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Kyungpook Mathematical Journal -0001; 56(3): 877-897

Published online November 30, -0001

Copyright © Kyungpook Mathematical Journal.

Szász-Kantorovich Type Operators Based on Charlier Polynomials

Arun Kajla, Purshottam Narain Agrawal

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India

Received: November 21, 2015; Accepted: April 7, 2016

In the present article, we study some approximation properties of the Kantorovich type generalization of Szász type operators involving Charlier polynomials introduced by S. Varma and F. Taşdelen (Math. Comput. Modelling, 56 (5–6) (2012) 108–112). First, we establish approximation in a Lipschitz type space, weighted approximation theorems and A-statistical convergence properties for these operators. Then, we obtain the rate of approximation of functions having derivatives of bounded variation.

Keywords: Kantorovich operator, Charlier polynomials, modulus of continuity, $A$-statistical convergence, bounded variation.

Szász ([31]) constructed the following linear positive operators

Sn(f;x)=e-nxk=0(nx)kk!f(kn),

where x ∈ [0, ∞) and f(x) is a continuous function on [0, ∞) whenever the above sum converges uniformly. Butzer ([7]) defined and studied an integral modification of the operators Sn. Several researchers have studied approximation properties of these operators and their iterates (cf. [6, 13, 16, 24, 25, 32, 35]).

Jakimovski and Leviatan ([21]) introduced a generalization of Szász operators involving the Appell polynomials and studied some approximation properties of these operators. Varma et al. ([33]) constructed a generalization of Szász operators defined by means of the Brenke type polynomials and studied convergence properties of these operators using the Korovkin theorem and the order of convergence by using the classical second order modulus of continuity and Peetre’s K-functional. Altomare et al. ([4]) defined a new kind of generalization of Szász-Mirakjan-Kantorovich operators and studied the rate of convergence by means of suitable moduli of smoothness. Very recently, Agrawal et al. ([2]) introduced a Kantorovich type generalization of the q-Bernstein-Schurer operators and gave some approximation properties of these operators. In [34], Varma and Taşdelen introduced a link between discrete orthogonal polynomials and certain linear positive operators. They have defined Szász type operators involving Charlier polynomials. These polynomials ([18]) have the generating functions of the form

et(1-ta)u=k=0Ck(a)(u)tkk!,t<a,

where Ck(a)(u)=r=0k(kr)(-u)r(1a)r and (m)0 = 1, (m)j = m(m+1) · · · (m+j−1) for j ≥ 1.

Varma and Taşdelen ([34]) defined the Szász type operators involving Charlier polynomials as

Ln(f;x,a)=e-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!f(kn),         a>1,x0.

Further, they considered Kantorovich type generalization of the operators Ln(f; x, a) for a function fC˜0,):={fC[0,):F(x)=0xf(s)dsKeBx,Band K+} as follows:

Ln,a*(f;x)=ne-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!knk+1nf(t)dt,

where a > 1 and x ≥ 0 and studied the uniform convergence of Ln,a*(f;x) to f on each compact subset of [0, ∞) and the degree of approximation in terms of the classical modulus of continuity.

The purpose of this paper is to establish some more approximation properties of the operators Ln,a* such as weighted approximation, A-statistical convergence and approximation of functions with a derivative of bounded variation. The outline of paper is as follows.

In Section 2, we present some moment estimates and a result needed to study approximation of functions with derivatives of bounded variation. In Section 3, we discuss the main results of the paper wherein we establish approximation in a Lipschitz type space, weighted approximation theorems and A-statistical convergence properties for the operators Ln,a*. Lastly, we obtain the rate of convergence for functions having a derivative of bounded variation on every finite subinterval of [0, ∞), for these operators.

In this section we collect some properties and examples of Charlier polynomials and some results about the operators Ln,a* useful in the sequel.

Since the Charlier polynomials play substantial role in the definition of the operators given by (1.3), we mention below some examples and properties of these non-classical polynomials:

Example 2.1

C0(a)(u)=1,         C1(a)(u)=1-ua,         C2(a)(u)=1-ua2(1+2a)+u2a2 and C3(a)(u)=1-ua3(3a2+3a+2)+3u2a3(a+1)-u3a3 etc.

