AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 867203 10.1155/2012/867203 867203 Research Article A Kantorovich Type of Szasz Operators Including Brenke-Type Polynomials Taşdelen Fatma Aktaş Rabia Altın Abdullah Bellouquid Abdelghani Department of Mathematics Faculty of Science Ankara University Tandoğan 06100 Ankara Turkey ankara.edu.tr 2012 11 12 2012 2012 27 09 2012 15 11 2012 2012 Copyright © 2012 Fatma Taşdelen 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 give a Kantorovich variant of a generalization of Szasz operators defined by means of the Brenke-type polynomials and obtain convergence properties of these operators by using Korovkin's theorem. We also present the order of convergence with the help of a classical approach, the second modulus of continuity, and Peetre's K-functional. Furthermore, an example of Kantorovich type of the operators including Gould-Hopper polynomials is presented and Voronovskaya-type result is given for these operators including Gould-Hopper polynomials.

1. Introduction

The Szasz operators (also called Szasz-Mirakyan operators) which are defined by  (1.1)Sn(f;x)=e-nxk=0(nx)kk!f(kn), where n, x0, and fC[0,) have an important role in the approximation theory, and their approximation properties have been investigated by many researchers.

In , Jakimovski and Leviatan proposed a generalization of Szasz operators by means of the Appell polynomials pk(x) which have the generating functions of the form: (1.2)g(t)etx=k=0pk(x)tk, where g(z)=k=0akzk(a00) is an analytic function in the disc |z|<R, (R>1) and g(1)0. Under the assumption that pk(x)0 for x[0,), Jakimovski and Leviatan , defined the following linear positive operators: (1.3)Pn(f;x)=e-nxg(1)k=0pk(nx)f(kn).

After that, Ismail  defined another generalization of Szasz operators involving the operators (1.1) and (1.3) by means of Sheffer polynomials. Let A(z)=k=0akzk(a00) and H(z)=k=1hkzk(h10) be analytic functions in the disc |z|<R,(R>1). Here, ak and hk are real. The Sheffer polynomials pk(x) are generated by (1.4)A(t)exH(t)=k=0pk(x)tk. With the help of these polynomials, Ismail constructed the following linear positive operators: (1.5)Tn(f;x):=e-nxH(1)A(1)k=0pk(nx)f(kn),n under the assumptions

for x[0,), pk(x)0,

A(1)0 and H(1)=1.

Later, Varma et al.  defined another generalization of Szasz operators by means of the Brenke-type polynomials. Suppose that (1.6)A(t)=r=0artr,a00,B(t)=r=0brtr,br0(r0) are analytic functions. The Brenke-type polynomials  have generating functions of the form (1.7)A(t)B(xt)=k=0pk(x)tk from which the explicit form of pk(x) is as follows: (1.8)pk(x)=r=0kak-rbrxr,k=0,1,2,.

Under the assumptions (1.9)(i)A(1)0,ak-rbrA(1)0,0rk,k=0,1,2,,(ii)B:[0,)(0,),(iii)(1.6)  and  (1.7)  converge  for  |t|<R,(R>1),

Varma et al. introduced the linear positive operators Ln(f;x) via (1.10)Ln(f;x):=1A(1)B(nx)k=0pk(nx)f(kn), where x0 and n.

The aim of this paper is to present a Kantorovich type of the operators given by (1.10) and to give their some approximation properties. We consider the Kantorovich version of the operators (1.10) under the assumptions (1.9) as follows: (1.11)Kn(f;x):=nA(1)B(nx)k=0pk(nx)k/n(k+1)/nf(t)dt, where n, x0, and fC[0,). It is easy to see that Kn defined by (1.11) is linear and positive.

In the case of B(t)=et and A(t)=1, with the help of (1.7), it follows that pk(x)=xk/k!, so the operators (1.11) reduce to the Szasz-Mirakyan-Kantorovich operators defined by  (1.12)Kn(f;x):=ne-nxk=0(nx)kk!k/n(k+1)/nf(t)dt. Various approximation properties of the Szasz-Mirakyan-Kantorovich operators and their iterates may be found in .

