Approximation Methods: Theory and ApplicationsView this Special Issue
Approximation Properties of New Modified Gamma Operators
This paper is aimed at constructing new modified Gamma operators using the second central moment of the classic Gamma operators. And we will compute the first, second, fourth, and sixth order central moments by the moment computation formulas, and their quantitative properties are researched. Then, the global results are established in certain weighted spaces and the direct results including the Voronovskaya-type asymptotic formula, and point-wise estimates are investigated. Also, weighted approximation of these operators is discussed. Finally, the quantitative Voronovskaya-type asymptotic formula and Grüss Voronovskaya-type approximation are presented.
In , Karsli et al. used analysis methods to obtain the rate of point-wise convergence for the operators (1). In , Karsli and Özarslan obtained some direct local and global approximation results for the operators (1). In , İzgi studied some direct results in asymptotic approximation about the operators (1). In , Krech gave a note about the results of İzgi in  and obtained an error estimate for the operators (1). In , Krech gave direct approximation theorems for the operators (1) in certain weighted spaces. In , Cai and Zeng constructed -Gamma operators and gave their approximation properties. In , Zhao et al. extended the works of Cai and Zeng and considered the stancu generalization of -Gamma operators. Recently, Cheng et al. constructed -Gamma operators using -Beta function of the second kind and discussed their approximation properties in . In , Zhou et al. extended the works of Cheng et al. in  and constructed -Gamma-Stancu operators. There are many papers about the research and application of other Gamma-type operators, and we mention some of them [12–17].
In this paper, we construct new modified Gamma operators using the second central moment of the operators (1) as follows:
Definition 1. For and , we construct new modified Gamma operators by where
The paper is organized as follows: In Section 1, we introduce the history of Gamma operators and construct new modified Gamma operators using the second central moment. In Section 2, we obtain the basic results by the moment computation formulas. And the first, second, fourth, and sixth order central moment computation formulas and limit equalities are also obtained. In Section 3, we establish the global approximation results for the operators (2) in certain weighted spaces. In Sections 4 and 5, we investigate the direct results including the Voronovskaya-type asymptotic formula and point-wise estimates in three different Lipschitz classes and discuss weighted approximation. In Section 6, we present a quantitative Voronovskaya-type asymptotic formula and a Grüss Voronovskaya-type approximation (for the quantitative Voronovskaya type theorem and Grüss-Voronovskaya theorem for the other operators, see also [18–24]).
2. Basic Results
In this section, we present certain auxiliary results which will be used to prove our main theorems for the operators (2).
Lemma 2 (see ). For any , , we have
Lemma 3. If we define , then there holds the following relation
where , .
Then, the following lemma can be obtained immediately:
Lemma 4. For any , , we have By the classical Korovkin theorem, we easily obtain the following lemma:
Lemma 5. For all and any finite interval , then the sequence converges to uniformly on , where denotes the set of all real-valued bounded and continuous functions defined on , endowed with the norm .
3. Global Results
In this section, we establish some global results by using certain Lipschitz classes. We first recall some basic definitions. Let and define the weighted function as follows:
Meantime, we consider the following subspace of generated by : endowed with the norm for . For every , , and , the usual weighted modulus of continuity, the second-order weighted modulus of smoothness, and the corresponding Lipschitz classes are, respectively, defined as
Theorem 6. Let be fixed. Then, there exists a positive constant such that Furthermore, for all , we have Thus, is a linear positive operator from to for any .
Theorem 7. For any fixed , , there exists a positive constant such that
Proof. The formula (11) implies (26) for . If , then we obtain which by (11) and (12) yield (26) for . Assuming and using (11) and (6), we obtain where and are two constants only depending on . This completes the proof.
Now, for , we consider the two spaces and , and we have the three following theorems:
Theorem 8. For any fixed , if , there exists a positive constant such that for all and .
Theorem 9. For any fixed , if , then there exists a positive constant such that for all and .
Proof. Let . We denote the Steklov means of by , : It is obvious that for . Hence, if , then for every fixed . Furthermore, we have By Using (23) and (37), we have for any . From (29) and (37), we have By (37), we have for any . Finally, we have for any . Choosing , the proof is proved.
Theorem 10. Defining a new operator, For any fixed , if , then there exists a positive constant such that for all and .
Theorem 11. For any fixed , if , then there exists a positive constant such that for all and . In particular, if for some , then holds.
Proof. Let , and the Steklov means of the second order of defined by for . By simple computation, we have Meantime, while . Using the following inequality, Combining (23) and (44), we have Hence, choosing , the first part of the proof is proved. The second part of the proof can be directly observed from the definition of the space .
4. Direct Results
4.1. Voronovskaya-Type Theorem
Theorem 12. If and exists at a point , then
Proof. By the Taylor’s expansion formula for , we have where Applying the L’Hospital’s Rule, Thus, . Consequently, we can write By the Cauchy-Schwarz inequality, we have We observe that and . Then, it follows in Lemma 5 that Hence, from (17), we can obtain Combining (15) and (16), we complete the proof of Theorem 12.
Corollary 13. If , then we have uniformly with respect to any finite interval .
4.2. Point-Wise Estimates
In this subsection, we establish three point-wise estimates of the operators (2). First, we obtain the rate of convergence locally by using functions belonging to the Lipschitz class. We denote that is in , , and if it satisfies the following condition: where is a positive constant depending only on and .
Theorem 14. If , then for any , we have where denotes the distance between and .
Proof. Let be the closure of . Using the properties of infimum, there is at least a point such that . By the triangle inequality we have Choosing and and using the well-known Hölder inequality, we have Next, we obtain the local direct estimate of the operators (2), using the Lipcshitz type maximal function of the order introduced by Lenze  as
Theorem 15. If , then for any , we have
Proof. From equation (70), we have Applying the well-known Hölder inequality, we have Finally, we establish point-wise estimate of the operators (2) in the following Lipschitz-type space (see ) with two distinct parameters : where , , is a positive constant depending only on and .
Theorem 16. If , then for any , we have
Proof. Applying the well-known Hölder inequality with and , we have Thus, the proof is completed.
5. Weighted Approximation
Let be the set of all functions defined on satisfying the condition with an absolute constant which depends only on . denotes the subspace of all continuous functions with the norm . By , we denote the subspace of all functions for which is finite.
Theorem 17. If and , we have
Proof. Let be arbitrary but fixed. Applying , we have Let . Since , there exists , such that for all , Hence, Thus, Next, for sufficiently large such that , then . Applying Lemma 5, there exists , such that for all , Let . Combining (80) (82), and (83), we have Hence, the proof of Theorem 17 is completed.
Theorem 18. If , then we have
Proof. Applying the Korovkin theorem , it is sufficient to show the following three conditions: Since , the condition (86) holds for . From Lemma (11), we have Thus, . Finally, we have which implies that .
6. Some Voronovskaya-Type Approximation Theorem
As is known, if is not uniform, the limit may be not true. In , Yüksel and Ispir defined the following weighted modulus of continuity: and proved the properties of monotone increasing about as , , and the inequality