# Approximation Properties of a New Type of Gamma Operator Defined with the Help ofk-Gamma Function

Authors: Gurhan Icoz; Seda Demir
Journal: Journal of Function Spaces
 (2022)

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## Abstract

With the help of thek-Gamma function, a new form of Gamma operator is given in this article. Voronovskaya type theorem, weighted approximation, rates of convergence, and pointwise estimates have been found for approximation features of the newly described operator. Finally, numerical examples have been provided to demonstrate that the operator is approaching the function.


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## Body

## 1. Introduction

One of the most important topics in mathematical analysis is approximation theory. The theory is studied in almost every subject, including engineering and physics. Many mathematicians have made investigation in this area. In 1885 [1], Weierstrass claimed that polynomials can approximate every function in the closed interval a,b. Besides, theorems about this subject are prepared by Korovkin around 1950 [2]. The Korovkin approximation theorem is one of the well-known theorems in mathematics. Their theorems indicate that a series of positive linear operators can converge to the identity operator under specific condition [2]. As a result using these theorems, some studies on linear and positive linear operators have been added to the literature. For example, King [3] introduced the Bernstein operator to preserve the function a2h=h2 in 2003. Then, King constructed a new set of operators with respect to the test functions 1,h,h2 and obtained their linear combinations. On the other hand, one of these operators is the Gamma operator which is constructed by Lupas and Müller [4]. The classical Gamma operator in [4] is expressed as follows:
(1)Kmφ;y=ym+1Γm+1∫0∞e−yvvmφmvdv,∀y∈0,∞,m∈ℕ.Then, in the literature, some researchers introduced the generalizations of Gamma and beta functions and also the extensions of Gamma-type operators and their extensions [5–14]. One of the studies of this topic was by Daz and Pariguan [15]; they introduced and researched k-Gamma function when they were assessing Feynman integrals. k-Gamma function has been showed up various effects on mathematics and applications. One of these effects has been working the Schrodinger equation for harmonium and related models in view of important operations in quantum chemistry [16]. The others have used k-Gamma function for combinatorial analysis in statistic.According to these studies, thek-Gamma function was defined by Daz and Pariguan as follows:
(2)Γkz=∫0∞tz−1e−tk/kdt,k∈ℝ,Rez>0.As can be seen from the definition,Γk is a one parameter deformation of the classical Gamma function such that Γk⟶Γ as k⟶1. For k=2, it reduces to an integral of Gaussian functions [17]. When we get u=−tk/k in equation (2), we find the expression Γkz=kz/k−1Γz/k, and all properties of the classic Gamma operator can be generalized into k-Gamma function. It also led to a few new conclusions for k-Gamma function. A few of them are given that
(3)Γkk=1,Γkz+k=zΓkz,zn,k=Γkz+nkΓkz,in [15]. For more such properties of k-Gamma and related functions, we can refer to the article [11, 15, 17].The primary goal of this research is to give thek-Gamma operator given by (4) and its approximation properties. For the operator in (4), in Section 2, we will use Korovkin theorem in [18]. Then, in Section 3, we will consider the Voronovskaya type theorem. In Section 4, we will examine the weighted approximation. Later, we will give the rates of convergence with Peetre’s K-functional and Lipschitz class in Section 5. Moreover, in Section 6, we will obtain pointwise estimates, and finally, in Section 7, we will show the numerical examples for the operators in (4).


