Journal of Function Spaces
Volume 2019 (2019), Article ID 9607517, 6 pages
https://doi.org/10.1155/2019/9607517

Research Article

On -Analogue of Gamma Operators

Wen-Tao Cheng¹ and Wen-Hui Zhang²

¹School of Mathematical Sciences, Anqing Normal University, Anhui 246133, China
²Qingdao Hengxing University of Science and Technology, Shandong 266041, China

Correspondence should be addressed to Wen-Tao Cheng; chengwentao_0517@163.com

Received 12 November 2018; Revised 15 January 2019; Accepted 21 January 2019; Published 12 February 2019

Academic Editor: Henryk Hudzik

Copyright © 2019 Wen-Tao Cheng and Wen-Hui Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, a kind of new analogue of Gamma type operators based on -integers is introduced. The Voronovskaja type asymptotic formula of these operators is investigated. And some other results of these operators are studied by means of modulus of continuity and Peetre functional. Finally, some direct theorems concerned with the rate of convergence and the weighted approximation for these operators are also obtained.

1. Introduction

In recent years, with the rapid development of -calculus, the study of approximation theory with -integer has been discussed widely. Afterwards, with the generalization from -calculus to -calculus, it has been used efficiently in many areas of sciences such as algebras [1, 2] and CAGD [3]. And, recently, approximation by sequences of linear positive operators has been transferred to operators with -integer. Some useful notations and definitions about -calculus and -calculus in this paper are reviewed in [4–6].

Let . For each nonnegative integer , the -integer and -factorial are defined as

and

Further, the power basis is defined as

And

Let be a nonnegative integer; the -Gamma function is defined as

Aral and Gupta [7] proposed -Beta function of second kind for as

And the relationship between -analogues of Beta and Gamma functions is as follows:

Particularly, when , . It may be observed that, in -setting, order is important, which is the reason why -variant of Beta function does not satisfy commutativity property; that is, .

In [8], Mazhar studied some approximation properties of the Gamma operators as follows:

Recently, Mursaleen first applied -calculus in approximation theory and introduced the -analogue of Bernstein operators [9], -Bernstein-Stancu operators [10], and -Bernstein-Schurer operators [11] and investigated their approximation properties. And many well-known approximation operators with -integer have been introduced, such as -Bernstein-Stancu-Schurer-Kantorovich operators [12], -Szász-Baskakov operators [13], and -Baskakov-Beta operators [14]. As we know, many researchers have studied approximation properties of the Gamma operators and their modifications (see [15–21], etc.). All this achievement motivates us to construct the -analogue of the Gamma operators (8). First, we introduce -analogue of Gamma operators as follows.

Definition 1. For , , , and , the -Gamma operators can be defined as

The paper is organized as follows. In the first section, we give the basic notations and the definition of -Gamma operators. In the second section, we present the moments of the operators. In the third section, we obtain Voronovskaja type asymptotic formula. In the fourth section, we present a direct result of -Gamma operators in terms of first- and second-order modulus of continuity. In the last section, we study the rate of convergence and the weighted approximation of the -Gamma operators.

2. Auxiliary Results

In order to obtain the approximation properties of the operators , we need the following lemma and remarks.

Lemma 2. The following equalities hold:

(1).(2), for .(3), for .(4), for .(5), for .

Proof. According to the properties of -Beta function and -Gamma function, we have

This proves Lemma 2.

Remark 3. Let , and ; then, for , we have the central moments as follows:

(1).(2).

Remark 4. The sequences and satisfy such that , , and , , and as , where , and ; then

(1).(2).

Proof. (1) Using Remark 3,

(2) Let ; we have

Similarly, we have

Using Lemma 2, we obtain

where and

Combining with

and

we can obtain .

3. Voronovskaja Type Theorem

We give a Voronovskaja type asymptotic formula for by means of the second and fourth central moments.

Theorem 5. Let be bounded and integrable on the interval ; second derivative of exists at a fixed point ; the sequences and satisfy such that , , and , , and as , where ; then

Proof. Let be fixed. In order to prove this identity, we use Taylor’s expansion:

where is bounded and . By applying the operator to the equality above, we obtain

Since , for all , there exists such that and it will imply for all fixed as is sufficiently large. Meanwhile, if , then , where is a constant. Using Remark 4, we have

and

The proof is completed.

4. Local Approximation

We denote the space of all real valued continuous bounded functions defined on the interval by . The norm on the space is given by

Let us consider the following -functional:

where and . By [22] (p. 177, Theorem 2.4), there exists an absolute constant such that

where

is the second-order modulus of smoothness of . By

we denote the usual modulus of continuity of .

Our first result is a direct local approximation theorem for the operators .

Theorem 6. Let ; ; then, for every and , we have

where is some positive constant.

Proof. For all , using Taylor’s expansion for , we have

Applying the operators to both sides of the equality above and using Remark 3, we get

Using , we have

Lastly, taking infimum on both sides of the inequality above over all ,

for which we have the desired result by (27).

Theorem 7. Let and let be any bounded subset of the interval . If is locally in , , the condition

holds, then, for each , we have

where is a constant depending on and ; and is the distance between and defined by

Proof. From the properties of infimum, there is at least one point in the closure of ; that is, , such that

Using the triangle inequality, we have

Choosing and and using the well-known Hölder inequality,

This completes the proof.

5. Rate of Convergence and Weighted Approximation

Let be the set of all functions defined on satisfying the condition , where is a constant depending only on and is a weight function. Let be the space of all continuous functions in with the norm and . We consider in the following two theorems. Meanwhile, we denote the modulus of continuity of on the closed interval , , by

Obviously, for the function , the modulus of continuity tends to zero. Then we establish the following theorem on the rate of convergence for the operators .

Theorem 8. Let , , and be its modulus of continuity on the finite interval , where . Then, for every , ,

Proof. For all and , we easily have ; therefore,

And, for all , , and , we have

From (43) and (44), we get

By Schwarz’s inequality and Remark 3, we have

By taking , we get the proof of Theorem 8.

The following is a direct estimate in weighted approximation.

Theorem 9. Let and satisfy such that , , , , and . Then, for , we have

Proof. Using the Korovkin theorem in [23], we see that it is sufficient to verify the following three conditions:

Since and , (48) holds true for . By Remark 3, for , we have

Thus the proof is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant no. 11626031).

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