The case of B(t)=et gives the Kantorovich version of the operators (1.3).

The structure of the paper is as follows. In Section 2, the convergence of the operators (1.11) is given by means of Korovkin's theorem. The order of approximation is obtained with the help of a classical approach, the second modulus of continuity, and Peetre's K-functional in Section 3. Finally, as an example, we present a Kantorovich type of the operators including Gould-Hopper polynomials and then we give a Voronovskaya-type theorem for the operators including Gould-Hopper polynomials.

2. Approximation Properties of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M47"><mml:mrow><mml:msub><mml:mrow><mml:mi>K</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> Operators

In this section, we give our main theorem with the help of Korovkin theorem. We begin with the following lemma which is necessary to prove the main result.

Lemma 2.1.

For all x[0,), the operators Kn defined by (1.11) verify (2.1)Kn(1;x)=1,(2.2)Kn(s;x)=B(nx)B(nx)x+2A(1)+A(1)2nA(1),(2.3)Kn(s2;x)=B′′(nx)B(nx)x2+2B(nx)[A(1)+A(1)]nA(1)B(nx)x+1n2A(1){A′′(1)+2A(1)+A(1)3}.

Proof.

Using the generating function of the Brenke-type polynomials given by (1.7), we can write (2.4)k=0pk(nx)=A(1)B(nx),k=0kpk(nx)=A(1)B(nx)+nxA(1)B(nx),k=0k2pk(nx)=n2x2A(1)B′′(nx)+nxB(nx){2A(1)+A(1)}+B(nx){A′′(1)+A(1)}. From these equalities, the assertions of the lemma are obtained.

Lemma 2.2.

For x[0,), one has (2.5)Kn((s-x)2;x)={B′′(nx)-2B(nx)+B(nx)B(nx)}x2+{2A(1)[B(nx)-B(nx)]+A(1)[2B(nx)-B(nx)]nA(1)B(nx)}x+1n2A(1){A′′(1)+2A(1)+A(1)3}.

Proof.

From the linearity of Kn, we get (2.6)Kn((s-x)2;x)=Kn(s2;x)-2xKn(s;x)+x2Kn(1;x). Next, we apply Lemma 2.1.

Theorem 2.3.

Let (2.7)E:={f:x[0,),f(x)1+x2  is  convergent  as  x},(2.8)limyB(y)B(y)=1,limyB′′(y)B(y)=1. If fC[0,)E, then (2.9)limnKn(f;x)=f(x), and the operators Kn converge uniformly in each compact subset of [0,).

Proof.

Using Lemma 2.1 and taking into account the equality (2.8) we get (2.10)limnKn(si;x)=xi,i=0,1,2. The above convergence is satisfied uniformly in each compact subset of [0,). We can then apply the universal Korovkin-type property (vi) of Theorem  4.1.4 in  to obtain the desired result.

3. The Order of Approximation

In this section, we deal with the rates of convergence of the Kn(f) to f by means of a classical approach, the second modulus of continuity, and Peetre's K-functional.

Let fC~[0,). If δ>0, the modulus of continuity of f is defined by (3.1)w(f;δ):=supx,y[0,)|x-y|δ|f(x)-f(y)|, where C~[0,) denotes the space of uniformly continuous functions on [0,). It is also well known that, for any δ>0 and each x[0,), (3.2)|f(x)-f(y)|w(f;δ)(|x-y|δ+1).

The next result gives the rate of convergence of the sequence Kn(f) to f by means of the modulus of continuity.

Theorem 3.1.

