# The Approximation Szász-Chlodowsky Type Operators Involving Gould-Hopper Type Polynomials

Authors: Behar Baxhaku; Artan Berisha
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

We introduce the Szász and Chlodowsky operators based on Gould-Hopper polynomials and study the statistical convergence of these operators in a weighted space of functions on a positive semiaxis. Further, a Voronovskaja type result is obtained for the operators containing Gould-Hopper polynomials. Finally, some graphical examples for the convergence of this type of operator are given.


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## Body

## 1. Introduction

In [1], Jakimovski and Leviatan introduced and investigated some approximation properties of the Favard-Szász type operator, by using Appell polynomials pk(x)≥0 which satisfy the identity(1)gtetx=∑k=0∞pkxtk,where g(z)=∑k=0∞akzk is an analytic function in the disc z<R, (R>1) and g(1)≠0,(2)Jf;x=enxg1∑k=0∞pknxfknbn,x∈0,∞.Varma et al. constituted a link between orthogonal polynomials and the positive linear operators. In [1], they proposed Szász operators involving the Brenke polynomials defined by (3)Tnf;x=1A1Bnx∑k=0∞pknxfkn,where x≥0 and n∈N.Recently, Büyükyazıcı et al. in [2] introduced the Chlodowsky variant of operators (3). Inspired by this work, we give the Szász-Chlodowsky type operators including Gould-Hopper polynomials. The generating functions for these Gould-Hopper polynomials are given by(4)ehtd+1ext=∑k=0∞gkd+1x,htkk!and the explicit representations(5)gkd+1x,h=∑k=0k/d+1k!s!k-d+1s!hsxk-d+1s,where · denotes the integer part. Now, let us define Szász-Chlodowsky type generalization of the Szász operators with the help of generating function (4), as follows:(6)Gn,hdf;x=e-nx/bn-h∑k=0∞gkd+1nx/bn,hk!fknbn,x∈0,∞,where h≥0 and bn is a positive increasing sequence with the properties(7)limn→∞⁡bn=∞,limn→∞⁡bnn=0.The generalization of Szász type operators has been studied in [3–10].The aim of the this paper is to study some direct results in terms of the modulus of continuity of the second order, convergence of derivative operators to derivative functions, the weighted space, and the degree of approximation off by Gn,h(d). We also study the statistical convergence. The rate of convergence of the operators Gn,h(d) to a certain function is also illustrated through graphics using Matlab.