## 2. A New Modification of Gamma Operators Defined with the Help ofk-Gamma Function

We shall see a new type of Gamma operators defined with the help of thek-Gamma function in this section, and some findings will be presented in the rest of the article. In this paper, we will use the expressions azh=hz and ψy,z=h−yz, y∈0,∞ as polynomial functions. The modified representation of the classical Gamma operator is shown as follows:
(4)Km∗φ;y=ym+1+1/kΓkmk+k+1∫0∞e−yvvkm+1/kφmvdv,∀y∈0,∞,k>0,m∈ℕ,where for v>0, φ∈Cγ0,∞=φ∈C0,∞:φu=Ouγ,asu⟶∞ for m>γ. Here C0,∞ is the set of continuous functions on 0,∞. This modified operator is clearly positive and linear in this case. Furthermore, the new Gamma operator defined with the help of the k-Gamma function is directly preserved constant, and test functions are provided in case of limit.We note that for special case ofk=1/pp∈ℕ in (4), we have Schurer variant of Gamma operators in (1).The following lemma will be presented without proof and used in fundamental theorems for the rest of the paper.Lemma 1.
Lety∈0,∞. The following are the moment values:
(5)Km∗a0h;y=a0y,Km∗a1h;y=mkmk+1a1y,Km∗a2h;y=mk2mk+1mk−k+1a2y,Km∗a3h;y=mk3mk+1mk−k+1mk−2k+1a3y,Km∗a4h;y=mk4mk+1mk−k+1mk−2k+1mk−3k+1a4y.
By generalizing the moment values, we have the following lemma.Lemma 2.
Lety∈0,∞ and z∈ℕ,Km∗a0h;y=a0y. Then, the general formula for the following moment values is obtained
(6)Km∗azh;y=mkz∏i=0z−1mk−ki+1azy,z=1,2,⋯.Lemma 3.
Lety∈0,∞. Using the equations in Lemma 1, the following are obtained:
(7)Km∗ψy,0h;y=1,Km∗ψy,1h;y=−1mk+1a1y,Km∗ψy,2h;y=mk2−k+1mk+1mk−k+1a2y,Km∗ψy,3h;y=m3k3mk+1mk−k+1mk−2k+1−3m2k2mk+1mk−k+1+3mkmk+1−1a3y,Km∗ψy,4h;y=m4k4mk+1mk−k+1mk−2k+1mk−3k+1−4m3k3mk+1mk−k+1mk−2k+1+6m2k2mk+1mk−k+1−4mkmk+1+1a4y.As a result of our research, the Schurer variant of Gamma operators have not been defined or used. Also, if it is realized thatk∈ℝ+, it is obtained that our operators are a generalization of the Schurer type operators.Throughout this paper, we use the normφ=supφy:y∈0,∞ for φ∈C0,∞.Lemma 4.
Letφ∈CB0,∞. Then, we get
(8)Km∗φ;y≤φ.Proof.
By using the result of Lemma1, we have
(9)Km∗φ≤ym+1+1/kΓkmk+k+1∫0∞e−yvvkm+1/kφmvdv≤φym+1+1/kΓkmk+k+1∫0∞e−yvvkm+1/kdv=φKm∗a0h;y=φ.
Thus, we obtain the desired result. Because the moments are conserved in the limit state of the Korovkin test functions,Km∗ is an approximation process on any compact T⊂0,∞, according to the Korovkin theorem in [18].Theorem 5.
Letφ∈C0,∞∩E, where E=φ:limy⟶∞φ/1+y2=kconstant. Then, consistently in each compact subset of 0,∞, we have
(10)limm⟶∞Km∗φ;y=φy.Proof.
By using Lemma1, when z=0,1,2, we get
(11)limm⟶∞Km∗azh;y=azyfor uniformly each compact subset of 0,∞. Then, using the Korovkin theorem in [18], we give limm⟶∞Km∗φ;y=φy for uniformly each compact subset of 0,∞.


