International Journal of Mathematics and Mathematical Sciences
Volume 2015 (2015), Article ID 874178, 7 pages
http://dx.doi.org/10.1155/2015/874178

Research Article

Some Applications of New Modified -Integral Type Operators

R. P. Pathak and Shiv Kumar Sahoo

Department of Mathematics, National Institute of Technology, G E Road, Raipur 492010, India

Received 23 March 2015; Revised 29 June 2015; Accepted 12 July 2015

Academic Editor: Ram N. Mohapatra

Copyright © 2015 R. P. Pathak and Shiv Kumar Sahoo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1], Dogru and Duman [2], Aral and Gupta [3], 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 [8] 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 by

In 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 [4] 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 [9].

Deo et al. [10] introduced new version of Bernstein-Durrmeyer type operators defined as follows: for , where ,

where

and they established some approximation results on it.

Sharma [6] introduced the following -Durrmeyer type operators defined as follows: for , where ,

where

and they established some approximation results on it.

In this paper motivated by Sharma [6, 7, 10–12] we introduce a -analogue of the -Baskakov-Durrmeyer type operators defined as follows: for ,

where we set .

Kasana et al. [13] obtained a sequence of modified Szâsz type operators for integrable function on defined as

where 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

We use Lemma 5 [6] for and, by the definition of -Beta function, we get .

Theorem 4. Let the sequence of positive linear operators be defined by (10). For all ; ; ; , one gets

Proof. By using Lemma 5 and letting in the operators be defined by (10), we get

Again, we set in the operators ; we get

Similarly, we set in the operators ; we get

This completes the proof.

Lemma 5. For the special case , one gets [14]

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 [14]:

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 [3]; 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 get

Since

and , this implies , and we get

Again, since

we get

By using -statistical convergence theorem given by Duman and Orhan [15], here we let ((21), (23), and (25)), and we get for if and only if

This 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 gets

where 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 by

We see that ; the modulus of continuity tends to zero. Consider

and we get

Again

and we get

By (31) and (33), we get for if and only if

This 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 gets

where 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 [16].

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 get

where 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 get

Applying on (38), we get

where . Then for every there exist such that for all , we get

here, Since is arbitrary and small, , we get

This 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 get

where 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 get

By 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.

5. Conclusion

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.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

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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