Let fC~[0,)E. The Kn operators satisfy the following inequality: (3.3)|Kn(f;x)-f(x)|2w(f;λn(x)), where (3.4)λ:=λn(x)=Kn((s-x)2;x)={B′′(nx)-2B(nx)+B(nx)B(nx)}x2+{2A(1)[B(nx)-B(nx)]+A(1)[2B(nx)-B(nx)]nA(1)B(nx)}x+1n2A(1){A′′(1)+2A(1)+A(1)3}.

Proof.

Using (2.1), (3.2), and the linearity property of Kn operators, we can write (3.5)|Kn(f;x)-f(x)|nA(1)B(nx)k=0Pk(nx)k/n(k+1)/n|f(s)-f(x)|dsnA(1)B(nx)k=0Pk(nx)k/n(k+1)/n(|s-x|δ+1)w(f;δ)ds{1+nA(1)B(nx)δk=0Pk(nx)k/n(k+1)/n|s-x|ds}w(f;δ). By using the Cauchy-Schwarz inequality for integration, we get (3.6)k/n(k+1)/n|s-x|ds1n(k/n(k+1)/n|s-x|2ds)1/2 which holds that (3.7)k=0Pk(nx)k/n(k+1)/n|s-x|ds1nk=0Pk(nx)(k/n(k+1)/n|s-x|2ds)1/2. By applying the Cauchy-Schwarz inequality for summation on the right-hand side of (3.7), we have (3.8)k=0Pk(nx)k/n(k+1)/n|s-x|dsA(1)B(nx)n(A(1)B(nx)nKn((s-x)2;x))1/2=A(1)B(nx)n(Kn((s-x)2;x))1/2=A(1)B(nx)n(λn(x))1/2, where λn(x) is given by (3.4). If we use this in (3.5), we obtain (3.9)|Kn(f;x)-f(x)|{1+1δλn(x)}w(f;δ). On choosing δ=λn(x), we arrive at the desired result.

Recall that the second modulus of continuity of fCB[0,) is defined by (3.10)w2(f;δ):=sup0<tδf(·+2t)-2f(·+t)+f(·)CB, where CB[0,) is the class of real valued functions defined on [0,) which are bounded and uniformly continuous with the norm fCB=supx[0,)|f(x)|.

Peetre's K-functional of the function fCB[0,) is defined by (3.11)K(f;δ):=infgCB2[0,){f-gCB+δgCB2}, where (3.12)CB2[0,):={gCB[0,):g,g′′CB[0,)}, and the norm gCB2:=gCB+gCB+g′′CB (see ). It is clear that the following inequality: (3.13)K(f;δ)M{w2(f;δ)+min(1,δ)fCB}, holds for all δ>0. The constant M is independent of f and δ.

Theorem 3.2.

Let fCB2[0,). The following (3.14)|Kn(f;x)-f(x)|ζfCB2 holds, where (3.15)ζζn(x)={B′′(nx)-2B(nx)+B(nx)2B(nx)}x2+{2A(1)[B(nx)-B(nx)]+A(1)[2(n+1)B(nx)-(2n+1)B(nx)]2nA(1)B(nx)}x+12n2A(1){A′′(1)+2A(1)+A(1)3}+2A(1)+A(1)2nA(1).

Proof.

From the Taylor expansion of f, the linearity of the operators Kn and (2.1), we have (3.16)Kn(f;x)-f(x)=f(x)Kn(s-x;x)+12f′′(η)Kn((s-x)2;x),η(x,s). Since (3.17)Kn(s-x;x)={B(nx)-B(nx)B(nx)}x+2A(1)+A(1)2nA(1)0 for sx, by considering Lemmas 2.1 and 2.2 in (3.16), we can write that (3.18)|Kn(f;x)-f(x)|{(B(nx)-B(nx)B(nx))x+2A(1)+A(1)2nA(1)}fCB+12[{B′′(nx)-2B(nx)+B(nx)B(nx)}x2+{2A(1)[B(nx)-B(nx)]+A(1)[2B(nx)-B(nx)]nA(1)B(nx)}x+1n2A(1){A′′(1)+2A(1)+A(1)3}]f′′CB[{B′′(nx)-2B(nx)+B(nx)2B(nx)}x2+{2A(1)[B(nx)-B(nx)]+A(1)[2(n+1)B(nx)-(2n+1)B(nx)]2nA(1)B(nx)}x+12n2A(1){A′′(1)+2A(1)+A(1)3}+2A(1)+A(1)2nA(1)]fCB2 which completes the proof.