## 2. Notations and Auxiliary Results

Let us denoteCE0,∞≔{f∈C0,∞:ft≤MeNt,t∈0,∞, for some M,N>0}. The following notations and lemmas are needed to prove the main results.In what follows, letei(t)=ti, i∈N0, be the test functions.Lemma 1.
From (4), one has (i)
∑k=0∞(gkd+1nx/bn,h/k!)=enx/bn+h=κ1(x);(ii)
∑k=0∞(kgkd+1nx/bn,h/k!)=enx/bn+hnx/bn+h(d+1)=κ2(x);(iii)
∑k=0∞(k2gkd+1nx/bn,h/k!)=enx/bn+hn2x2/bn2+(nx/bn)2h(d+1)+1+h(d+1)2(h+1)=κ3(x);(iv)
∑k=0∞(k3gkd+1nx/bn,h/k!)=enx/bn+h(n3x3/bn3+(3n2x2/bn2)hd+1+1+(3nx/bn)h(d+1)h(d+1)+d+2+1/3+h(d+1)3(h2+3h+1))=κ4(x);(v)
∑k=0∞(k4gkd+1nx/bn,h/k!)=enx/bn+h(n4x4/bn4+(2n3x3/bn3)2hd+1+3+(6n2x2/bn2)h2(d+1)2+hd+1d+3+7/6 + (2nx/bn)3h2(d+1)2(2d+3)+h(d+1)(2d2+7d+7)+2h3(d+1)3+1/2+h(d+1)4(h3+6h2+7h+1))=κ5(x).Lemma 2.
For the operatorsGn,h(d), one has (i)
Gn,h(d)(e0;x)=1;(ii)
Gn,hd(e1;x)=x+(bn/n)h(d+1);(iii)
Gn,hd(e2;x)=x2+(bnx/n)2hd+1+1+(bn2/n2)h(h+1)(d+1)2;(iv)
Gn,hd(e3;x)=x3+(3bnx2/n)hd+1+1+(3bn2x/n2)hd+1hd+1+d+2+1/3+(3bn3/n3)h(d+1)2(d+1)(h2+1)+h(2d+1);(v)
Gn,hd(e4;x)=x4+(2bnx3/n)2hd+1+3+(6bn2x2/n2)h2(d+1)2+hd+1d+3+7/6 + (2bn3x/n3)3h2(d+1)2(2d+3)+h(d+1)(2d2+7d+7)+2h3(d+1)3+1/2+(bn4/n4)h(d+1)4(h3+6h2+7h+1).Proof.
From Lemma2 and by definition of Gn,h(d), we have(8)Gn,hde0;x=e-nx/bn-hκ1x=1,Gn,hde1;x=bnne-nx/bn-hκ2x=x+bnnhd+1.Now, we consider the case Gn,h(d)(e2;x) as follows: (9)Gn,hde2;x=bn3n3e-nx/bn-hκ4x=x2+bnxn2hd+1+1+bn2n2hh+1d+12.For e3, we have (10)Gn,hde3;x=bn2n2e-nx/bn-hκ3x=x3+3bnx2nhd+1+1+3bn2xn2hd+1hd+1+d+2+13+3bn3n3hd+12d+1h2+1+h2d+1,and, finally, (11)Gn,hde4;x=bn4n4e-nx/bn-hκ5x=x4+2bnx3n2hd+1+3+6bn2x2n2h2d+12+hd+1d+3+76+2bn3xn33h2d+122d+3+hd+12d2+7d+7+2h3d+13+12+bn4n4hd+14h3+6h2+7h+1.Lemma 3.
Letηn,sd(x)=Gn,h(d)((e1-x)s;x), and then, for every x∈0,∞, one has (i)
ηn,1d(x)=(bn/n)h(d+1);(ii)
ηn,2d(x)=bnx/n+(bn2/n2)h(h+1)(d+1)2;(iii)
ηn,4d(x)=3bn2x2/n2+(2bn3x/n3)(d+1)3h(-2h2+3h-3)+h(d+1)2-3h+7-12hd+2d2+1/2+(bn4/n4)h(d+1)4(h3+6h2+7h+1).Theorem 4.
Letf∈CE0,∞. Then, limn→∞⁡Gn,h(d)(f;x)=f(x), uniformly on each compact subset of 0,∞.Proof.
By Lemma2, limn→∞⁡Gn,h(d)(ei;x), i=0,1,2, uniformly on every compact subset of 0,∞. So, by Bohman-Korovkin theorem, the result follows.Example 5.
Forn=50, 100, and 500, d=0.5, and bn=n, the convergence of Gn,h(d)(f;x) to fx=xcos⁡(x+1) is illustrated in Figure 1(a).Figure 1
(a)
(b)Example 6.
Forn=50, 100, and 500, d=0.5, and bn=n, the convergence of Gn,h(d)(f;x) to f(x)=x2/x2+1 is illustrated in Figure 1(b).Theorem 7.
Letf∈CE0,∞, and then for any x∈0,c one has (12)Gn,hdf;x-fx≤1+c+bnnhh+1d+12ωf;bnn.Proof.
By using Lemma3 case (ii) and the well-known properties of the modulus of continuity, we have(13)Gn,hdf;x-fx≤Gn,hdfknbn-fx;x≤1+1δGn,hdknbn-x;xωf;δ.Recalling the Cauchy-Schwarz inequality, we obtain the formula below:(14)Gn,hdf;x-fx≤1+1δGn,hdknbn-x2;x1/2ωf;δ=1+1δGn,hdt-x2;x.By means of Lemma 3 case (ii), for 0≤x≤c, one gets (15)Gn,hde1-x2;x≤bncn+bn2n2hh+1d+12.Using (22) and taking δ=bn/n in (19), we obtain the desired result.The theorem below shows that the derivative(dr/dxr)Gn,h(d)(f;x) is also an approximation process for drf/dxr.Theorem 8.
Letf∈CE0,∞. If f(r) exists at a point x∈(0,∞), then one has (16)drdxrGn,hdf;x-drdxrfx≤r!1+c+bnnhh+1d+12ωf;bnn+ωdrdxrf;rbnn,where ω(drf/dxr,·) is the modulus of continuity of drf/dxr.Proof.
By simple calculations, the following formula is obtained:(17)drdxrGn,hdf;x=nbnrGn,hdΔbn/nrf;x,where Δbn/nrf((k/n)bn) is the difference of order r of f corresponding to the increment bn/n. Using the relation between finite difference and divided difference, the derivative of order r of the operators is represented as follows: (18)drdxrGn,hdf;x=r!Gn,hdΔbn/nrfr!bn/nr;x=r!e-nx/bn-h∑k=0∞gkd+1nx/bn,hk!knbn,k+1nbn,…,k+rnbn;f=r!Gn,hdμ;x,where μ(x)=x,x+bn/n,…,x+r(bn/n);f. Then, using Theorem 4, we have(19)drdxrGn,hdf;x-drdxrfx≤r!Gn,hdμ;x-μx+r!μx-drdxrfx≤r!1+c+bnnhh+1d+12ωf;bnn+r!μx-drdxrfx.By using the mean value theorem and some known classical properties of the modulus of continuity [11], one has (20)μx+δ-μx=x+δ,x+δ+bnn,…,x+δ+rbnn;f-x,x+bnn,…,x+rbnn;f+1r!drdxrfx+δ+rbnnϕ1-drdxrfx+rbnnϕ2+1r!ωdrdxrf;δ+rbnnϕ1-ϕ2≤1r!ωdrdxrf;δ+rbnn,where ϕ1,ϕ2∈(0,1). Hence, we obtain (21)ωμ,δ≤1r!ωdrdxrf;δ+rbnn.On the other hand, (22)r!μx-drdxrfx≤r!x,x+bnn,…,x+rbnn;f-drdxrfx≤drdxrfx+rbnnϕ3-drdxrfx≤ωdrdxrf;ϕ3rbnn≤ωdrdxrf;rbnn,where ϕ3∈(0,1). Using the estimates in (15), we have the desired result.


