Sequence Spaces, Function Spaces and Approximation TheoryView this Special Issue
Rate of Approximation for Modified Lupaş-Jain-Beta Operators
The main intent of this paper is to innovate a new construction of modified Lupaş-Jain operators with weights of some Beta basis functions whose construction depends on such that and . Primarily, for the sequence of operators, the convergence is discussed for functions belong to weighted spaces. Further, to prove pointwise convergence Voronovskaya type theorem is taken into consideration. Finally, quantitative estimates for the local approximation are discussed.
In 1972, Jain  with the help of Poisson distribution introduced a famous linear positive operators as follows: where , defined on , and
If we put in (1), then it becomes Szász-Mirakyan-type.
In 1995, Lupaş  introduced a sequence of linear positive operators; later on in 1999, it was modified by Agratini  as follows: and also discussed the Kantorovich and Durrmeyer variant of operator (1).
In 2018, Tunca et al.  modified operator (3) in such a way that in the construction, authors take the negative subscript -1 of the Pochhammer symbol into consideration; due to this, the calculations become simpler in a remarkable degree just as
In order to approximate Lebesgue integrable functions, the most important modifications are Kantorovich and Durrmeyer integral operators. The Durrmeyer variant of operator (1) is introduced by Tarabie  and Mishra and Patel  with some beta basis functions.
In , Cárdenas-Morales et al.  defined the Bernstein-type operators by and also presents a better degree of approximation depending on . This type of approximation operators generalizes the Korovkin set from to . The Durrmeyer variant of is defined in . In , Aral et al.  defined a similar modification of Szász-Mirakyan-type operators by using a suitable function .
be a continuously differentiable function on
The new formulated operators are defined as for , , and suitable functions defined on .
As we know, in order to approximate Lebesgue integrable functions, the most important modifications are Kantorovich and Durrmeyer integral operators. Motivated by the above mentioned Durrmeyer type generalizations of various operators and also from [11–23], in this paper, Durrmeyer-type modification of generalized Lupaş-Jain operators (5) by taking weights of some beta basis function is defined as follows: where and is defined as where is the beta basis function and is a function satisfying the conditions and given above.
The rest of the work is organized as follows: in the second section, moments and central moments for are calculated. In the third section, we study convergence properties of in the light of weighted space. In the fourth section, we obtain the order of approximation of new constructed operators associated with the weighted modulus of continuity. In the fifth section, we shall prove Voronovskaya-type theorem in quantitative form. These kinds of results are very useful to describe the rate of point-wise convergence. Finally, in the last section, we obtain some local approximation results related to -functional.
2. Basic Results
In this section, we prove some lemmas for which are required to prove our main results.
Lemma 1. Let be given by (6). Then for each , , and , we have (i)(ii)(iii)(iv)(v)
Lemma 2. For operators , we have the following properties: (i)(ii)(iii)
3. Convergence of
Here, we prove the convergence of by using weight function. Let be a function satisfying the conditions and given above. Also, let be a weight function and the weighted space is defined as follows: where is a constant which depends only on , with the norm
Also, we mention some subspaces of as
It is obvious that
We have the following results for the weighted Korovkin-type theorems due to Gadjiev 
Lemma 3. .
The positive linear operators act from to if and only if the inequality holds, where is a constant depending on .
Theorem 4 (see ). Let the sequence of positive linear operators acting from to and satisfying Then, for each , we have
Remark 5. Examining Lemma 1 based on the famous Korovkin theorem , it is clear that does not form an approximation process. Now, in order to obtain convergence properties, we replace the constant by such that
Theorem 6. Let such that , and also, let be the sequence of positive linear operators. Then, for each function , we have
4. Rate of Convergence
In this part, we would like to determine the rate of convergence for by weighted modulus of continuity which was introduced by Holho  in 2008, as follows: where , with the following properties: (i)(ii), , for (iii)
Theorem 7 (see ). Let be a sequence of positive linear operators with where the sequences , , , and converge to zero as . Then for all , where
Theorem 8. Let such that , and also, let be the sequence of positive linear operators. Then for all , we have where
Remark 9. For in Theorem 8, we obtain
5. Pointwise Convergence of
In this section, we shall analyze pointwise convergence of by obtaining the Voronovskaya theorem in a quantitative form by using the same technique in .
Theorem 10. Let such that , and also, let , and suppose that and exist at . If is bounded on , then, we have
Proof. By using Taylor expansion of at , we have
Therefore, (28) together with the assumption on ensures that
and is convergent to zero as . Now applying the operators (6) to the equality (27), we obtain
From Lemma 2, we get
By estimating the equality (30), we will get the proof.
Since from (28), for every , . Let such that for every . By Cauchy-Schwartz inequality, we get Since we obtain Thus, by taking into account equations (31), (32), and (35) to equation (30), the proof is completed.
Remark 11. If we choose with , then, one can easily see that and
6. Local Approximation
In this section, for the operators , we shall present local approximation theorems. Let denote the space of real-valued continuous and bounded functions defined on the interval . The norm on the space is defined by
-functional is defined as where and . By Devore and Lorentz (, p. 177, Theorem 6.4), there exists an absolute constant such that
The second order modulus of smoothness is as follows: where . The usual modulus of continuity of is defined by
Theorem 12. Let such that and for all Also, let be a function satisfying the conditions and , and is finite. Then, there exists an absolute constant such that where
Proof. Let and By Taylor’s formula, we have By using the equality, Now, putting in the last term in equality (43), we get By applying operator (6) to the both sides of equality (43) and from Lemma 1, we deduce As we know, is strictly increasing on , and with condition (), we get where Also, Hence, we have if ; then Taking infimum over all , we obtain Theorem 12 is proved.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
We declare that there is no conflict of interest.
The first author is grateful to the Council of Scientific and Industrial Research (CSIR), India, for providing the Senior Research Fellowship with file no. 09/1172(0001)/2017-EMR-I. This work is also supported by the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant No. 2018C087R), the Program for New Century Excellent Talents in Fujian Province University, and Sponsoring Agreement for Overseas Studies in Fujian Province.
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