Proposition 2.2

([8], Ch. VI, p.170) For the functionCk(a)(u), there hold the following:

  • Ck(a)(u)is a polynomial in u of degree k with the coefficient of ukas(-1a)k,

  • Ck(a)(u)can be expressed in terms of Laguerre polynomialsLk(u-k)(a)asCk(a)(u)=k!(-1a)kLk(u-k)(a), whereLk(α)(a)=r=0k(k+αk-r)(-a)rr!,

  • Ck(a)(u)satisfies the recursion relation-aCk+1(a)(u)=(u-k-a)Ck(a)(u)+KCk-1(a)(u),         k1,

  • Ck(a)(u)satisfies the discrete orthogonality propertyu=0ω(u)Cm(a)(u)Cn(a)(u)=an(n!)         δmn,

    whereω(u)=e-aauu!and δmnis the Kronecker delta.

Lemma 2.3

For the operatorsLn,a*(f;x), we have

  • Ln,a*(1;x)=1,

  • Ln,a*(t;x)=x+32n,

  • Ln,a*(t2;x)=x2+xn(4+1a-1)+103n2,

  • Ln,a*(t3;x)=x3+x2n(152+3a-1)+xn2(232+5a-1+2(a-1)2)+374n3 ,

  • Ln,a*(t4;x)=x4+2x3n(6+3a-1)+x2n2(63-24a-1+11(a-1)2)+xn3(98+59a-1-16(a-1)2+6(a-1)3)+151n4.

Proof

The proofs of the parts (i), (ii) and (iii) are given in ([34], Lemma 2). The moments Ln,a*(t3;x) and Ln,a*(t4;x) can be computed following the same idea of proof of ([34], Lemma 2).

Lemma 2.4

The central moments for the operatorsLn,a*(f;x)are given by

  • Ln,a*(t-x;x)=32n;

  • Ln,a*((t-x)2;x)=a(a-1)xn+103n2;

  • Ln,a*((t-x)3;x)=xn2(32+5a-1+2(a-1)2)+374n3;

  • Ln,a*((t-x)4;x)=x2n2(37-44a-1+3(a-1)2)+xn3(61+59a-1-16(a-1)2+6(a-1)3)+151n4.

Remark 2.5

From Lemma 2.4, for each x ∈ [0, ∞), η(a) > 1 and n sufficiently large, we have

Ln,a*(t-x;x)(Ln,a*((t-x)2;x))1/2η(a)xn,

where η(a) is some positive constant depending on a.

The operators Ln,a*(f;x) also admit the integral representation

Ln,a*(f;x)=0Kn*(x,t)f(t)dt

and Kn*(x,t):=ne-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!χn,k(t), where χn, k(t) is the characteristic function of the interval [kn,k+1n] with respect to [0, ∞).

Lemma 2.6

For a fixed x ∈ (0, ∞) and sufficiently large n, we have

  • βn*(x,y)=0yKn*(x,t)dtη(a)xn(x-y)2, 0 ≤ y < x,

  • 1-βn*(x,z)=zKn*(x,t)dtη(a)xn(z-x)2, x<z <∞.

Proof

(i) Using Remark 2.5, we get

βn*(x,y)=0yKn*(x,t)dt0y(x-tx-y)2Kn*(x,t)dt=Ln,a*((t-x)2;x)(x-y)-2η(a)xn(x-y)2.

The assertion (ii) can be proved in a similar manner hence the details are omitted.

In what follows, let CB[0, ∞) be the space of all real valued bounded and uniformly continuous functions f on [0, ∞), endowed with the norm fCB[0,)=supx[0,)f(x).

Further, let us define the following Peetre’s K-functional:

K2(f,δ)=infgW2{f-gCB[0,)+δgCB[0,)},δ>0,

where W2 = {gCB[0, ∞) : g′, g″ ∈ CB[0, ∞)} and the norm

fW2=fCB[0,)+fCB[0,)+fCB[0,).

By ([9], p.177, Theorem 2.4) there exists an absolute constant M >0 such that

K2(f,δ)M{ω2(f,δ)+min(1,δ)fCB[0,)},

where the second order modulus of smoothness is defined as

ω2(f,δ)=sup0<hδsupx[0,)f(x+2h)-2f(x+h)+f(x).

The usual modulus of continuity of fCB[0, ∞) is defined as

ω(f,δ)=sup0<hδsupx[0,)f(x+h)-f(x).

In this section we establish approximation properties in several settings. For the reader’s convenience we split up this section in more subsections.

3.1. Lipschitz-type space

Let us consider the Lipschitz-type space with two parameters [29]: for a1, a2 > 0, we define

LipM(a1,a2)(α):={fC[0,):f(t)-f(x)Mt-xα(t+a1x2+a2x)α2;x,t(0,)},

where M is a positive constant and α ∈ (0, 1].