## 3. Weighted Approximation

LetDψ0,∞ be the space of all functions f defined on 0,∞ satisfying the condition |f(x)|≤Mfψ(x), where Mf is a positive constant depending only on f and ψ(x)=1+x2 is a weight function. By Cψ0,∞, we denote the subspace of all continuous functions f∈Dψ0,∞ with the norm fψ=supx∈0,∞⁡|f(x)|/(1+x2) and Cψ∗0,∞=f∈Cψ0,∞:limn→∞⁡f(x)/(1+x2)<∞.Theorem 9.
Letf∈Cψ0,∞, and let ψ=1+x2 be a weighted function; then, the inequality (23)Gn,hdψ;xψ≤1+Mfis satisfied.Proof.
Using Lemma2, one has (24)Gn,hdf;xψ=supx≥011+x21+x2+bnxn2hd+1+1+bn2n2hh+1d+12≤1+bnn2hd+1+1+bn2n2hh+1d+12.Since limn→∞⁡(bn/n)=0, there exists constant Mf such that (25)Gn,hdψ;xψ≤1+Mf.This proof is complete.From [12], for f∈Cψ∗0,∞, the weighted modulus of continuity of f is defined by (26)Ωψf;δ=sup0≤ψx-ψt<δ,x,t∈0,∞⁡fx-ftψx-ψt+1ψx,where ψ is a continuously differentiable function on 0,∞, ψ(0)=0, and infx≥0ψ′(x)≥1. Now, we define following sequence of positive linear operators Pn,h(d) defined with the help of Gn,h(d) defined in (6): (27)Pn,hdf;x=e-nx/bn-hψ2x∑k=0∞gkd+1nxbn,h1ψ2k/nbnk!fknbn,x∈0,∞.Theorem 10 (see [12]).
Let(Ln) be the sequences of linear positive operators and η(x)≤ψk(x), k=1,2,3. If(28)Ln1-1ψ1=αn,Lnρ-ρψ2=βn,Lnρ2-ρ2ψ3=γn,where ψ(x)=max⁡{ψ1,ψ2,ψ3} and αn, βn, and γn tend to zero as n→∞, then, for all functions f∈Cψ∗0,∞, the inequality (29)Lnf;x-fψρ2≤Ωρf;αn+βn+γn+fραnholds for n large enough.Theorem 11.
LetPn,h(d) be the sequences of linear positive operators defined by (27) and η(x)=1+x2. If f∈Cψ∗0,∞, then the inequality (30)Pn,hdf;x-fψ4η≤16Ωψf;αn+2βn+fψαnis satisfied, where αn=(2h(d+1)+1)(bn/n)+(bn/n)2h(h+1)(d+1)2 and βn=(bn/n)h(d+1).Proof.
By simple calculation, we have(31)Pn,hd1;x-1=ψ2xe-nx/bn-h∑k=0∞gkd+1nxbn,h1ψ2k/nbnk!fknbn-1ψ2x,(32)Pn,hdt;x-x=ψ2xe-nx/bn-h∑k=0∞gkd+1nxbn,h1ψk/nbnk!fknbn-1ψx,(33)Pn,hdψ2;x-ψ2x=0.From Lemma 2, we have (34)limn→∞⁡e-nx/bn-h∑k=0∞gkd+1nxbn,h1ψ2k/nbnk!fknbn-1ψ2xη=0,limn→∞⁡e-nx/bn-h∑k=0∞gkd+1nxbn,h1ψk/nbnk!fknbn-1ψxψ=0.Using Lemma 2 and (31), we obtain (35)Pn,hd1;x-1ψ2η=limn→∞⁡e-nx/bn-h∑k=0∞gkd+1nxbn,h1ψ2k/nbnk!fknbn-1ψ2xη≤2hd+1+1bnn+bnn2hh+1d+12=αn.By means of Lemma 2 and (32), one gets (36)Pn,hdψ;x-ψψ2η=limn→∞⁡e-nx/bn-h∑k=0∞gkd+1nxbn,h1ρk/nbnk!fknbn-1ψxη≤bnnhd+1=βn.Finally, from (33), we obtain (37)Pn,hdψ2;x-ψ2ψ2η=0=γn.If we apply Theorem 10, we obtain the desired result.