## 3. Voronovskaya Type Theorem

By establishing Voronovskaya’s theorem below, we will illustrate the asymptotic behavior ofKm∗m≥1 operators in this section.Theorem 6.
Letφ∈C0,∞∩E such that φ′,φ′′∈C0,∞∩E. The following limit is valid:
(12)limm⟶∞mKm∗φ;y−φy=−1kyφ′y+12y2φ′′y.Proof.
From the definition of Taylor formula(13)φh=φy+φ′yh−y+12φ′′yh−y2+Ωh,yh−y2,where
(14)Ωh,y=φ′′δ−φ′′y2,such that δ lying between y and h and
(15)limh⟶yΩh,y=0.
When theKm∗m≥1 operator is applied to (13), we get
(16)Km∗φ;y=φy+φ′yKm∗h−y;y+12φ′′yKm∗h−y2;y+Km∗Ωh,yh−y2;y.
To get the formula(17)mKm∗φ;y−φy=φ′ymKm∗h−y;y+12φ′′ymKm∗h−y2;y+mKm∗Ωh,yh−y2;y,multiply both sides of the last inequality by m. In the limit case, this equation is
(18)limm⟶∞mKm∗φ;y−φy=φ′ylimm⟶∞mKm∗h−y;y+12φ′′ylimm⟶∞mKm∗h−y2;y+limm⟶∞mKm∗Ωh,yh−y2;y.
We know the values(19)limm⟶∞mKm∗h−y;y=limm⟶∞m−1mk+1y=−1ky,limm⟶∞mKm∗h−y2;y=limm⟶∞mmk2−k+1mk+1mk−k+1y2=y2,using Lemma 3. So, we have
(20)limm⟶∞mKm∗φ;y−φy=−1kyφ′y+12φ′′yy2+limm⟶∞mKm∗Ωh,yψy,2h;y.
We show that the limit to the right of the equation in (20) is equal to zero. It can easily be said from the Cauchy-Schwarz inequality that
(21)mKm∗Ωh,yψy,2h;y≤Km∗Ω2h,y;ym2Km∗ψy,4h;y.
Then, using Korovkin theorem, we have(22)limm⟶∞Km∗Ω2h;y,y=Ω2y,y=0,since Ω2y,y=0 and Ω.,y∈C0,∞∩E and bounded as h⟶∞ and in view of fact that
(23)Km∗ψy,4h;y=O1m2,where Km∗ψy,4h;y=3m2k4+m18k4−22k3+6k2−6k3+11k2−6k+1/mk+1mk−k+1mk−2k+1mk−3k+1. The proof is completed when equations (21) and (22) are written in (13).


## 4. Weighted Approximation

The Korovkin theorem for weighted approximation of the operators in (4) is given in this section. To demonstrate this, we will follow the theorems given by Gadjiev [19].Considerϑy=1+y2 as continuous weighted function on ℝ, with limy⟶∞ϑy=∞, ϑy≥1 for all y∈0,∞. Let us have a look at the weighted spaces below. The property φy≤Nφϑy represents the weighted space of real-valued functions φ on ℝ. This subspace is denoted by
(24)Bϑ0,∞=φ:0,∞⟶0,∞:φy≤Nφϑy,y∈0,∞.Nφ is a constant depending on the functions φ.Since, the weighted subspaces ofBϑ0,∞ is given by
(25)Cϑ0,∞=φ∈Bϑ0,∞:φiscontinousonℝ=C0,∞∩Bϑ0,∞.Eventually, additional subspace for allφ∈Cϑ0,∞ for which limy⟶∞φy/ϑy exists finitely defined as
(26)Cϑκ0,∞=φ∈Cϑ0,∞:limy⟶∞φyϑy=κφexistsanditisfinite.Thisκφ is a constant dependent on the φ functions. All three mapping spaces above are normed spaces endowed with
(27)φϑ=supy∈0,∞φyϑy.Lemma 7.
Letφ∈Cϑ0,∞. Then, for the modified operator Km∗φ, we have
(28)Km∗φϑ≤Cφϑ,which imply that the sequence of the modified operators Km∗φ is an approximation process from Cϑ0,∞ to Bϑ0,∞.Proof.
The desired result of this lemma is easily obtained from properties of the modified Gamma operator and Lemma1.Gadjiev proposed a weighted approach to linear positive operator sequences for unbounded intervals in [19]. The following theorem is similar to the Gadjiev theorem.Theorem 8.
Letφ∈Cϑκ0,∞. For the modified Gamma operator, the following equality holds:
(29)limm⟶∞Km∗φ;y−φyϑ=0.Proof.
It will be enough to show that equivalence is attained forlimm⟶∞Km∗az;y−azϑ=0,z=0,1,2 using the theorem in [19]. For z=0, we have Km∗a0;y−a0ϑ=0. Now, let us examine the cases z=1,2. When the necessary results for these situations are used,
(30)Km∗a1;y−a1ϑ=supy∈0,∞Km∗a1;y−a11+y2=supy∈0,∞mk/mk+1y−y1+y2≤mkmk+1−1supy∈0,∞y1+y2.≤1mk+1is obtained. If we take the limit of this expression, it becomes
(31)limm⟶∞1mk+1=0.
Then, we have(32)Km∗a2;y−a2ϑ=supy∈0,∞Km∗a2;y−a21+y2=supy∈0,∞m2k2/mk+1mk−k+1y2−y21+y2≤m2k2mk+1mk−k+1−1supy∈0,∞y21+y2≤mk2−2mk+k−1mk+1mk−k+1.
If we take the limit of this expression, it becomes(33)limm⟶∞mk2−2mk+k−1mk+1mk−k+1=0.
As a result of the equations obtained above, the evidence is finished.


