Abstract
In the present paper, we will introduce -Gamma operators based on -integers. First, the auxiliary results about the moments are presented, and the central moments of these operators are also estimated. Then, we discuss some local approximation properties of these operators by means of modulus of continuity and Peetre -functional. And the rate of convergence and weighted approximation for these operators are researched. Furthermore, we investigate the Voronovskaja type theorems including the quantitative -Voronovskaja type theorem and -Grüss-Voronovskaja theorem.
1. Introduction
Gamma operators are very important positive linear operators and have been widely used in probability theory and computational mathematics. For , where and be the space of all continuous functions on the interval , the Gamma operators were introduced in [1] by
We can learn some properties of Gamma operators and their modified operators in [2–7]. In [8], Qi et al. defined new Gamma operators as follows: where , . Obviously, if , then . Meantime, (while ). In order to preserve the constant, we defined -Gamma operators as follows:
Definition 1. For , , , the -Gamma operators are defined by
Let us recall some useful concepts and notations from -calculus, which can be founded in [9–11]. For nonnegative integer , the -integer and -factorial are defined by
Further, -power basis can be defined by
The -derivative of a function can be defined by and provided exists. High-order -derivatives can be defined by , , . The formula for the -derivative of a product is . We easily know that if a function is continuous on an interval which does not include 0, then is continuous -differentiable.
The -improper integral of function can be defined by
The -analogue of the classical exponential function is
The -Gamma function is defined by and satisfies the functional relation: , . Moreover, for any nonnegative integer , the relation holds: .
Now, we construct the -analogue of -Gamma operators using -Gamma function as follows.
Definition 2. For , , , , the -analogue of -Gamma operators (3) are defined as
The paper is organized as follows: In Section 1, we introduce the history of Gamma operators, recall some basic notations about the -calculus, and construct -Gamma operators based on -integers with -Gamma function. In Section 2, we obtain the auxiliary results about the moment computation formula. The second- and fourth-order central moments computation formula and other quantitative properties are also presented. In Section 3, we discuss local approximation about the operators by means of modulus of continuity and Peetre -functional. In Section 4 and Section 5, the rate of convergence and weighted approximation for these operators are researched. In the last section, we firstly prove quantitative -Voronovskaja type theorems in terms of weighted modulus of continuity, and then the -Grüss-Voronovskaja theorem in the quantitative mean is also presented (for the quantitative -Voronovskaja type theorem0 and the -Grüss-Voronovskaja theorem for the other operators, see also [12, 13]).
2. Auxiliary Results
In this section, we will give some lemmas and corollaries, which are necessary to obtain the approximation properties of the operators .
Lemma 3. For , , , , the following formula holds:
Proof. According to the properties of -Gamma function, we have Lemma 3 is proved.
Corollary 4. By the lemma given above and some elementary calculations, we can get the results
Lemma 5. Let be a sequence satisfying , and . Then, for each , , ,we can obtain
Proof. By Lemma 3, we can easily get (14). Without loss of generality, we only prove equation (15). Equation (16) and equation (17) can be proved in some way. Set , , and . Using , , we can easily get Combining we have . This means that equation (15) is obtained. Thus, the proof of Lemma 5 is accomplished.
Lemma 6. Let be a sequence satisfying , and . Then, for each , the following relations hold.
Proof. By the definition of -power basis, we have . Thus, we can write . Using the monotonicity of the operators and the Cauchy-Schwarz inequality, we can get The inequality (21) can be get in the same way. Using the monotonicity of the operators , (16)and (17), Cauchy-Schwarz inequality, respectively, we can obtain Thus, we complete the proof.
3. Local Approximation
Let be the space of all real-valued continuous bounded functions on , endowed with the norm . Moreover, the Peetre’s -functional is defined by where . By ([14], p. 177, Theorem 2.4), there exists an absolute constant such that where and the second-order modulus of smoothness is defined by
The usual modulus of smoothness is defined by
Theorem 7. Let , , . Then for all and , there exists an absolute such that
Proof. Using Definition 2, we easily obtain for all . Next, we define new operators by We can get and for all . For and , using Taylor’s expansion, we can write Hence, Further, for all , we can write Taking infimum over all and using (25), we can get the desired conclusion.