## 4.A-Statistical Convergence

Now, letA=ajn,  j,n∈N, be an infinite summability matrix. For a given sequence (xn), the A-transform of x, denoted by ((Ax)j), is given by (38)Axj=∑n=1∞ajnxnwhich provides the converging series for each j∈N. We say that A is a regular if limj⁡(Ax)j=L whenever limn⁡(xn)=L. A sequence (xn) is called A-statistically convergent to L if, for every ϵ>0, limj⁡∑n:|xn-L|≥ϵajn=0. This limit is denoted by stA-limn⁡xn=L. Replacing A by C1, the Cesàro matrix of order one in (6), from A-statistical convergence, is reduced to statistical convergence. Similarly, if we take A=I, the identity matrix, then A-statistical convergence coincides with the ordinary convergence. Kolk [13] proved that, in the case of limj⁡maxn⁡=0, A-statistical convergence is stronger than ordinary convergence.Further, we will first obtain the following weighted Korovkin theorem viaA-statistical convergence.Theorem 12.
Let(ank) be a nonnegative regular infinite summability matrix and x∈0,∞. Let ψγ≥1 be a continuous function such that (39)limn→∞⁡ψxψγ-1x=0.Then, for all f∈Cψ∗0,∞, we have(40)stA-limn→∞⁡Gn,hdf;x-fψγ=0.Proof.
From [14], for any f∈Cψ∗0,∞, it is enough to prove that (41)stA-limn→∞⁡Gn,hdei;x-eiψγ=0,forei=ti,i=0,1,2.So, by Lemma 2, we can easily get (42)stA-limn→∞⁡Gn,hde0;x-e0ψγ=0.Again, by using Lemma 2, we have(43)Gn,hde1;x-e1ψ=bnnhd+1supx∈0,∞⁡11+x2≤bnnhd+1.Now, for every given ϵ>0, let us define the following sets: (44)S=n:Gn,hde1;x-e1ψ≥ϵ,S1=n:bnnhd+1≥ϵ.It is clear that S⊆S1. Hence, for all n∈N, we get (45)∑k∈Sank≤∑k∈S1ank.Therefore, stA-limn→∞⁡Gn,hde1;x-e1ψγ=0.
Similarly, we have(46)Gn,hde2;x-e2ψ=supx∈0,∞⁡x1+x2bnn2hd+1+1+bn2n2hh+1d+12supx∈0,∞⁡11+x2≤2hd+1+1bnn+bnn2hh+1d+12.Now, we define the following sets: (47)U=n:Gn,hde2;x-e2ψ≥ϵ,U1=n:2hd+1+1bnn≥ϵ2,U2=n:bnn2hh+1d+12≥ϵ2.In view of (46), it is clear that U⊆U1∪U2, which yields (48)∑k∈Uank≤∑k∈U1ank+∑k∈U2ank.Thus, we get stA-limn→∞⁡Gn,hde2;x-e2ψγ=0.
Similarly, from Lemma3, we have (49)stA-limn→∞⁡ηn,sdxψ=0,s=1,2,3,4.Now, we give a Voronovskaja type relation for the operators Gn,h(d).Theorem 13.
LetA=(ank) be a nonnegative regular infinite summability matrix. Then, for every f∈Cψ∗0,∞ such that f′,f′′∈Cψ∗0,∞, one has (50)stA-limn→∞⁡nbnGn,hdf;x-fx=hd+1f′x+x2f′′xuniformly with respect to x∈0,E, (E>0).Proof.
Letx≥0, and f′,f′′∈Cψ∗0,∞. Define the function θ by (51)θt,x=ft-fx-t-xf′x-1/2t-x2f′′xt-x2,ift≠x,0,ift=x.Then, by assumption, we get θ(x,x)=0 and θ(·,x)∈Cψ∗0,∞.
