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

With the help of the k -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.


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 ½a, 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 a 2 ðhÞ = h 2 in 2003. Then, King constructed a new set of operators with respect to the test functions f1, h, h 2 g 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: 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][6][7][8][9][10][11][12][13][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.

A New Modification of Gamma Operators
Defined with the Help of k-Gamma Function We shall see a new type of Gamma operators defined with the help of the k-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 a z ðhÞ = h z and ψ y,z = ðh − yÞ z , y ∈ ð0,∞Þ as polynomial functions. The modified representation of the classical Gamma operator is shown as follows: where for v > 0, φ ∈ C γ ð0,∞Þ = fφ ∈ Cð0,∞Þ: φðuÞ = Oðu γ Þ, asu⟶∞g for m > γ. Here Cð0, ∞Þ is the set of continuous functions on ð0, ∞Þ: 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 of k = ð1/pÞðp ∈ ℕÞ 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. Let y ∈ ð0,∞Þ: The following are the moment values: By generalizing the moment values, we have the following lemma.

Lemma 2.
Let y ∈ ð0,∞Þ and z ∈ ℕ, K * m ða 0 ðhÞ ; yÞ = a 0 ðyÞ: Then, the general formula for the following moment values is obtained Lemma 3. Let y ∈ ð0,∞Þ: Using the equations in Lemma 1, the following are obtained: As a result of our research, the Schurer variant of Gamma operators have not been defined or used. Also, if it 2 Journal of Function Spaces is realized that k ∈ ℝ + , it is obtained that our operators are a generalization of the Schurer type operators. Throughout this paper, we use the norm kφk = sup fφð yÞ: y ∈ ð0,∞Þg for φ ∈ Cð0,∞Þ: Proof. By using the result of Lemma 1, we have Thus, we obtain the desired result. Because the moments are conserved in the limit state of the Korovkin test functions, K * m is an approximation process on any compact T ⊂ ð0,∞Þ, according to the Korovkin theorem in [18].

Voronovskaya Type Theorem
By establishing Voronovskaya's theorem below, we will illustrate the asymptotic behavior of ðK * m Þ m≥1 operators in this section.
The following limit is valid: Proof. From the definition of Taylor formula where such that δ lying between y and h and When the ðK * m Þ m≥1 operator is applied to (13), we get To get the formula multiply both sides of the last inequality by m. In the limit case, this equation is We know the values 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- Then, using Korovkin theorem, we have since Ω 2 ðy, yÞ = 0 and Ωð:,yÞ ∈ Cð0,∞Þ ∩ E and bounded as h ⟶ ∞ and in view of fact that where K * m ðψ y,4 ðhÞ ; yÞ = ð3m 2 k 4 + mð18k 4 − 22k 3 + 6k 2 Þ − 6 k 3 + 11k 2 − 6k + 1Þ/ððmk + 1Þðmk − k + 1Þðmk − 2k + 1Þðmk − 3k + 1ÞÞ: The proof is completed when equations (21) and (22) are written in (13).

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].
Consider ϑðyÞ = 1 + y 2 as continuous weighted function on ℝ, with lim jyj⟶∞ ϑðyÞ = ∞, ϑðyÞ ≥ 1 for all y ∈ ½0,∞Þ: Let us have a look at the weighted spaces below. The property jφðyÞj ≤ N φ ϑðyÞ represents the weighted space of realvalued functions φ on ℝ. This subspace is denoted by N φ is a constant depending on the functions φ. Since, the weighted subspaces of B ϑ ½0, ∞Þ is given by Eventually, additional subspace for all φ ∈ C ϑ ½0,∞Þ for which lim jyj⟶∞ φðyÞ/ϑðyÞ exists finitely defined as This κ φ is a constant dependent on the φ functions. All three mapping spaces above are normed spaces endowed with Lemma 7. Let φ ∈ C ϑ ð0,∞Þ: Then, for the modified operator K * m ðφÞ, we have which imply that the sequence of the modified operators K * m ðφÞ is an approximation process from C ϑ ð0, ∞Þ to B ϑ ð0, ∞Þ : Proof. The desired result of this lemma is easily obtained from properties of the modified Gamma operator and Lemma 1.
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.
Let φ ∈ C κ ϑ ð0,∞Þ: For the modified Gamma operator, the following equality holds: Proof. It will be enough to show that equivalence is attained for lim m⟶∞ kK * m ða z ; yÞ − a z k ϑ = 0, z = 0, 1, 2 using the theorem in [19]. For z = 0, we have kK * m ða 0 ; yÞ − a 0 k ϑ = 0: Now, let us examine the cases z = 1, 2. When the necessary results for these situations are used, is obtained. If we take the limit of this expression, it becomes Journal of Function Spaces If we take the limit of this expression, it becomes As a result of the equations obtained above, the evidence is finished.

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 that K * m ðφÞ outperforms the classical operator in terms of error estimation. Let us define the following in light of this goal.
The modulus of continuity of w is denoted by ω y 0 ðφ, δÞ for interval ð0, y 0 , y 0 ≥ 0 and can be described as follows: The modulus of continuity ω y 0 ðφ, δÞ ⟶ 0 is easily understood as δ ⟶ 0 for the function φ ∈ C B ð0,∞Þ, where C B ð0, ∞Þ is defined as space of all continuous and bounded functions on the interval ð0, ∞Þ: Now, let us look at the rates of convergence theorem for ðK * m Þ m≥1 .
The Peetre's K-functional is expressed by The second-order modulus of continuity is defined by in [20]. The relation ω 2 and K * 2 is as follows: in [21].

Numerical Example
In this section of the article, we provide some numerical examples to verify the rates of convergence of K * m ðφ ; yÞ 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 K * m ðφ ; yÞ with the classical Gamma operator. In this example, K * m ðφ ; yÞ and φðyÞ = y 2 e −y applied for φ : ½0, 10 ⟶ ½0,∞Þ: In Figure 1, 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).

Concluding Remarks
We have defined a new form of Gamma operator by considering k-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. 8 Journal of Function Spaces

Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.