Some Applications of New Modified -Integral Type Operators
We introduce a new sequence of -integral operators. We show that it is a weighted approximation process in the polynomial space of continuous functions defined on unit interval. Weighted statistical approximation theorem, Korovkin type theorems for fuzzy continuous functions, and an estimate for the rate of convergence for these operators.
The study of -Calculus is a generalization of any subjects, such as hyper geometric series, complex analysis, and particle physics. Currently it continues being an important subject of study. It has been shown that positive linear operators constructed by -numbers are quite effective as far as the rate of convergence is concerned and we can have some unexpected results, which are not observed for classical case. In the last decade, some new generalizations of well-known positive linear operators, based on -integers, were introduced and studied by several authors. For example, -Meyer Konig and Zeller operators were studied by Trif , Dogru and Duman , Aral and Gupta , and so forth. In 20011, Aral and Gupta [4, 5] introduced a -generalization of the classical Baskakov operators. In 2012, Sharma [6, 7] introduced the -Durrmeyer type operators. Orkcu and Dogru  introduced Kantorovich type generalization of -Szasz-Mirakjan operators and discussed their -statistical approximation properties. In this paper motivated by Sharma we introduced a -analogue of the -Durrmeyer operators and we study better rate of convergence and statistical approximation properties.
We mention some important definitions of -Calculus.
Definition 1. For any fixed real numbers and , the -integers are defined byIn this way for real number one may write
Definition 2. The -factorial is defined by
Definition 3. For any number , the -binomial coefficient is defined by
Aral and Gupta  introduced a -generalization of the classical Baskakov operators. For all , , and each positive integer , the operators are defined as For , the above operators become classical Baskakov operators .
Deo et al.  introduced new version of Bernstein-Durrmeyer type operators defined as follows: for , where , whereand they established some approximation results on it.
Sharma  introduced the following -Durrmeyer type operators defined as follows: for , where ,whereand they established some approximation results on it.
Kasana et al.  obtained a sequence of modified Szâsz type operators for integrable function on defined aswhere and belong to and is fixed.
In this paper, motivated by Kasana and Sharma, we introduce a -analogue of the -Baskakov-Durrmeyer type operators defined as follows: for , where and belong to and is fixed.
The aim of this paper is to study some approximation properties of a new generalization of operators based on -integers. We estimate moments for these operators. Also, we study statistical convergence and Korovkin type theorems for fuzzy continuous functions. Finally, we give better error estimations for operators (10) and (12).
2. Estimation of Moments
Theorem 4. Let the sequence of positive linear operators be defined by (10). For all ; ; ; , one gets
Lemma 5. For the special case , one gets 
Lemma 6. For the sequence of positive linear operators , one gets the following central moments: let , ;
Lemma 7. For the special case , one has the following central moment :
3. Weighted Statistical Approximation Theorem
The aim of this section is to use statistical convergence to study Korovkin type approximation of function by means of sequence of positive linear operators from a weighted space into a weighted subspace. Let be the space of the continuous and bounded functions defined on such that , where is a constant depending on . Our operators acting from similar methods of ; we obtain the following results.
Theorem 8. Let sequence , , such that , and the sequence of positive linear operators , , be defined by (10). Then, for all and , one gets
Proof. The weight functions and weighted subspace are defined by ; ; ; ; and such that is continuous on with norm; here and are Banach space. By using Theorem 4, we getSinceand , this implies , and we getAgain, sincewe getBy using -statistical convergence theorem given by Duman and Orhan , here we let ((21), (23), and (25)), and we get for if and only ifThis completes the proof.
Theorem 9. Let sequence , , such that , and the sequence of positive linear operators , , be defined by (12). Then, for all functions , one getswhere and belong to and is fixed.
Proof. The proof is analogous as Theorem 8.
Theorem 10. Let sequence , , such that , and the sequence of positive linear operators , , be defined by (10). Then, for all and functions , we get
Proof. To prove the theorem, we use modulus of continuity of on closed interval given byWe see that ; the modulus of continuity tends to zero. Consider and we getAgain and we getBy (31) and (33), we get for if and only ifThis completes the proof.
Theorem 11. Let sequence , , such that , and the sequence of positive linear operators , , be defined by (12). Then, for all functions , one getswhere and belong to and is fixed.
Proof. The proof is analogous as Theorem 10.
4. Korovkin Type Theorems for Fuzzy Continuous Functions
In this section we mention some important definitions given by Burgin and Duman .
Definition 12. A number a is called an -limit of a sequence , if for any , the inequality is valid for almost all , that is, there is such that for any , we have It is denoted by .
Definition 13. A sequence that has an -limit is called -convergent and it is said that , -converges to its -limit. It is denoted by .
Definition 14. A function is called -continuous in if and is called fuzzy continuous in if , where defined as:
For example the functions when , and are fuzzy continuous in each finite interval of the real line , but they are not continuous in any interval with the length larger than 1. To define the Riemann integral for a continuous function , step functions are utilized. If the integral of exists, then any such step function is fuzzy continuous.
Theorem 15. Let a sequence ; such that and let the sequence of positive linear operators ; be defined by (10). If for Then for all functions , we getwhere is any real number such that for some
Proof. Let the functions defined as: for all . Now, for each , there corresponds such that whenever . Again for , then there exist a positive number such that . Thus for all and , we getApplying on (38), we get where . Then for every there exist such that for all , we get here, Since is arbitrary and small, , we getThis completes the proof.
Theorem 16. Let a sequence ; such that and let the sequence of positive linear operators ; be defined by (12). If for Then for all functions , we getwhere is any real number such that for some
Proof. The proof is analogous as Theorem 15.
Theorem 17. Let be the integrable and bounded in the interval and let if exists at a point . Let a sequence ; such that and let the sequence of positive linear operators ; be defined by (10). Then, one gets that
Proof. Let if exists at a point , then by using Taylor’s expansion, we write where as . Applying , we get By using Theorem 4, and multiplying both sides, we get Here we write,Since is arbitrary and small, and whenever , we getBy using (48), in (46), we get This completes the proof.
Theorem 18. Let be the integrable and bounded in the interval and let if exists at points ; . Let a sequence ; such that and let the sequence of positive linear operators ; be defined by (12). Then, one gets that
Proof. The proof is analogous as Theorem 17.
In this way we conclude that weighted statistical approximation theorem, Korovkin type theorems for fuzzy continuous functions, an estimate for the rate of convergence, and some properties are obtained for these operators. For , the operators defined by (8) reduce generalized Baskakov operators. For , the operators defined by (6) reduce generalized Durrmeyer operators (5). By using maple programming, Honey Sharma concludes that Deo’s modified operator [10, 11] does not improve the approximation process. In this paper we conclude that, by Deo’s modified operators, (10) and (12) improve the approximation process when the value of is very large, that is, when tends to infinity.
Conflict of Interests
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.
The authors are thankful to Director of National Institute of Technology, Raipur (CG), for encouragement. This work was supported and grant funded by the Chhattisgarh Council of Science Technology, Raipur 492001, India. They are also thankful to Director of CCOST, Chhattisgarh, India, for time-to-time eminent support and encouragement.
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