Theorem 3.1

LetfLipM(a1,a2)(α). Then, for all x > 0, we have

Ln,a*(f;x)-f(x)M(una(x)(a1x2+a2x))α2,

whereuna(x)=Ln,a*((t-x)2;x).

Proof

First, we prove the theorem for the case α = 1. We may write

Ln,a*(f;x)-f(x)ne-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!knk+1nf(t)-f(x)dtMne-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!knk+1nt-xt+a1x2+a2xdt.

Using the fact that 1t+a1x2+a2x<1a1x2+a2x, and the Cauchy-Schwarz inequality, the above inequality implies that

Ln,a*(f;x)-f(x)Mna1x2+a2xe-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!knk+1nt-xdt=Ma1x2+a2xLn,a*(t-x;x)M(una(x)a1x2+a2x).

Thus, the result holds for α = 1. Now, let 0 < α < 1, then applying the Hölder inequality with p=1α and q=11-α, we have

Ln,a*(f;x)-f(x)ne-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!knk+1nf(t)-f(x)dt{e-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!(nknk+1nf(t)-f(x)dt)1α}α{ne-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!knk+1nf(t)-f(x)1αdt}αM{ne-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!knk+1nt-xt+a1x2+a2xdt}αM(a1x2+a2x)α2{ne-1(1-1a)(a-1)nxk=0Ck(a)(-(a-1)nx)k!knk+1nt-xdt}αM(a1x2+a2x)α2(Ln,a*(t-x;x))αM(una(x)(a1x2+a2x))α2.

This completes the proof.

Let Hφ[0, ∞) be the space of all functions f defined on [0, ∞) with the property that |f(x)| ≤ Mfφ(x), where Mf is a positive constant depending only on f and φ(x) = 1+x2 is a weight function. Let Cφ[0, ∞) be the subspace of Hφ[0, ∞) of all continuous functions with the norm fφ=supx[0,)f(x)1+x2 and Cφ*[0,)={fCφ[0,):limxf(x)1+x2<}. The usual modulus of continuity of f on [0, b] is defined as

ωb(f,δ)=supt-xδsupx,t[0,b]f(t)-f(x).

In what follows, let || · ||C[0, d] denote the sup-norm over [0, d], d > 0.

Theorem 3.2

Let fCφ[0, ∞). Then, we have

Ln,a*(f)-fC[0,b]4Mf(1+b2)una(b)+2ωb+1(f,una(b)),

whereuna(b)=abn(a-1)+103n2.

Proof

From [17], for x ∈ [0, b] and t ≥ 0, we have

f(t)-f(x)4Mf(1+x2)(t-x)2+(1+t-xδ)ωb+1(f,δ),δ>0.

Hence applying Cauchy-Schwarz inequality, we get

Ln,a*(f;x)-f(x)4Mf(1-x2)Ln,a*((t-x)2;x)+ωb+1(f,δ)(1+1δLn,a*(t-x;x))4Mf(1+x2)una(x)+ωb+1(f,δ)(1+1δuna(x))4Mf(1+b2)una(b)+ωb+1(f,δ)(1+1δuna(b)).

Choosing δ=una(b), we get the desired result.

Next we give a theorem to approximate all functions in Cφ[0, ∞). This type of result is discussed in [14] for locally integrable functions.

Theorem 3.3

For each fCφ[0, ∞) and β > 0, we have

limnsupx[0,)Ln,a*(f;x)-f(x)(1+x2)1+β=0.
Proof

For any fixed x0 > 0,

supx[0,)Ln,a*(f;x)-f(x)(1+x2)1+βsupxx0Ln,a*(f;x)-f(x)(1+x2)1+β+supxx0Ln,a*(f;x)-f(x)(1+x2)1+βLn,a*(f)-fC[0,x0]+fφsupxx0Ln,a*(1+t2;x)(1+x2)1+β+supxx0f(x)(1+x2)1+β,=I1+I2+I3,say.

Since |f(x)| ≤ ||f||φ(1 + x2), we have

I3=supxx0f(x)(1+x2)1+βsupxx0fφ(1+x2)βfφ(1+x02)β.

Let ε > 0 be arbitrary. In view of ( [34], Theorem 3) there exists n1 ∈ ℕ such that

fφLn,a*(1+t2;x)(1+x2)1+β<1(1+x2)1+βfφ((1+x2)+ɛ3fφ),         nn1<fφ(1+x2)β+ɛ3,nn1.