Theorem 3.3.

Let fCB[0,). Then (3.19)|Kn(f;x)-f(x)|2M{w2(f;δ)+min(1,δ)fCB}, where (3.20)δ:=δn(x)=12ζn(x) and M>0 is a constant which is independent of the functionsf and δ. Also, ζn(x) is the same as in Theorem 3.2.

Proof.

Suppose that gCB2[0,). From Theorem 3.2, we can write (3.21)|Kn(f;x)-f(x)||Kn(f-g;x)|+|Kn(g;x)-g(x)|+|g(x)-f(x)|2f-gCB+ζgCB2=2[f-gCB+δgCB2]. The left-hand side of inequality (3.21) does not depend on the function gCB2[0,), so (3.22)|Kn(f;x)-f(x)|2K(f;δ), where K(f;δ) is Peetre's K-functional defined by (3.11). By the relation between Peetre's K-functional and the second modulus of smoothness given by (3.13), inequality (3.21) becomes (3.23)|Kn(f;x)-f(x)|2M{w2(f;δ)+min(1,δ)fCB} whence we have the result.

Remark 3.4.

Note that when n, then λn, ζn, and δn tend to zero in Theorems 3.13.3 under the assumption (2.8).

4. Special Cases and Further Properties

Gould-Hopper polynomials gkd+1(x,h) , which are d-orthogonal polynomial sets of Hermite type , are generated by (4.1)ehtd+1exp(xt)=k=0gkd+1(x,h)tkk! from which it follows that (4.2)gkd+1(x,h)=m=0[k/(d+1)]k!m!(k-(d+1)m)!hmxk-(d+1)m, where, as usual, [·] denotes the integer part.

In , the authors showed that the Gould-Hopper polynomials are Brenke-type polynomials with A(t)=ehtd+1 and B(t)=et, and the restrictions (1.9) and condition (2.8) for the operators given by (1.10) are satisfied under the assumption h0. These operators including the Gould-Hopper polynomials are as follows: (4.3)Ln*(f;x):=e-nx-hk=0gkd+1(nx,h)k!f(kn), where x[0,).

The special case A(t)=ehtd+1 and B(t)=et of (1.11) gives the following Kantorovich version of Kn(f;x) including the Gould-Hopper polynomials: (4.4)Kn*(f;x):=ne-nx-hk=0gkd+1(nx,h)k!k/n(k+1)/nf(t)dt under the assumption h0.

Remark 4.1.

For h=0, we find gkd+1(nx,0)=(nx)k and the operators given by (4.4) reduce to the Szasz-Mirakyan-Kantorovich operators given by (1.12).

Now, we give a Voronovskaya-type theorem for the operators (4.4). In order to prove this theorem, we need the following lemmas.

Lemma 4.2.

For the operators Kn*, one has (4.5)Kn*(1;x)=1,Kn*(s;x)=x+h(d+1)n+12n,Kn*(s2;x)=x2+2n(h(d+1)+1)x+1n2[h(d+1){h(d+1)+d+2}+13],Kn*(s3;x)=x3+3x2n{h(d+1)+32}+xn2{3h2(d+1)2+3h(d+1)(d+3)+72}+1n3{h2(h+3)(d+1)3+h(d+1)3+32h(h+1)(d+1)2+h(d+1)+14},Kn*(s4;x)=x4+4x3n{h(d+1)+2}+3x2n2{2h(h+1)(d+1)2+6h(d+1)+5}+xn3{12h2(d+1)2(d+2)+4h3(d+1)3+2h(d+1)(2d2+10d+15)+6}+1n4{15h3(h+6)(d+1)4+8h(1+3h)(d+1)3-9h(h+1)(d+1)2+7h(d+1)+2h3(d+1)3+4d(d-1)(d+1)2h2+3h2d2(d+1)2+(d-2)(d-1)d(d+1)h+15}.

