Abstract
With the help of the -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.
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 . 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 in 2003. Then, King constructed a new set of operators with respect to the test functions and obtained their linear combinations. On the other hand, one of these operators is the Gamma operator which is constructed by Lupa 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 [5–14]. One of the studies of this topic was by Daz and Pariguan [15]; they introduced and researched -Gamma function when they were assessing Feynman integrals. -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 -Gamma function for combinatorial analysis in statistic.
According to these studies, the -Gamma function was defined by Daz and Pariguan as follows:
As can be seen from the definition, is a one parameter deformation of the classical Gamma function such that as For , it reduces to an integral of Gaussian functions [17]. When we get in equation (2), we find the expression , and all properties of the classic Gamma operator can be generalized into -Gamma function. It also led to a few new conclusions for -Gamma function. A few of them are given that in [15]. For more such properties of -Gamma and related functions, we can refer to the article [11, 15, 17].
The primary goal of this research is to give the -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 -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 of -Gamma Function
We shall see a new type of Gamma operators defined with the help of the -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 and , as polynomial functions. The modified representation of the classical Gamma operator is shown as follows: where for , for . Here is the set of continuous functions on This modified operator is clearly positive and linear in this case. Furthermore, the new Gamma operator defined with the help of the -Gamma function is directly preserved constant, and test functions are provided in case of limit.
We note that for special case of 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 The following are the moment values: By generalizing the moment values, we have the following lemma.
Lemma 2. Let and Then, the general formula for the following moment values is obtained
Lemma 3. Let 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 is realized that , it is obtained that our operators are a generalization of the Schurer type operators.
Throughout this paper, we use the norm for
Lemma 4. Let Then, we get
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, is an approximation process on any compact , according to the Korovkin theorem in [18].
Theorem 5. Let where . Then, consistently in each compact subset of we have
Proof. By using Lemma 1, when we get for uniformly each compact subset of . Then, using the Korovkin theorem in [18], we give for uniformly each compact subset of .
3. Voronovskaya Type Theorem
By establishing Voronovskaya’s theorem below, we will illustrate the asymptotic behavior of operators in this section.
Theorem 6. Let such that The following limit is valid:
Proof. From the definition of Taylor formula where such that lying between and and When the operator is applied to (13), we get To get the formula multiply both sides of the last inequality by . In the limit case, this equation is We know the values using Lemma 3. So, we have 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 Then, using Korovkin theorem, we have since and and bounded as and in view of fact that where 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].
Consider as continuous weighted function on , with , for all Let us have a look at the weighted spaces below. The property represents the weighted space of real-valued functions on . This subspace is denoted by
is a constant depending on the functions .
Since, the weighted subspaces of is given by
Eventually, additional subspace for all for which exists finitely defined as
This is a constant dependent on the functions. All three mapping spaces above are normed spaces endowed with
Lemma 7. Let Then, for the modified operator , we have which imply that the sequence of the modified operators is an approximation process from to
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 For the modified Gamma operator, the following equality holds:
Proof. It will be enough to show that equivalence is attained for using the theorem in [19]. For we have Now, let us examine the cases . When the necessary results for these situations are used, is obtained. If we take the limit of this expression, it becomes Then, we have If we take the limit of this expression, it becomes 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 that outperforms the classical operator in terms of error estimation. Let us define the following in light of this goal.
The modulus of continuity of is denoted by for interval and can be described as follows:
The modulus of continuity is easily understood as for the function where is defined as space of all continuous and bounded functions on the interval Now, let us look at the rates of convergence theorem for .
Theorem 9. For and let be the modulus of continuity on the finite interval Then, the following inequality exists: where is a constant only according as
Proof. Now, let , and Then, we can conclude that for Then, again let So, the following inequality holds for As a result, from the above inequalitiy, we deduce that for and Applying and Cauchy-Schwarz inequality to (38), we obtain By choosing , we can conclude the proof.
Let with the norm also in [20].
Theorem 10. Let be the operator defined in (4). Then, for any , we have where is in Lemma 3.
Proof. Let . When referring to the Taylor series, obtain where between and , from which it follows: where because of (41). Thus, we have Since and we get The desired result is obtained.
The Peetre’s -functional is expressed by
The second-order modulus of continuity is defined by in [20]. The relation and is as follows: in [21].
Theorem 11. Let be the operator defined in (4). Then, for any we have where is a positive constant and and
Proof. We prove this by using Theorem 10. Let . Since By taking infimum over all on the right side of the last inequality and by using (50), we get This completes the proof, by using equation (52).
6. Pointwise Estimates
Let us look at some pointwise estimates of rates of convergence of . 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 and In this case, a function can be called on if the following condition holds: where is a constant that relies on and mentioned above.
Theorem 12. Let such that and given as above. In the circumstances, we give where given above and is the distance between and . This distance is described as:
Proof. Let us define the closure of the set as . Then, one can argue that at least one point occurs where Then, due to the monotonicity properties of , we deduce that Then, from the definition of Hölder’s inequality, we have 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 order presented in [22] for this goal, that is, where and
Theorem 13. For and the following inequality holds: for
Proof. Thanks to the definition of given above and well-known Hölder inequality, we deduce that As a result, the desired outcome is achieved.
Now, finally, let us consider the following Lipschitz type space with two parameters, such that introduced in [23] where and is a positive constant.
Theorem 14. For and then, we have where
Proof. The proof is divided into two parts. For the first, we use , which means for and . We conclude that by using the well-known Cauchy-Schwarz inequality, which validates the theory for Then, let us consider For and we obtain that We derive that with the help of the well-known Hölder inequality. Finally, we have which completes the proof by applying the well-known Cauchy-Schwarz inequality.
For the case of and , we have the following corollary.
Corollary 15. The local estimate in parametric Lipschitz space is obtained for special fixed parameters and . for and
7. Numerical Example
In this section of the article, we provide some numerical examples to verify the rates of convergence of in two dimensions ( is fixed for Figure 1 and is fixed for Figure 2). In our first example, we compare the operator with the classical Gamma operator.
In this example, and applied for
In Figure 1, it is seen that the operator puts closer to the function as the value of gets larger ( is fixed). In Figure 2, it is seen that the operator puts closer to the function as the value of gets larger ( is fixed).
8. Concluding Remarks
We have defined a new form of Gamma operator by considering -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.
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.
Acknowledgments
I would like to thank the Scientific and Technological Research Council of Turkey (TUBTAK) for the graduate scholarship that supported the second author.