## 5. The Rates of Convergence

Now, we can concentrate on the rates of convergence the modified Gamma operator in terms of the modulus continuity. We shall now show thatKm∗φ outperforms the classical operator in terms of error estimation. Let us define the following in light of this goal.The modulus of continuity ofw is denoted by ωy0φ,δ for interval 0,y0,y0≥0 and can be described as follows:
(34)ωy0φ,δ=suph−y≤δ;y,h∈0,y0φh−φy.The modulus of continuityωy0φ,δ⟶0 is easily understood as δ⟶0 for the function φ∈CB0,∞, where CB0,∞ is defined as space of all continuous and bounded functions on the interval 0,∞. Now, let us look at the rates of convergence theorem for Km∗m≥1.Theorem 9.
Fory0>0 and φ∈CB0,∞, let ωy0+1φ,δ be the modulus of continuity on the finite interval 0,y0+1⊂0,∞. Then, the following inequality exists:
(35)Km∗φ;y−φy≤3Nφmk2−k+1mk+1mk−k+1y021+y02+2ωy0+1φ,mk2−k+1mk+1mk−k+1y02,where Nφ is a constant only according as φ.Proof.
Now, letφ∈CB0,∞,0<y≤y0, and h>y0+1.Then, we can conclude that
(36)φh−φy≤φh+φy≤3Nφh−y21+y02for h−y>1. Then, again let φ∈CB0,∞,0<y≤y0. So, the following inequality holds
(37)φh−φy≤ωy0+1φ,h−y≤ωy0+1φ,δ1+1δh−yfor h≤y0+1. As a result, from the above inequalitiy, we deduce that
(38)φh−φy≤3Nφh−y21+y02+ωy0+1φ,δ1+1δh−yfor 0<y≤y0 and 0<h<∞. Applying Km∗ and Cauchy-Schwarz inequality to (38), we obtain
(39)Km∗φ;y−φy≤3NφKm∗h−y2;y1+y02+ωy0+1φ,δ1+1δKm∗h−y2;y≤3Nφmk2−k+1mk+1mk−k+11+y02+2ωy0+1φ,mk2−k+1mk+1mk−k+1y02.By choosing δ=mk2−k+1/mk+1mk−k+1y02, we can conclude the proof.Let(40)CB20,∞=φ∈CB0,∞:φ′,φ′′∈CB0,∞,with the norm
(41)φCB20,∞=φCB0,∞+φ′CB0,∞+φ′′CB0,∞also
(42)φCB0,∞=supy∈0,∞φyin [20].Theorem 10.
LetKm∗ be the operator defined in (4). Then, for any φ∈CB20,∞, we have
(43)Km∗φ;y−φy≤12τ2+τφCB20,∞,where τ is Km∗ψy,2;y in Lemma 3.Proof.
Letφ∈CB20,∞. When referring to the Taylor series, obtain
(44)φh=φy+φ′yh−y+12φ′′ξh−y2,where ξ between y and h, from which it follows:
(45)φh−φy≤N1h−y+12N2h−y2,where
(46)N1=supy∈0,∞φ′y=φ′CB0,∞≤φCB20,∞,N2=supy∈0,∞φ′′y=φ′′CB0,∞≤φCB20,∞,because of (41). Thus, we have
(47)φh−φy≤h−y+12h−y2φCB20,∞.
Since(48)Km∗φ;y−φy=Km∗φh−φy;y≤Km∗φh−φy;y,and Km∗h−y;y≤Km∗h−y2;y1/2=τ, we get
(49)Km∗φ;y−φy≤Km∗h−y;y+12Km∗h−y2;yφCB20,∞≤12τ2+τφCB2;0,∞.
The desired result is obtained.The Peetre’sK-functional is expressed by
(50)K2∗φ,δ=infCB20,∞φ−ùCB0,∞+δùCB20,∞:ù∈CB20,∞.The second-order modulus of continuity is defined by(51)ω2φ,δ=sup0<ù<δsupy∈0,∞φy+2ù−2φy+ù+φùin [20]. The relation ω2 and K2∗ is as follows:
(52)K2∗φ,δ≤Nω2φ;δ+min1,δφCB0,∞in [21].Theorem 11.
LetKm∗.;. be the operator defined in (4). Then, for any φ∈CB0,∞, we have
(53)Km∗φ;y−φy≤2Nω2φ;Δm,y+min1,Δm,yφCB0,∞,where N is a positive constant and Δm,y=τ2+τ/2 and τ=Km∗ψy,2;y.Proof.
We prove this by using Theorem10. Let ù∈CB20,∞. Since
(54)Km∗φ;y−φy≤Km∗φ−ù;y+Km∗φ;y−ùy+φy−ùy≤2φ−ùCB0,∞+12τ2+τùCB20,∞≤2φ−ùCB0,∞+14τ2+τùCB20,∞.
By taking infimum over allù∈CB20,∞ on the right side of the last inequality and by using (50), we get
(55)Km∗φ;y−φy≤2K2∗φ;τ2+τ4.
This completes the proof, by using equation (52).


