Approximation Properties of a New Gamma Operator

Abstract

This paper deals with a kind of modification of the classical Gamma operators defined on the semiaxis which holds fixed functions 1 and . We study the uniform approximation effect and the direct results. We also investigate the weighted A-statistical convergence. Finally, the Voronovskaja type asymptotic formula is given.

1. Introduction

For , the Gamma operators are given by [1].

These operators play important role in many fields inside applied mathematics [1–11]. A considerable amount of literature has been published on approximation processes. Various operators were conveniently modified, which preserve the test functions that emerged in this field and a lot of advancements have been made regarding this subject to acquire better approximation [10–17]. For some recent studies on linear positive operators preserving exponential functions, we refer the readers to the literature [2, 5, 10, 11, 13, 14, 16, 17]. Following the idea of King [12], further developed in references [2, 5, 10–17], we introduce the operators which preserve constants and . We study their uniform convergence properties, asymptotic formula, and the weighted-statistical convergence results.

The improved Gamma operators are defined bywhere is a positive real parameter,

Remark 1. By calculating, we have .

Remark 2. denotes the space of all continuous functions bounded on . .

Remark 3. In this paper, the norm of the function is defined as a uniform norm: .

2. Auxiliary Results

According to the definition of the operators , we have as follows:

Lemma 1. If , , then

Lemma 2. If , , then

Meanwhile, let , , we have

Proof. From the definition of the operators, one hasusing the software Mathematica, we have the estimate of , and .

Lemma 3. For any , and , we have

Proof. By definition of the operators , one has

Lemma 4. For any , then .

Lemma 5. For , one has .

Proof. From the condition: , we know that exists and is finite. For , . Without loss of generality, let , and . For every , there exists a constant , such that for . Let , if , noting that , we haveFor , , i.e. . Therefore, .

Lemma 6. [5, 9] The sequence of operators converges to as uniformly in for all , if and only if

3. Properties of Approximation

In this section, we will give the main approximation properties of the operators .

Theorem 1. Let , , then the sequence of the operators converges to uniformly in the interval .

Proof. Due to Lemma 6, we only need to check the relations:Firstly, using Lemma 1, we have,Secondly, using Mathematica, we can write the expansion :Let , , , from the relation (14), we haveThen,Let , , , from the relation (14), we haveTherefore,One hasFrom the relations (13), (16), and (19), the proof of Theorem 1 is complete.

Definition 1. The modulus of continuity is defined by Holhos [18]. For any and ,

Definition 2. The classical moduli of continuity are defined by [7, 8]:The relationship between the two moduli defined above is [18] , whereFrom Holhos ([18], theorem 3), we can get the following results.

Theorem 2. The are positive linear operators and let

If all the vanish at infinity, then for any , we have the following conclusion: .

Corollary 1. For any , by Theorem 2, we have

Remark 4. For , then , respectively. As a consequence of Theorem 2, we have

Theorem 3. If , then

Proof. Let’s use Taylor’s formula for , we getwhere and is between and . Next, we apply the operators to both sides of the equality,By Lemma 2, we can getLet , from above definition of , we know thatHence, we obtainUsing the linearity of , we getChoosing , .
So, we have the desired result.

Definition 3. Let and . We said ,if the inequality holds.

Remark 5. For , , by Hölder inequality and Theorem 3, we have

Theorem 4. Let . If there exists such thatthen,

On the other hand, for a given point , if inequality (35) holds, then there exists such that , for each .

Proof. By the inequality (34), we haveBy Theorem 3, we get .
We assume that the inequality (35) holds, then .
The proof of the theorem is complete.

Remark 6. Theorem 4 shows that the new Gamma operators have superior properties.

4. Properties of Statistical Approximation

The concept of statistical convergence, which was first introduced by Fast [19], is a generalization of ordinary convergence. Several extensions of statistical approximation have appeared in the literature [20–24] and references therein. In this section, we will give some weighted-statistical approximation properties of the operators . First, we will recall the following definitions, and notations can be found in ref. [22, 24].

Definition 4. [22] Let , . The natural density of E is denoted byhere is the cardinality of . A sequence is said to be statistically convergent to L, if, for every , . In symbol, we write .

Definition 5. [24] A given non-negative infinite summability matrix is said to be regular if , whenever . Then, the sequence is said to be A-statistically convergent to L, denoted by , provided that for each , .

Definition 6. [24] Let be a non-negative regular summability matrix, be a sequence of non-negative numbers such that and as . Then, is said to be weighted A-statistically convergent to , if, for any , . In this case, we write .

Definition 7. [24] Let be a non-negative regular summability matrix, be a sequence of non-negative numbers such that and as , and be a positive non-increasing sequence. Then, is weighted A-statistically convergent to L with the rate , if for each ,This relation is denoted by .

Theorem 5. Let be a non-negative regular summability matrix. For any and , we have

Proof. For , there is a constant , such that . Therefore, , . For any , there is a such that , . Consider of the form , we obtainwhere . For , by a direct computation, we writeThe term can be written asFor a given , such that . If we define the following sets,we see that ,Taking the limit and noting the conditions, we obtain

Remark 7. Here, we use the Korovkin test functions . We can also use the usual test functions .

Theorem 6. Let be a non-negative regular summability matrix. If the following condition yields,

Let , then for , we have

Proof. For , one hasLet , taking supermum over on both sides, we obtainDefining the following sets for a given ,it is easy to see that , andHence,

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is partially supported by the Science and Technology Project of Hebei Education Department (No. ZD2019053) and Science Foundation of Hebei Normal University (No. L2020203).