Abstract
In this article, we introduce a new Durrmeyer-type generalization of -Szász-Mirakjan operators using the -gamma function of the second kind. The moments and central moments are obtained. Then, the Voronovskaja-type asymptotic formula is investigated and point-wise estimates of these operators are studied. Also, some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre -functional. Finally, the rate of convergence and weighted approximation of these operators are presented.
1. Introduction
In recent years, -analogues of well-known positive operators have been widely constructed and researched since Mursaleen et al. first introduced -Berstein operators [1] and -Bernstein-Stancu operators [2]. In [3], Acar first introduced -Szász-Mirakjan operators and gave a recurrence relation for the moments of these operators. In [4], Mursaleen et al. proposed a Kantorovich variant of the -analogue of Szász-Mirakjan operators under the nondecreasing condition. In [5], Sharma and Gupta introduced -Szász-Mirakjan-Kantorovich operators and studied their approximation properties. In [6], Acar et al. constructed King’s type -Szász-Mirakjan operators preserving and discussed the order of approximation and weighted approximation properties of these operators. In [7], Aral and Gupta constructed -Szász-Mirakjan-Durrmeyer operators using -gamma function of the first kind and estimated moments and established some direct results of these operators. In [8], Mursaleen et al. introduced a new modification of Szász-Mirakjan operators based on the -calculus and investigated their approximation properties including weighted approximation and Voronovskaya-type theorem. In [9], Mursaleen et al. proposed two different Kantorovich-type -Szász-Mirakjan operators and discussed their error estimated. In [10], Kara and Mahmudov constructed a new -Szász-Mirakjan operators as
Definition 1. Let and . For , -Szász-Mirakjan operators can be defined as where .
Meantime, quantitative estimates for the convergence in the polynomial weighted spaces and Voronovskaya theorem for new -Szász-Mirakjan operators (1) were given. All these achievement motivates us to construct the Durrmeyer analogue of the -Szász-Mirakjan operators defined by (1).
Definition 2. Let and . For , we construct the -Szász-Mirakjan-Durrmeyer operators as Let us recall the basic notations of -calculus which can be found in [11]. For any fixed real number and , the -integers are defined as where denotes the -integers and . Also -factorial is defined as follows: Now, we introduce two types of -analogues of exponential function and (see [7]): Let be an arbitrary function. The improper -integral of on was defined as (see [12]) The -gamma function of the second kind was defined in [12] as follows: Meantime, the -gamma function fulfills the following relation: moreover, for any nonnegative integer , the following relation holds:
2. Auxiliary Results
In order to discuss the approximation properties of the operators , we need the following lemmas.
Lemma 3 (see ([10], Lemma 4)). For , , and , we have
Lemma 4. For , , , and , we have where , , , , , , and .
Proof. Using the -gamma function of the second kind, we can obtain and next using , we have Using the equality and Lemma 3, we have Using , we have Thus, Similarly, using , we can obtain
Lemma 5. Using Lemma 4, we immediately have the following explicit formulas for the central moments:
Lemma 6. The sequences , satisfy such that , and , as , , then
Proof. Applying Lemma 5, we can easily obtain (19) and (20). While , we can rewrite Using and , we can get (21).
3. Voronovskaja-Type Theorem
Theorem 7. Let , be the sequences defined in Lemma 6 and . Supposing that exists at a point , then, we can obtain where denotes the set of all real-valued bounded and continuous functions defined on , endowed with the norm .
Proof. By the Taylor’s expansion theorem of function , we can obtain where , is bounded and . Applying the operator to the equality above, we can obtain Since , then for all , there exists such that and it will imply for all fixed as sufficiently large. While if , then , where is a constant. Using Lemma 6The proof is completed.
4. Point-Wise Estimate
In this section, we establish two point-wise estimate of the operators . First, a function is said to satisfy the Lipschitz condition on (named ), , if
where is an absolute positive constant depending only on and .
Theorem 8. Let , and be any bounded subset on . If, then, for all , we have where denotes the distance between and defined by
Proof. Let be the closure of . Using the properties of infimum, there is at least a point such that . By the triangle inequality we can obtain Choosing and and using the well-known Hölder inequality, we have Next, we obtain the local direct estimate of the operators , using the Lipcshitz-type maximal function of the order introduced by Lenze [13] as
Theorem 9. Let and . Then, for all , we have
Proof. From equation (33), we have Applying the well-known Hölder inequality, we can get
5. Local Approximation
In this section, we establish local approximation theorem for -Szász-Mirakjan-Durrmeyer operators. 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 ([14], p.177, Theorem 2.4), there exists an absolute constant such
Theorem 10. Let , be the sequences defined in Lemma 6 and . Then, for all , there exists an absolute positive such that
Proof. For a given function , let us define the following new operators: By Lemma 4 and Lemma 5, we obtain Let and . By Taylor’s expansion formula, we get Applying to the above equality, we can write On the other hand, since . Hence, Taking infinum on the right hand side over all and using (39), we obtain the desired assertion.
Corollary 11. Let , be the sequences defined in Lemma 6 and . Then, for any finite interval , the sequence converges to uniformly on .
6. Rate of Convergence
Let where is an absolute constant depending only on . is equipped with the norm . As is known, if is not uniform, we cannot obtain . In [15], 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 12. Let , , and , we have
Proof. For any and , we easily have , thus and for any , and , we have For (51) and (52), we can get By Schwarz’s inequality, for any , we can get By taking and supremum over all , we accomplish the proof of Theorem 12.
7. Weighted Approximation
In this section, we will discuss the following three theorems about weighted approximation for the operators .
Theorem 13. Let and the sequences , be the sequences defined in Lemma 6; then, for any , there exists such that for all , the inequality holds.
Proof. Using (47) and (48), we can write For any and , (56) can be rewritten Using (20) and (21), there exists such that for any , By Schwarz’s inequality, we can obtain Since is linear and positive, using (58)–(60), we can obtain Choosing , we have the conclusion holds.
Theorem 14. Let , be the sequences defined in Lemma 6. Then, for any , we have
Proof. By weighted Korovkin theorem in [16], we see that it is sufficient to verify the following three conditions: Since , then (64) holds true for . By Lemma 4, we can obtain Thus, the proof of Theorem 14 is completed.
Theorem 15. Let , be the sequences defined in Lemma 6. Then, 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 Lemma 4, while , we can obtain Hence, we can choose and to be so large such that for any the inequality holds. Also, the first term of the above inequality tends to zero by Theorem 12, that is Thus, combining (68)–(71), we obtain the desired result.
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 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).