Hence, fφsupxx0Ln,a*(1+t2;x)(1+x2)1+β<fφ(1+x02)β+ɛ3, ∀nn1.

Thus, I2+I3<2fφ(1+x02)β+ɛ3, ∀nn1.

Now, let us choose x0 to be so large that fφ(1+x02)β<ɛ6.

Then,

I2+I3<2ɛ3,nn1.

By Theorem 3.2, there exists n2 ∈ ℕ such that

I1=Ln,a*(f)-fC[0,x0]<ɛ3,nn2.

Let n0 = max(n1, n2). Then, combining (3.1)–(3.4)

supx[0,)Ln,a*(f;x)-f(x)(1+x2)1+β<ɛ,         nn0.

This completes the proof.

Let fCφ*[0,). The weighted modulus of continuity is defined as :

Ω(f;δ)=supx[0,),0<hδf(x+h)-f(x)1+(x+h)2,

was defined by Yüksel and Ispir in [36].

Lemma 3.4([36])

LetfCφ*[0,), then

  • Ω(f, δ) is a monotone increasing function of δ,

  • limδ0+Ω(f;δ)=0,

  • for each m ∈ ℕ, Ω(f, mδ) ≤ mΩ(f; δ),

  • for each λ ∈ [0, ∞), Ω(f; λδ) ≤ (1 + λ)Ω(f; δ).

Theorem 3.5

LetfCφ*[0,). Then there exists a positive constant M(a) depending on a such that

supx(0,)Ln,a*(f;x)-f(x)(1+x2)52M(a)Ω(f;n-1/2).
Proof

For t > 0, x ∈ (0, ∞) and δ > 0, by the definition of Ω(f; δ) and Lemma 3.4, we can write

f(t)-f(x)(1+(x+x-t)2)Ω(f;t-x)2(1+x2)(1+(t-x)2)(1+t-xδ)Ω(f;δ).

Since Ln,a* is linear and positive, we have

Ln,a*(f;x)-f(x)2(1+x2)Ω(f;δ)×{1+Ln,a*((t-x)2;x)+Ln,a*((1+(t-x)2)t-xδ;x)}.

From Lemma 2.4 (ii), we have

Ln,a*((t-x)2;x)M1(a)(1+x2)n,

where M1(a) is some positive constant depending on a. Formally applying Cauchy-Schwarz inequality, we have

Ln,a*((1+(t-x)2)t-xδ;x)1δLn,a*((t-x)2;x)+1δLn,a*((t-x)4;x)Ln,a*((t-x)2;x).

By using Lemma 2.4 (ii), there exists a positive constant M2(a) depending on a such that

(Ln,a*(t-x)4;x)M2(a)(1+x2)n.

Collecting the estimates (3.6)–(3.9) and taking M(a)=2(1+M1(a)+M1(a)+M2(a)M1(a)),δ=1n, we get the required result (3.5).

Let A = (ank), (n, k ∈ ℕ), be a non-negative infinite summability matrix. For a given sequence x := (xk), the A-transform of x denoted by Ax : ((Ax)n) is defined as

(Ax)n=k=1ankxk,

provided the series converges for each n. A is said to be regular if limn(Ax)n=L whenever limnxn=L. The sequence x = (xn) is said to be a A-statistically convergent to L i.e. stA-limnxn=L if for every ε > 0, limnk:xk-Lɛank=0.

Replacing A by C1, the Cesáro matrix of order one, the A-statistical convergence reduces to the statistical convergence. Similarly, if we take A = I, the identity matrix, then A-statistical convergence coincides with the ordinary convergence. It is to be noted that the concept of A–statistical convergence may also be given in normed spaces. Many researchers have studied the statistical convergence of different types of operators (cf. [5, 10, 11, 12, 15, 20, 26, 27, 30]). In the following result we prove a weighted Korovkin theorem via A-statistical convergence.

Throughout this section, let us assume that ei(t) = ti, i = 0, 1, 2.

Theorem 3.6

Let (ank) be a non-negative regular infinite summability matrix and x ∈ [0, ∞). Let φγ ≥ 1 be a continuous function such that

limxφ(x)φγ(x)=0.

Then, for allfCφ*[0,), we have

stA-limnLn,a*(f)-fφγ=0.
Proof

From ([12], p. 195, Th. 6), it is enough to show that

stA-limnLn,a*(ei)-eiφ=0.

From Lemma 2.3, we get

stA-limnLn,a*(e0)-e0φ=0.

