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Wen-Tao Cheng, Qing-Bo Cai, "Generalized -Gamma-type operators", Journal of Function Spaces, vol. 2020, Article ID 8978121, 10 pages, 2020. https://doi.org/10.1155/2020/8978121
Generalized -Gamma-type operators
In the present paper, the generalized -gamma-type operators based on -calculus are introduced. The moments and central moments are obtained, and some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre -functional. Also, the rate of convergence, weighted approximation, and pointwise estimates of these operators are studied. Finally, a Voronovskaja-type theorem is presented.
In , Mazhar introduced gamma operators preserving linear functions as follows: where , , . In , Karsli considered new gamma operators preserving as follows: where . In , Mao defined generalized gamma operators as follows: where . Obviously, and . In [4, 5], some approximation properties of operators ((1)–(3)) were discussed.
In Bernstein polynomials, some of their modifications and corresponding operators have been studied in many papers (see [6–10]). The -analogue of well-known positive probability operators were widely studied and discussed (see books [11–13]) since Bernstein polynomials were proposed by Lupa  and Phillips . In , the -analogue of the operators (1) was defined and discussed. In , Cai and Zeng constructed and studied a -analogue of the operators (2). Meantime, modifications and generalizations of the operators (2) were introduced and researched in [18–20]. In , Karsli constructed a -analogue of the operators (3) and extended the works of [16, 17, 20]. Recently, many operators are constructed with two-parameter -integers based on postquantum calculus (-calculus) which have been used widely in many areas of sciences such as Lie group, different equations, hypergeometric series, and physical sciences. First, we recall some useful concepts and notations from -calculus, which can be found in [11–13]. The -integers are defined by
By some simple calculation, for any , we have the following relation:
The -factorial is defined by
The -power basis is defined by
Let be an arbitrary function. The improper -integral of on is defined as (see )
Let be a nonnegative integer. The -gamma function is defined as
Aral and Gupta  proposed a -beta function of the second kind for , as and gave the relation of the -analogues of beta and gamma functions:
As a special case, if , . It is obvious that the order is important for the -setting, which is the reason why the -variant of beta function does not satisfy commutativity property, i.e., .
Since Mursaleen et al. firstly introduced -calculus in approximation theory and constructed the -analogue of Bernstein operators  and -Bernstein-Stancu operators , generalizations of many well-known approximation operators based on -calculus were widely introduced and discussed by several authors (see -Szász-Mirakjan operators , -Baskakov-Durrmeyer-Stancu operators , -Bernstein-Stancu-Schurer-Kantorovich operators , -Baskakov-beta operators , -Lorentz polynomials , -Szász-Mirakjan Kantorovich operators , -Bleimann-Butzer-Hahn operators , -Bernstein operators [33, 34], and so on). In , Cheng and Zhang constructed a -analogue of the operators (1) using the -beta function of the second kind and studied their approximation properties. Later, Cheng et al. defined the -analogue of the operators (2) and researched their approximation properties in . All these achievements motivate us to construct the -analogue of the gamma operator (3) and generalize the works of [35, 36]. Now, we construct generalized -gamma-type operators as follows:
Definition 1. Let, , , and . For, then the -analogue of the gamma operator (3) can be defined by In the case , we obtain the operators (1); in the case , we obtain the operators (2); in the case , we obtain the operators (3); in the case , we obtain the operators ; in the case , we obtain the operators ; in the case , we obtain the operators ; in the case , we obtain the operators ; and in the case , we obtain the operators .
The paper is organized as follows: In Section 1, we introduce the history of gamma-type operators and recall some basic notations about -calculus; then, we construct the generalized -gamma operators with the -beta function. In Section 2, we obtain the auxiliary lemmas and corollaries about the moment computation formulas. And the second- and fourth-order central moment computation formula and limit equalities are also obtained. In Section 3, we discuss the local approximation about the operators by means of modulus of continuity and Peetre -functional. In Sections 4 and 5, the rate of convergence and weighted approximation for these operators are researched. In Section 6, two pointwise estimates are given by using the Lipschitz-type maximal function. In Section 7, the Voronovskaja-type asymptotic formula is presented.
2. Moment Estimates
In order to obtain the approximation properties of the operators , we need the following lemmas and corollaries.
Lemma 2. For , , , , and , we have
Proof. Set , , and , we have
Then, the following corollary can be obtained immediately.
Corollary 3. For , , and , the following equalities hold:
Corollary 4. For , , and , using Corollary 3, we can easily obtain the following explicit formulas for the first and second central moments:
Corollary 5. The sequences satisfy such that and as ; then, for any , we have
Proof. Using (5), we have . Hence, Using and , we have Thus, Now, we prove the limit equality (19) while is similar. Set and , by (5); we can obtain Hence, we can rewrite Further, we can easily get For any , . We can obtain we obtain the required result.
Corollary 6. Let us denote the norm on (the class of real valued continuous bounded functions on ). For any , we have
3. Local Approximation
For any , let us consider the following -functional: where and . The usual modulus of continuity and the second-order modulus of smoothness of can be defined by
By  (p.177, Theorem 2.4), there exists an absolute constant such that
Theorem 7. Let be the sequences defined in Corollary 5 and . Then, for all , there exists an absolute positive such that
Proof. Define the following new operators: Let and . By Taylor’s expansion formula, we get Applying to the above equality and , we can obtain By Corollary 6 and (32), we easily know . Hence, Taking the infimum on the right-hand side over all and using (30), we obtain the desired assertion.
Corollary 8. Let be the sequences defined in Corollary 5 and . Then, for any finite interval , the sequence converges to uniformly on .
4. Rate of Convergence
Let where is the weighted function given by and is an absolute constant depending only on . is equipped with the norm . As is known, if is not uniform, we cannot obtain . In , Ispir defined the following weighted modulus of continuity: and proved the properties of monotone increasing about as , , and the inequality while and . Meantime, we recall the modulus of continuity of on the interval by
Theorem 9. Let , , and , we have