## 6. Pointwise Estimates

Let us look at some pointwise estimates of rates of convergence ofKm∗φ;y. At first, the relationship between the local approximation and the local smoothness of the function is given. In this direction, let us give the following definitions. Let s∈0,1 and I^⊂0,∞. In this case, a function φ∈CB0,∞ can be called LipNφs on I^ if the following condition holds:
(56)φv−φy≤Nφ,sv−ys,v∈0,∞andy∈I^,where Nφ,s is a constant that relies on φ and s mentioned above.Theorem 12.
Letφ∈CB0,∞∩LipNφs such that s and I^ given as above. In the circumstances, we give
(57)Km∗φ;y−φy≤Nφ,smk2−k+1mk+1mk−k+1a2ys/2+2dy,I∧s,y∈0,∞,where Nφ,s given above and dy,I^ is the distance between y and I^. This distance is described as:
(58)dy,I^=infv−y,v∈I^Proof.
Let us define the closure of the setI^ as I¨. Then, one can argue that at least one point v0∈I¨ occurs where
(59)dy,I^=y−v0.
Then, due to the monotonicity properties ofKm∗m≥1, we deduce that
(60)Km∗φ;y−φy≤Km∗φv−φv0;y+Km∗φy−φv0;y≤Nφ,sKm∗v−v0s;y+y−v0s≤Nφ,sKm∗v−ys;y+2y−v0s.
Then, from the definition of Hölder’s inequality, we have(61)Km∗φ;y−φy≤Nφ,sKm∗v−y2;yS/2+2dy,I∧s=Nφ,smk2−k+1mk+1mk−k+1a2yS/2+2dy,I∧s,which concludes the theorem.Now, let us try to determine the local direct approximation of the new Gamma operator modification. Let us start with the Lipschitz type maximum function of orders presented in [22] for this goal, that is,
(62)ω~sφ,y=sup0<v<∞,v≠yφv−φyv−ys,where s∈0,1 and y∈0,∞.Theorem 13.
Forφ∈CB0,∞ and ω~s∈0,1, the following inequality holds:
(63)Km∗φ;y−φy≤ω~sφ,yKm∗mk2−k+1mk+1mk−k+1a2ys/2
fory∈0,∞.Proof.
Thanks to the definition ofω~sϕ,y given above and well-known Hölder inequality, we deduce that
(64)Km∗φ;y−φy≤Km∗φv−φy;y≤ω~sφ,yKm∗v−ys;y≤ω~sφ,yKm∗v−y2;ys/2≤ω~sφ,yKm∗mk2−k+1mk+1mk−k+1a2ys/2.
As a result, the desired outcome is achieved.Now, finally, let us consider the following Lipschitz type space with two parameters,c,d>0, such that
(65)LipNc,ds=φ∈C0,∞:φv−φy≤Nv−yscy2+dy+vs/2;y,v∈0,∞introduced in [23] where s∈0,1 and N is a positive constant.Theorem 14.
Forφ∈LipNc,ds and y∈0,∞, then, we have
(66)Km∗φ;y−φy≤Nmk2−k+1/mk+1mk−k+1a2ycy2+dy,
wherec,d>0.Proof.
The proof is divided into two parts. For the first, we uses=1, which means
