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 asand
Further, the power basis is defined asAnd
Let be a nonnegative integer; the -Gamma function is defined as
Aral and Gupta [7] proposed -Beta function of second kind for asAnd 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 haveThis 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 haveSimilarly, we haveUsing Lemma 2, we obtainwhere andCombining withandwe 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 obtainSince , 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 haveandThe 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 thatwhereis the second-order modulus of smoothness of . Bywe 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 havewhere is some positive constant.
Proof. For all , using Taylor’s expansion for , we haveApplying the operators to both sides of the equality above and using Remark 3, we getUsing , we haveLastly, 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 conditionholds, then, for each , we havewhere 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 thatUsing the triangle inequality, we haveChoosing 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 , , byObviously, 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 haveFrom (43) and (44), we getBy Schwarz’s inequality and Remark 3, we haveBy 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 haveThus 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).