+rc,kx;cii×0pn-r-1c,k+r-1t;ct-x2dtii+fx+n-rck=0pn+rc,kx;cii×2xpn-r-1c,k+r-1t;ct-xdtii+Cx1+cxnx2f2x-x-xfx+ii+C1+cxnk=1[n]xx+x/kfxxii+xnxx+(x/n)fxx.

For estimating the integral(44)n-rck=0pn+rc,kx;c2xpn-r-1c,k+r-1t;cC1t2qdt,we proceed as follows: since t2x implies that t2t-x so by Schwarz inequality and Lemma 1,(45)n-rck=0pn+rc,kx;c2xpn-r-1c,k+r-1t;cC1t2qdtC12qn-rck=0pn+rc,kx;c×0pn-r-1c,k+r-1t;cC1t-x2qdtC12qUn,r,2qx=On-qas  n.By using Hölder’s inequality and Remark 2 (α=β=0), we get the estimate as follows: (46)fx+n-rck=0pn+rc,kx;c×2xpn-r-1c,k+r-1t;ct-xdtfx+×n-rck=0pn+rc,kx;ckkk×0pn-r-1c,k+r-1t;ct-x2dtk=01/2fx+Cx(1+cx)n.Collecting the estimates from (43)–(46), we obtain(47)An,rα,βf,xOn-q+fx+×Cx(1+cx)n+C(1+cx)nx×f2x-fx-xfx++fx+C1+cxnk=1nxx+x/kfxx+xnxx+(x/n)fxx.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=0xxtfxudtλn,rx,t0y+yxfxtλn,rx,tdtCx1+cxn0ytxfx1x-t2dt+yxtxfxdt=Cx1+cxn1nx-x/uxfxdu+xnx-(x/n)xfxCx1+cxnk=1[n]x-(x/k)xfx+xnx-(x/n)xfx,where u=x/x-t.

Combining (41), (47), and (48), we get the desired result.

Corollary 7.

Let fsDBq(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-rC(1+cx)nk=1[n]x-(x/k)x+(x/k)Ds+1fx+xnx-(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-,0tx0,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 .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research paper. Special thanks are due to Professor Józef Banaś, Editor of the Journal of Function Spaces, for his efforts to send the reports of the paper timely. The authors are also grateful to all the editorial board members and reviewers of esteemed journal, that is, Journal of Function Spaces. The second author Lakshmi Narayan Mishra acknowledges the Ministry of Human Resource Development (MHRD), New Delhi, India, for supporting this research paper at the Department of Mathematics, National Institute of Technology (NIT), Silchar, Assam. The third author Vishnu Narayan Mishra acknowledges that this paper project was supported by Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat (Gujarat), India. All the authors carried out the proof of theorems. Each author contributed equally in the development of the paper. Vishnu Narayan Mishra conceived of the study and participated in its design and coordination. All the authors read and approved the final version of paper.

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