Corollary 8. Let , . Then for all and , there exists an absolute such that
Corollary 9. Let be a sequence satisfying , and as , the limit holds for all .
4. Rate of Convergence
As is known, if is not uniformly continuous on , we cannot get as . To research the rate of convergence of the operators on , we recall the weighted modulus of continuity (see [15] or [16]). First, we shall consider the following three classes of functions: where is a positive constant which depends only on ,
The space is a linear normed space endowed with the norm . For any , is defined by if , then has the following properties: (i)(ii),
In [17–19], the following inequality was introduced and used
Meanwhile, we introduce the modulus of continuity of by .
The following is a theorem of the rate of convergence for the operators :
Theorem 9. Let , , , , we have
Proof. For any , , we can easily obtain , therefore If , for any , we can obtain Combining (39) with (40), we can get By Cauchy-Schwarz’s inequality and Corollary 4, for all , we have By choosing and taking supremum over all , we can get the desired results.
Theorem 10. be a sequence satisfying , and as and ; then, there exists a positive integer such that for all and , the inequality holds.
Proof. Using (14) and (16), there exists a positive integer such that for all , By Cauchy-Schwarz’s inequality, we can get Since is linear and positive, using (38), (46), and (47), for any , we can obtain Taking , we complete the proof.
5. Weighted Approximation
In this section, we will discuss the weighted approximation theorems for the operators .
Theorem 11. Let be a sequence satisfying , , and as and , we have
Proof. Using Korovkin’s theorem (see [20]), it is sufficient to verify the following three conditions: Since , (51) holds for . By Lemma 3 and , we can easily obtain We can draw the final conclusion through all the estimates above.
Theorem 12. Let be a sequence satisfying , and as and . For any and , we have
Proof. Let be arbitrary but fixed. Then, Since , we have . Let be arbitrary, we can choose to be so large that In view of Corollary 9, while , we obtain Using Theorem 9, we can see that the first term of the inequality (53) implies that Combining (53)–(56), we get the desired result.
6. Voronovskaja Type Theorems
As is known, Voronovskaja type theorems of many positive operators are widely researched and discussed (see [21–28]). In this section, we will discuss the quantitative -Voronovskaja theorem and -Grüss-Voronovskaja theorem.
6.1. Quantitative -Voronovskaja Theorem
In this subsection, we will obtain the Quantitative -Voronovskaja theorem and Voronovskaja type asymptotic formula for the operators .
Theorem 13. Let be a sequence satisfying , and as and satisfy . Then, the inequality holds for any .
Proof. Using the -Taylor expansion formula (58), we have where is a number between and and Applying the operators to both sides of (58) and using , we have Multiplying the above inequality by , we have Furthermore, for all . Hence, Using (20), (21), for any , we can write If we choose , we can easily get which completes the proof of Theorem 13.
Corollary 14. Let be a sequence satisfying , and as and satisfy . Then, we can obtain
6.2. -Grüss-Voronovskaja Theorem
In this subsection, we will obtain the -Grüss-Voronovskaja theorem and its quantitative version for the operators .
Theorem 15. Let be a sequence satisfying , and as and satisfy . Then, the following inequality holds for any .
Proof. Using the equalities by simple computations, for and , we can obtain Hence, we can write By Theorem 13, for any fixed , we can easily have the following estimates Using (14), (20), and , we have Using (14), (20), and Theorem 13, we can get hence, we can know Combining (71)–(76), we complete the proof of Theorem 15.
Corollary 16. Let be a sequence satisfying , and as and satisfy . Then, the following limit equality holds for any .
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.
Authors’ Contributions
All authors read and approved the final manuscript.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No. 11626031), the Science and Technology Project of Department of Education of Jiangxi Province (No. GJJ190551), the Doctoral Research Startup Project of Jinggangshan University (No. JZB17002), the Key Natural Science Research Project in Universities of Anhui Province (Grant No. KJ2019A0572), the Philosophy and Social Sciences General Planning Project of Anhui Province of China (Grant No. AHSKYG2017D153), and the Natural Science Foundation of Anhui Province of China (Grant No. 1908085QA29).