Again by using Lemma 2.3, we have

Ln,a*(e1)-e1φ=32nsupx[0,)11+x232n.

For ε > 0, we define the following sets:

S:={n:Ln,a*(e1)-e1φɛ}S1:={n:32nɛ},

which yields us SS1, hence for all n ∈ ℕ, we have kSankkS1ank.

Therefore, we get stA-limnLn,a*(e1)-e1φ=0.

Similarly, we have

Ln,a*(e2)-e2φ1n(4+1a-1)supx[0,)x1+x2+103n2supx[0,)11+x214(4+1a-1)+103n2.

Now, we define the following sets:

T:={n:Ln,a*(e2)-e2φɛ},T1:={n:1n(4+1a-1)ɛ2},T2:={n:103n2ɛ2}.

In view of (3.10), it is clear that TT1T2, which yields us

kTankkT1ank+kT2ank.

Thus, we get stA-limnLn,a*(e2)-e2φ=0.

Similarly, from Lemma 2.4, we have

stA-limnLn,a*((e1-xe0)j)φ=0,j=0,1,2,3,4.

Next, we prove a Voronovskaja type theorem for the operators Ln,a*.

Theorem 3.7

Let A = (ank) be a nonnegative regular infinite summability matrix. Then, for everyfCφ*[0,)such that f′, fCφ*[0,), we have

stA-limnn(Ln,a*(f;x)-f(x))=32f(x)+12ax(a-1)f(x),

uniformly with respect to x ∈ [0, E], (E > 0).

Proof

Let f, f′, fCφ*[0,). For each x ≥ 0, define a function

Θ(t,x)={f(t)-f(x)-(t-x)f(x)-12(t-x)2f(x)(t-x)2iftx0,ift=x.

Then

Θ(x,x)=0and Θ(·,x)Cφ*[0,).

Thus, we have

f(t)=f(x)+(t-x)f(x)+12(t-x)2f(x)+(t-x)2Θ(t,x).

Operating by Ln,a* on the above equality, we obtain

n(Ln,a*(f;x)-f(x))=f(x)nLn,a*((t-x);x)+12f(x)nLn,a*((t-x)2;x)+nLn,a*((t-x)2Θ(t,x);x).

In view of Lemma 2.4, we get

stA-limnnLn,a*((t-x);x)=32,stA-limnnLn,a*((t-x)2;x)=ax(a-1),

and

stA-limnn2Ln,a*((t-x)4;x)=x2(37-44a-1+3(a-1)2),

uniformly with respect to x ∈ [0, E].

Applying Cauchy-Schwarz inequality, we have

nLn,a*((t-x)2Θ(t,x);x)n2Ln,a*((t-x)4;x)Ln,a*(Θ2(t,x);x).

Let η(t, x) = Θ2(t, x), we observe that η(x, x) = 0 and η(·,x)Cφ*[0,). It follows from [12] that

stA-limnLn,a*(Θ2(t,x);x)=stA-limnLn,a*(η(t,x);x)=η(x,x)=0,

uniformly with respect to x ∈ [0, E]. Hence, using (3.14), we obtain

stA-limnnLn,a*((t-x)2Θ(t,x);x)=0,

uniformly in x ∈ [0, E]. Combining (3.12), (3.13) and (3.15), we get the desired result.

Now, we obtain the rate of A-statistical convergence for the operators Ln,a* with the help of Peetre’s K-functional.

Theorem 3.8

Let fW2. Then, we have

stA-limnLn,a*(f)-fCB[0,)=0.
Proof

By our hypothesis, from Taylor’s expansion we find that

Ln,a*(f;x)-f(x)=f(x)Ln,a*((e1-x);x)+12f(χ)Ln,a*((e1-x)2;x);

where χ lies between t and x. Thus, we get

Ln,a*(f)-fCB[0,)fCB[0,)Ln,a*((e1-·),·)CB[0,)+fCB[0,)Ln,a*((e1-·)2,·)CB[0,)=C1+C2,         say.

Using (3.11) for ε > 0, we have

limnk:C1ɛ2ank=0,limnk:C2ɛ2ank=0.

From (3.17), we may write

k:Lk,a*(f)-fCB[0,)ɛankk:C1ɛ2ank+k:C2ɛ2ank.

Hence taking the limit as n→∞, we get the desired result.

Now we give an estimate of the rate of convergence by means of ω2(f, δ).

Theorem 3.9

Let fCB[0, ∞), we have

Ln,a*(f)-fCB[0,)Mω2(f,δn,a),

whereδn,a=Ln,a*((e1-·),·)CB[0,)+Ln,a*((e1-·)2,·)CB[0,).

