Abstract
In this article, our main purpose is to define the -variant of Szász-Durrmeyer type operators with the help of Dunkl generalization generated by an exponential function. We estimate moments and establish some direct results of the aforementioned operators. Moreover, we establish some approximation results in weighted spaces.
1. Introduction and Preliminaries
The well-known Bernstein operators [1] and the -Bernstein operators have become very important tools in the study of approximation theory and several branches of applied sciences and engineering [2, 3]. A good approach to introduce the -analogues in approximation theory is given by Mursaleen et al. [4] by an idea of newly introduced integers known as-integers and which is . In -calculus, there are generally two types of exponential functions which are defined as follows:
In 1950, Szász [5] defined positive linear operators on , and the Dunkl modification of these operators were given by Sucu [6] who was motivated by the work of Cheikh et al. [7]. The new generalization of these Szász operators [6] in quantum calculus (via -analogue) was introduced in [8] by Içöz and Çekim. Very recent work on the quantum Dunkl analogue in postquantum calculus studied in [9] for a set of all continuous functionsdefined ondenote it as; for parameter, they designed the following operators:
Lemma 1. For , we have
Moreover, for every and where exponential functions and recursion relations are given by
For , the number denotes the greatest integer functions.
The -analogues of Szász operators on the Dunkl type have been studied by several authors in [10–12] and for postquantum calculus in [9, 13–15]. We also refer some useful research articles on these topic (see [16–33]). Some convergence properties of operators through summability techniques can be examined in [34–39].
2. New Operators and Estimations of Moments
Here, with the motivational work of [9, 28], we design a different version of the-Szász-Durrmeyer operators compared to the previous one, and we define it by ((9)). To obtain a generalized version of the approximation in Dunkl form generally, we take positive sequences and for every and and also satisfy the following results: where the numbers and belong to .
Definition 2. Let and be defined by (7). Then, for every and , we have where and . Moreover, for all , the gamma functions in the postquantum calculus are defined as follows: and
Note that
For more detailed properties of the -analogue of the beta and gamma functions, see [40, 41].
Lemma 3. For the operators in (9), we have : and
Proof. We prove this Lemma by using the results obtained in (11), (12), (13), and (14). Therefore, for , we easily see that Take . Then, we have Similarly, If , then We apply the results defined by (7) and separate it into even and odd terms, i.e., take and for all , and applying (2) and Lemma 1, we easily see that and These conclusions complete the proof of Lemma 3.
Lemma 4. Let , for ; then, we have
3. Approximation in Weighted Spaces
To obtain the approximation in weighted Korovkin spaces, we take the weight function and on consider , , and such that wheredepends on , where is a constant, is the set of continuous functions on , is the set of all bounded and continuous functions on equipped with the norm , and on , a norm is given by .
Theorem 5. Let , , and . Then, for each , the sequence converges uniformly to on each compact subset of if and only if and .
Proof. Since the operators defined by (9) are positive and linear on , if and , then . Therefore, from Korovkin’s theorem for every , the operators converge uniformly to as if and only if In another way for all , if we assume converges uniformly to , when approaches to , then clearly and . Suppose in the case where the sequences and do not converge to and as . Thus, from Lemma 3, we have and which leads to contradiction, and hence, and as .
Theorem 6. Let the sequences of positive numbers and satisfy and as approaches to . Then, for every on , we have
Proof. Take for . Since by Theorem 5, converges to uniformly for , from Lemma 3, we conclude that If , Then, we have Similarly, if we take , we have which implies that These explanations complete the proof of Theorem 6.
Let the modulus of the continuity of for any and be defined as follows: where it is obvious that and for
Theorem 7. Take the numbers with the positive sequences , satisfying and as . Let be defined on the interval for . Then, for every on , we have where is a constant depending only on .
Proof. Let and for Then, clearly one has Also, when and for , then for a given From (38) and (39), we easily see that which implies that The Cauchy-Schwartz inequality gives us From an easy calculation, this leads us to Therefore, in view of (41)–(43), clearly we get Finally, if we take , then we use a denumerable to get the result.
4. Pointwise Approximation
In an approximation process for measuring the smoothness of a continuous function, we need Peetre’s -functional [42] defined as follows.
Definition 8. Let, and for a givenof the-functional, we have
Now, from [43], there exists a positive constant such that where the modulus of continuity of order two is given by
Moreover, the classical modulus of continuity is given by
Theorem 9. Suppose and are the sequences of positive numbers satisfying such that as . Let us define an auxiliary operator such that . Then, for every , we have where , and is defined in Theorem 7.
Proof. Let We easily get and where we easily know that Therefore, In view of the Taylor series expansion, we have On operating , we conclude that Since we know that we get Hence, the above discussion completes the proof.
Theorem 10. Let be defined by (9); then, for every , there exists an absolute constant such that where is defined by Theorem 9.
Proof. In the view of the result asserted by Theorem 9, we prove this theorem. For all and we have By taking the infimum over all and using (45), we get We consider the Lipschitz-type maximal function by [44] and obtain the local approximation such as for , , and . We recall that
Theorem 11. For all and we have where theLipschitz maximal function is defined by (60), andis defined by Theorem 7.
Proof. We prove Theorem11 by applying (60) and the well-known Hölder inequality: The desired results are proven.
We next denote
Theorem 12. Let Then, defined by (9) satisfies
Proof. From the Taylor series expansion of order two, we have for . Let Then, we have Therefore, we have This completes the proof of Theorem 12.
Data Availability
Not applicable.
Conflicts of Interest
The author declares there are no conflicts of interest.