By the linearity ofGn,h(d) applied to the last equality, we obtain(52)nbnGn,hdf;x-fx=nbnηn,1dxf′x+12nbnηn,2dxf′′x+nbnGn,hdθt,xt-x2;x.In view of Lemma 2, we get (53)stA-limn→∞⁡nbnηn,1dx=hd+1,stA-limn→∞⁡nbnηn,2dx=x,stA-limn→∞⁡n2bn2ηn,4dx=3x2.For the last term of the right hand side in (52), using Cauchy-Schwarz inequality, we are led to(54)nbnGn,hdθt,xe1-x2;x≤Gn,hdξ2t,x;xn2bn2ηn,4dx.We observe that θ2(x,x)=0, and θ2·,x∈Cγ∗0,∞. In view of Theorem 4,(55)stA-limn→∞⁡Gn,hdθ2t,x;x=θ2x,x=0.Then, it follows from (32) that(56)stA-limn→∞⁡nbnGn,hdθt,xe1-x2;x=0uniformly in x∈a,b. Combining (27), (31), and (46), we get the desired result.
LetCB0,∞ denote the space of all real-valued bounded and uniformly continuous functions f on 0,∞, endowed with the norm f=supx∈0,∞⁡|f(x)|. Further, the appropriate Peetre K-functional is given by (57)K2f;δ=infg∈W2⁡f-g+δg′′,where Δ2={g∈CB0,∞:g′,g′′∈C~B0,∞}. By [15, Theorem 4] we can find a constant C>0 such that(58)K2f;δ≤Cω2f;δ,where ω2(f;δ) is the second-order modulus of continuity of f∈CB0,∞. Now, we obtain the rate of A-statistical convergence for the operators Gn,h(d) with the help of Peetre’s K-functional.Theorem 14.
Letf∈Δ2. Then, one has (59)stA-limn→∞⁡Gn,hdf;x-fCB0,∞=0.Proof.
By the linearity property ofGn,h(d) operators and using the local Taylor formula to the function f∈Δ2, we have(60)Gn,hdf;x-fx=ηn,1dxf′x+12ηn,2dxf′′ξ,where ξ∈(t,x).
Thus, we get(61)Gn,hdf;x-fCB0,∞=f′CB0,∞ηn,1dxCB0,∞+12f′′CB0,∞ηn,2dxCB0,∞.In view of (49) for ϵ>0, we have(62)limn⁡∑k∈N:I1≥ϵ/2ank=0,limn⁡∑k∈N:I2≥ϵ/2ank=0.From (61), one can write the following: (63)∑k∈N:Gn,hdf;x-fCB0,∞ank≤∑k∈N:I1≥ϵ/2ank+∑k∈N:I2≥ϵ/2ank=0.Hence, taking the limit as n→∞, we get the desired result.The following theorem contains quantitative estimates by means of Peetre’sK-functional.Theorem 15.
Lettingf∈CB0,∞, one has the estimate (64)Gn,hdf;x-fCB0,∞≤Mω2f;δn,where δn=ηn,1dxCB0,∞+ηn,2dxCB0,∞.Proof.
Lettingg∈Δ2, by (61), we have (65)Gn,hdf;x-fCB0,∞=f′CB0,∞ηn,1dxCB0,∞+12f′′CB0,∞ηn,s2xCB0,∞≤ηn,1dxCB0,∞+12ηn,2dxCB0,∞fΔ2.Using the last inequality for f∈CB0,∞, and g∈Δ2, we get(66)Gn,hdf;x-fCB0,∞=Gn,hdf;x-Gn,hdgCB0,∞+Gn,hdg;x-gCB0,∞+g-fCB0,∞≤g-fCB0,∞+Gn,hdg;x-gCB0,∞≤2g-fCB0,∞+δngW2.Taking the infimum on the right sides of the above inequality for all functions g∈Δ2, so (67)Gn,hdf;x-fCB0,∞≤K2f;δn.By using the relation between Peetre’s K-functional and the second modulus of smoothness given in [15], we get (68)Gn,hdf;x-fCB0,∞≤Mω2f;δn+min⁡1,δngCB0,∞.From (49), we get stA-limn→∞⁡δn=0, and hence stA-limn→∞⁡ω2(f;δn)=0. Therefore, we get the rate of A-statistical convergence of the sequence Gn,h(d) to f(x) in the space CB0,∞.



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