Proof.

From the generating function (4.1) for the Gould-Hopper polynomials, one can easily find the above equalities.

Lemma 4.3.

For x[0,), one has (4.6)Kn*((s-x)2;x)=xn+1n2[h(d+1){h(d+1)+d+2}+13]Kn*((s-x)4;x)=3x2n2+xn3{6h(h+2)(d+1)2+2h(d+1)(-3d+2)+5}+1n4{15h3(h+6)(d+1)4+2h(h2+12h+4)(d+1)3+7h(d+1)-9h(h+1)(d+1)2+4d(d-1)(d+1)2h2+3h2d2(d+1)2+(d-2)(d-1)d(d+1)h+15}.

Proof.

It is enough to use Lemma 4.2 to obtain above equalities.

Theorem 4.4.

Let fC2[0,a]. Then one has (4.7)limnn[Kn*(f;x)-f(x)]=f(x){h(d+1)+12}+xf′′(x)2!.

Proof.

By Taylor's theorem, we get (4.8)f(s)=f(x)+(s-x)f(x)+(s-x)22!f′′(x)+(s-x)2η(s;x), where η(s;x)C[0,a] and limsxη(s;x)=0. If we apply the operator Kn* to the both sides of (4.8), we obtain (4.9)Kn*(f;x)=f(x)+f(x)Kn*(s-x;x)+f′′(x)2!Kn*((s-x)2;x)+Kn*((s-x)2η(s;x);x). In view of Lemmas 4.2 and 4.3, the equality (4.9) can be written in the form (4.10)n[Kn*(f;x)-f(x)]=n{h(d+1)n+12n}f(x)+n{xn+1n2[h(d+1){h(d+1)+d+2}+13]}f′′(x)2!+nKn*((s-x)2η(s;x);x), where (4.11)Kn*((s-x)2η(s;x);x)=ne-nx-hk=0gkd+1(nx,h)k!k/n(k+1)/n(s-x)2η(s;x)ds. Applying Cauchy-Schwarz inequality, we get (4.12)nKn*((s-x)2η(s;x);x)n2e-nx-hk=0gkd+1(nx,h)k!(k/n(k+1)/n(s-x)4ds)1/2(k/n(k+1)/nη2(s;x)ds)1/2. If we use Cauchy-Schwarz inequality again on the right-hand side of the inequality above, then we conclude that (4.13)nKn*((s-x)2η(s;x);x)(n3e-nx-hk=0gkd+1(nx,h)k!k/n(k+1)/n(s-x)4ds)1/2·(ne-nx-hk=0gkd+1(nx,h)k!k/n(k+1)/nη2(s;x)ds)1/2=n2Kn*((s-x)4;x)Kn*(η2(s;x);x). In view of Lemma 4.3, (4.14)limnn2Kn*((s-x)4;x)=3x2 holds. On the other hand, since η(s;x)C[0,a] and limsxη(s;x)=0, then it follows from Theorem 2.3 that (4.15)limnKn*(η2(s;x);x)=η2(x;x)=0. Considering (4.13), (4.14), and (4.15), we immediately see that (4.16)limnnKn*((s-x)2η(s;x);x)=0. Then, taking limit as n in (4.10) and using (4.16), we have (4.17)limnn[Kn*(f;x)-f(x)]=f(x){h(d+1)+12}+xf′′(x)2! which completes the proof.

Remark 4.5.

Getting h=0 in Theorem 4.4 gives a Voronovskaya-type result for the Szasz-Mirakyan-Kantorovich operators given by (1.12).

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