Proof

Let gW2, by (3.17), we have

Ln,a*(g)-gCB[0,)Ln,a*((e1-·),·)CB[0,)gCB[0,)+12Ln,a*((e1-·)2,·)CB[0,)gCB[0,)δn,agW2.

Using (3.18), for every fCB[0, ∞) and gW2, we get

Ln,a*(f)-fCB[0,)Ln,a*(f)-Ln,a*(g)CB[0,)+Ln,a*(g)-gCB[0,)+g-fCB[0,)2g-fCB[0,)+Ln,a*(g)-gCB[0,)2g-fCB[0,)+δn,agW2.

Taking the infimum on the right hand side over all gW2, we obtain

Ln,a*(f)-fCB[0,)2K2(f,δn,a).

Using (2.2), we have

Ln,a*(f)-fCB[0,)M{ω2(f,δn,a)+min(1,δn,a)fCB[0,)}.

From (3.11), we get stA-limnδn,a=0, hence stA-ω2(f,δn,a)=0. Therefore, we get the rate of A-statistical convergence of the sequence Ln,a*(f;x) to f(x) in the space CB[0, ∞). If we take A = I, we obtain the ordinary rate of convergence of these operators.

Now, we shall estimate the rate of convergence for the operators Ln,a* for functions with derivatives of bounded variation defined on (0, ∞) at points x where f′(x+) and f′(x−) exist, we shall prove that the operators (1.3) converge to the limit f(x). In this direction, the significant contributions have been made by (cf. [1, 3, 19, 22, 23, 28] etc).

Let fDBVγ(0, ∞), γ ≥ 0 be the class of all functions defined on (0, ∞), having a derivative of bounded variation on every finite subinterval of (0, ∞) and |f(t)| ≤ Mtγ,t > 0.

We notice that the functions fDBVγ(0, ∞) possess a representation

f(x)=0xg(t)dt+f(0),

where g(t) is a function of bounded variation on each finite subinterval of (0, ∞).

The following theorem is our main result.

Theorem 3.10

Let fDBVγ(0, ∞). Then, for every x ∈ (0, ∞) and sufficiently large n, we have

Ln,a*(f;x)-f(x)34nf(x+)+f(x-)+η(a)xn|f(x+)-f(x-)2|+η(a)nk=1[n]x-(x/k)x(fx)+xnx-(x/n)x(fx)+η(a)nk=0[n]xx+(x/k)(fx)+xnxx+(x/n)(fx),

whereab(fx)denotes the total variation offxon [a, b] andfxis defined by

fx(t)={f(t)-f(x-),0t<x0,t=xf(t)-f(x+)x<t<.
Proof

Since Ln,a*(1;x)=1, for every x ∈ (0, ∞) we get (see(2.1))

Ln,a*(f;x)-f(x)=0Kn*(x,t)(f(t)-f(x))dt=0Kn*(x,t)xtf(u)dudt.

For any fDBVγ(0, ∞), by (3.19) we may write

f(u)=fx(u)+12(f(x+)+f(x-))+12(f(x+)-f(x-))sgn(u-x)+δx(u)[f(u)-12(f(x+)+f(x-))],

where

δx(u)={1,u=x0,ux.

Obviously,

0(xt(f(u)-12(f(x+)+f(x-)))δx(u)du)Kn*(x,t)dt=0.

We may write

0(xt12(f(x+)+f(x-))du)Kn*(x,t)dt=12(f(x+)+f(x-))0(t-x)Kn*(x,t)dt=12(f(x+)+f(x-))Ln,a*((t-x);x)

and on an application of Cauchy-Schwarz inequality

0Kn*(x,t)(xt12(f(x+)-f(x-))sgn(u-x)du)dt12f(x+)-f(x-)0t-xKn*(x,t)dt12f(x+)-f(x-)Ln,a*(t-x;x)12f(x+)-f(x-)(Ln,a*((t-x)2;x))1/2.

Using Remark 2.1, and (3.203.21) we obtain the following estimate

Ln,a*(f;x)-f(x)34nf(x+)+f(x-)+12η(a)xnf(x+)-f(x-)+|0x(xtfx(u)du)Kn*(x,t)dt+x(xtfx(u)du)Kn*(x,t)dt|.

Now, let

An(fx,x)=0x(xtfx(u)du)Kn*(x,t)dt,

and

Bn(fx,x)=x(xtfx(u)du)Kn*(x,t)dt.

