Abstract
In this article, we introduce Stancu-type modification of generalized Baskakov-Szász operators. We obtain recurrence relations to calculate moments for these new operators. We study several approximation properties and -statistical approximation for these operators.
1. Introduction
In 1912, Bernstein [1] proposed the famous polynomial known as the Bernstein polynomial to give a simple, short, and most elegant proof of the Weierstrass approximation theorem. Since then, several papers have appeared to study approximation properties in different settings and spaces. Many new operators were constructed, e.g., Szász [2], Mirakjan [3], Kantorovic [4], Durrmeyer [5], Stancu [6], and many more [7–9]. These operators provide the improvement of approximating functions of different classes and give better and better estimates. For example, the Baskakov operators were given in [10]: For , the space of all continuous functions on normed with standard sup-norm
Devore and Lorentz [11] introduced a generalization of operators (1) dependent on a constant and independent of as follows: where and
Recently, Agrawal et al. [12] studied the following operators (2): for , for some , where
Inspired by Stancu’s work [6], we have studied recently the Stancu-type generalization in [13]. Now, we propose the Stancu-type generalization of operators (4) as follows: for any bounded and integrable function defined on , where . For , the operators (5) reduce to operators (4).
We establish recurrence relations to find moments and central moments. We study some approximation properties and the Voronovskaja-type asymptotic formula. We also study weighted approximation.
2. Auxiliary Results
Our first result is the recurrence formula for moments.
Theorem 1. The order moment for (5) is defined by . Then, and
Proof. We use the identity Then, where where where Therefore, Substituting (17), (14), and (13) in (12), we get Further, substituting (18) in (10), we get the result.
Corollary 2. From the above theorem, we get (i)(ii)(iii)
Theorem 3. The order central moment is defined by . The following recurrence relation holds:
Corollary 4. From the above theorem, we get (a)(b)
Corollary 5. We further get (a)(b)
3. Main Results
Peetre’s -functional is defined as for , where is bounded on and Note that where is the second-order modulus of continuity [11].
The usual modulus of continuity of is defined as
Theorem 6. For , where and
Proof. Put Note that and Let Then, by using Taylor’s theorem, we may write which gives Hence, Since and we have Now, by Corollary 4 (b), we get By (27), we get Since we get From (33), we get Now, taking the infimum over all , we obtain Hence, by using (22), we get the result.
For our next result, we consider the functions belonging to the Lipschitz class:
where and ;
Theorem 7. For , we have where .
Proof. First, we prove for . For , we get Since , we get by the Cauchy-Schwarz inequality: For , applying Hölder’s inequality, we get Since , we have Therefore, we get (39).
Next, we obtain a Voronovskaja-type asymptotic formula.
Theorem 8. If exists at a point for , then
Proof. From Taylor’s expansion of , we may write where By operating , we obtain By the Cauchy-Schwarz inequality, we get Since , we get Now, combining the above equations and using Corollary 5, we get
Let , where is a constant which depends only on , and
Also, let be continuous on , and where
The weighted modulus of continuity [14] is defined by
Lemma 9 (see [14]). Let . Then, (i) is a monotone increasing function of (ii) as (iii) for each (iv) for each
Theorem 10. For , we have
Proof. By Lemma 9, we have Operating , we get Using a second-order central moment, we get Applying the Cauchy-Schwarz inequality, we obtain Again, using the central moment of order 4, we get Combining the estimates (55)–(58) and choosing , , we get the required result.
4. -Statistical Convergence
Defining a -analog of the Cesàro matrix is not unique (see [15, 16]). Here, we consider the -Cesàro matrix, , defined by which is regular for .
Let (the set of natural numbers). Then, is called the asymptotic density of , where denotes the cardinality of the enclosed set. A sequence is called statistically convergent to the number if for each , where (see [17]).
Recently, Aktuğlu and Bekar [16] defined -density and -statistical convergence. The -density is defined by
A sequence is said to be -statistically convergent to the number if , where for every . That is, for each , and we write .
If for an infinite set , then ; hence, statistical convergence implies -statistical convergence but not conversely (c.f. [16, Example 15]). Recently in [18], authors proved Korovkin’s type theorem via q-statistical convergence. Using the same technique we prove the following theorem.
Theorem 11. For all , we have
Proof. It is sufficient to show that , for , where It is clear that By Corollary 2 (ii), we have For , define the sets: Then, Since , we have . Hence, Again, by Corollary 2 (iii), we obtain For , define the sets: Then, Since which implies that , Hence, the proof is completed.
Example 12. Let be defined by
That is, occurs times and occurs times , respectively. Let . Then, , i.e., , but does not exist, so is not statistically convergent.
Define , where it is defined by (71). Then, obviously . Applying the above theorem, we have
On the other hand, since is -statistically convergent but neither convergent nor statistically convergent, the sequence can not be convergent, while it is -statistically convergent.
5. Graphical Analysis
In this section, we will give some numerical examples with illustrative graphics with the help of MATLAB.
Example 13. Let , , and . The convergence of the operator towards the function is shown in Figure 1.
Example 14. Let , , and . The convergence of the operator towards the function is shown in Figure 2.
From these examples, we observe that the approximation of function by the operators becomes better when we take larger values of .
Notice that for , the operators (5) reduce to operators (4).
Example 15. Let . For , comparison of convergence of the constructed operator (5) (green and pink) with the previously defined operator (4) (blue) is shown in Figure 3. From this figure, it is clear that the constructed operator gives a better approximation to than the previously defined operator.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to writing this paper. All authors read and approved the manuscript.
Acknowledgments
This work is supported by the Natural Science Foundation of Fujian Province of China (Grant No. 2020J01783), the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant No. 2018C087R), and the Program for New Century Excellent Talents in Fujian Province University.