(67)Km∗φ;y−φy≤Km∗φv−φy;y,≤NKm∗v−ycy2+dy+1;y,≤Ncy2+dyKm∗v−y;y,for φ∈LipNc,ds and y∈0,∞. We conclude that
(68)Km∗φ;y−φy≤Ncy2+dyKm∗v−y2;y1/2≤Nmk2−k+1/mk+1mk−k+1a2ycy2+dy1/2,by using the well-known Cauchy-Schwarz inequality, which validates the theory for s=1. Then, let us consider s∈0,1. For φ∈LipNc,ds and y∈0,∞, we obtain that
(69)Km∗φ;y−φy≤Ncy2+dys/2Km∗v−ys;y.
We derive that(70)Km∗φ;y−φy≤Ncy2+dys/2Km∗v−ys;y≤Ncy2+dys/2Km∗v−y;yswith the help of the well-known Hölder inequality. Finally, we have
(71)Km∗φ;y−φy≤Ncy2+dys/2Km∗v−y2;ys/2≤Nmk2−k+1/mk+1mk−k+1a2ycy2+dys/2,which completes the proof by applying the well-known Cauchy-Schwarz inequality.For the case ofc=1 and d=0, we have the following corollary.Corollary 15.
The local estimate in parametric Lipschitz space is obtained for special fixed parametersc=1 and d=0.
(72)Km∗φ;y−φy≤Nmk2−k+1mk+1mk−k+1
forφ∈LipN1,0s and y∈0,∞.


## 7. Numerical Example

In this section of the article, we provide some numerical examples to verify the rates of convergence ofKm∗φ;y in two dimensions (m=10 is fixed for Figure 1 and k=3 is fixed for Figure 2). In our first example, we compare the operator Km∗φ;y with the classical Gamma operator.Figure 1
Graphics of theKm∗ operator for k=1/3,1,3,30, respectively, and m=10 is fixed (blue for k=1/3, orange for k=1, green for k=3, red for k=30, and purple for φy).Figure 2
Graphics of theKm∗ operator for m=10,20,60, respectively, and k=3 is fixed (blue for m=10, orange for m=20, green for m=60, and red for φy).In this example,Km∗φ;y and φy=y2e−y applied for φ:0,10⟶0,∞.In Figure1, it is seen that the operator puts closer to the function as the value of k gets larger (m=10 is fixed). In Figure 2, it is seen that the operator puts closer to the function as the value of m gets larger (k=3 is fixed).


## 8. Concluding Remarks

We have defined a new form of Gamma operator by consideringk-Gamma function. With the operator defined, the conditions of the Korovkin theorem are completed. Later, Voronovskaya type theorem, weighted approximation, the rates of convergence, and pointwise estimates are obtained. Finally, we give numerical example to confirm its approximation.



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