To complete the proof, it is sufficient to estimate the terms An(fx,x) and Bn(fx,x). From Lemma 2.6, since abdtβn*(x,t)1 for all [a, b] ⊆ (0, ∞), using integration by parts and applying Lemma 2.6 with y=x-(x/n), we have

An(fx,x)=|0xxt(fx(u)du)dtβn*(x,t)|=|0xβn*(x,t)fx(t)dt|(0y+yx)fx(t)βn*(x,t)dtη(a)xn0ytx(fx)(x-t)-2dt+yxtx(fx)dtη(a)xn0ytx(fx)(x-t)-2dt+xnx-(x/n)x(fx).

By the substitution of u = x/(xt), we obtain

η(a)xn0x-(x/n)(x-t)-2tx(fx)dt=η(a)xnx-11nx-(x/u)x(fx)duη(a)nk=1[n]kk+1x-(x/u)x(fx)duη(a)nk=1[n]x-(x/k)x(fx).

Thus,

An(fx,x)η(a)nk=1[n]x-(x/k)x(fx)+xnx-(x/n)x(fx).

Using integration by parts and applying Lemma 2.6 with z=x+(x/n), we have

|Bn(fx,x)|=|x(xtfx(u)du)Kn*(x,t)dt|=|xz(xtfx(u)du)dt(1-βn*(x,t))+z(xt(fx)(u)du)dt(1-βn*(x,t))|=|[(xtfx(u)du)(1-βn*(x,t))]xz-xzfx(t)(1-βn*(x,t))dt+z(xtfx(u)du)dt(1-βn*(x,t))|=|xzfx(u)du(1-βn*(x,z))-xzfx(t)(1-βn*(x,t))dt+[xtfx(u)du(1-βn*(x,t))]z-zfx(t)(1-βn*(x,t))dt|=|xzfx(t)(1-βn*(x,t))dt+zfx(t)(1-βn*(x,t))dt|η(a)xnzxt(fx)(t-x)-2dt+xzxt(fx)dt=η(a)xnx+(x/n)xt(fx)(t-x)-2dt+xnxx+(x/n)(fx).

By the substitution of v = x/(tx), we get

Bn(fx,x)η(a)n0nxx+(x/v)(fx)dv+xnxx+(x/n)(fx)η(a)nk=0[n]kk+1xx+(x/k)(fx)dv+xnxx+(x/n)(fx)=η(a)nk=0[n]xx+(x/k)(fx)+xnxx+(x/n)(fx).

Collecting the estimates (3.22)–(3.24), we get the required result. This completes the proof of the theorem.

The first author is thankful to the “University Grant Commission” India for financial support to carry out the above research work.

  1. Acar, T, Gupta, V, and Aral, A (2011). Rate of convergence for generalized Szász operators. Bull Math Sci. 1, 99-113.
    CrossRef
  2. Agrawal, PN, Finta, Z, and Kumar, AS (2015). Bernstein-Schurer-Kantorovich operators based on q-integers. Appl Math Comput. 256, 222-231.
  3. Agrawal, PN, Gupta, V, Kumar, AS, and Kajla, A (2014). Generalized Baskakov-Szász type operators. Appl Math Comput. 236, 311-324.
  4. Altomare, F, Montano, MC, and Leonessa, V (2013). On a generalization of Szász-Mirakjan-Kantorovich operators. Results Math. 63, 837-863.
    CrossRef
  5. Anastassiou, GA, and Duman, O (2008). A Baskakov type generalization of statistical Korovkin theory. J Math Anal Appl. 340, 476-486.
    CrossRef
  6. Başcanbaz-Tunca, G, and Erençin, A (2011). A Voronovskaya type theorem for q-Szász-Mirakyan-Kantorovich operators. Revue d’Analyse Numerique et de Theorie de l’Approximation. 40, 14-23.
  7. Butzer, PL (1954). On the extensions of Bernstein polynomials to the infinite interval. Proceedings of the American Mathematical Society. 5, 547-55.
    CrossRef
  8. Chihara, TS (1978). An Introduction to Orthogonal Polynomials. NewYork: Gordon and Breach
  9. Devore, RA, and Lorentz, GG (1993). Constructive Approximation. Berlin: Springer
    CrossRef
  10. Duman, EE, and Duman, O (2011). Statistical approximation properties of high order operators constructed with the Chan-Chayan-Srivastava polynomials. Appl Math Comput. 218, 1927-1933.
  11. Duman, EE, Duman, O, and Srivastava, HM (2006). Statistical approximation of certain positive linear operators constructed by means of the Chan-Chayan-Srivastava polynomials. Appl Math Comput. 182, 231-222.
  12. Duman, O, and Orhan, C (2004). Statistical approximation by positive linear operators. Studia Math. 161, 187-197.
    CrossRef
  13. Duman, O, özarslan, MA, and Della Vecchia, B (2009). Modified Szász-Mirakjan-Kantorovich operators preserving linear functions. Turkish Journal of Mathematics. 33, 151-158.
  14. Gadjiev, AD, Efendiyev, RO, and Ibikli, E (2003). On Korovkin type theorem in the space of locally integrable functions. Czech Math J. 53, 45-53.
    CrossRef
  15. Gadjiev, AD, and Orhan, C (2002). Some approximation theorems via statistical convergence. Rocky Mountain J Math. 32, 129-138.
    CrossRef
  16. Gupta, V, and Agarwal, RP (2014). Convergence Estimates in Approximation Theory: Springer
    CrossRef
  17. Ibikli, E, and Gadjieva, EA (1995). The order of approximation of some unbounded function by the sequences of positive linear operators. Turkish J Math. 19, 331-337.
  18. Ismail, MEH (2005). Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge: Cambridge University Press
    CrossRef
  19. Ispir, N (2007). Rate of convergence of generalized rational type Baskakov operators. Math Comput Modelling. 46, 625-631.
    CrossRef
  20. Ispir, N, and Gupta, V (2008). A-statistical approximation by the generalized Kantorovich-Bernstein type rational operators. Southeast Asian Bull Math. 32, 87-97.
  21. Jakimovski, A, and Leviatan, D (1969). Generalized Szász operators for the approximation in the infinite interval. Mathematica (Cluj). 34, 97-103.
  22. Kajla, A, and Agrawal, PN (2015). Szász-Durrmeyer type operators based on Charlier polynomials. Appl Math Comput. 268, 1001-1014.
  23. Karsli, H (2007). Rate of convergence of new gamma type operators for functions with derivatives of bounded variation. Math Comput Modelling. 45, 617-624.
    CrossRef
  24. Miclaus, D (2010). The Voronovskaja type theorem for the Szász-Mirakjan-Kantorovich operators. Journal of Science and Arts. 2, 257-260.
  25. Nowak, G, and Sikorska-Nowak, A (2009). Some approximation properties of modified Szasz-Mirakyan-Kantorovich operators. Revue d’Analyse Numerique et de Theorie de l’Approximation. 38, 73-82.
  26. Örkçü, M, and Doğru, O (2012). Statistical approximation of a kind of Kantorovich type q-Szász-Mirakjan operators. Nonlinear Anal. 75, 2874-2882.
    CrossRef
  27. Örkçü, M, and Doğru, O (2011). Weighted statistical approximation by Kantorovich type q-Szász-Mirakjan operators. Appl Math Comput. 217, 7913-7919.
  28. özarslan, MA, Duman, O, and Kaanoğlu, C (2010). Rates of convergence of certain King-type operators for functions with derivative of bounded variation. Math Comput Modelling. 52, 334-345.
    CrossRef
  29. Özarslan, MA, and Aktuğlu, H (). Local approximation for certain King type operators. Filomat. 27, 173-181.
  30. Radu, C (2009). On statistical approximation of a general class of positive linear operators extended in q-calculus. Appl Math Comput. 215, 2317-2325.
  31. Szász, O (1950). Generalization of S. Bernsteins polynomials to the infinite interval. J Res Nat Bur Standards. 45, 239-245.
    CrossRef
  32. Totik, V (1983). Approximation by Szasz-Mirakjan-Kantorovich operators in Lp(p > 1). Analysis Mathematica. 9, 147-167.
    CrossRef
  33. Varma, S, Sucu, S, and Icoz, G (2012). Generalization of Szász operators involving Brenke type polynomials. Comput Math Appl. 64, 121-127.
    CrossRef
  34. Varma, S, and Taşdelen, F (2012). Szász type operators involving Charlier polynomials. Math Comput Modelling. 56, 118-122.
    CrossRef
  35. Walczak, Z (2002). On approximation by modified Szasz-Mirakyan operators. Glasnik Matematicki. 37, 303-319.
  36. Yüksel, I, and Ispir, N (2006). Weighted approximation by a certain family of summation integral-type operators. Comput Math Appl. 52, 1463-1470.
    CrossRef