- Research
- Open Access
- Published:
Statistical approximation by Kantorovich-type discrete q-Betaoperators
Advances in Difference Equations volume 2013, Article number: 345 (2013)
Abstract
The aim of the present paper is to introduce a Kantorovich-type modification ofthe q-discrete beta operators and to investigate their statistical andweighted statistical approximation properties. Rates of statistical convergenceby means of the modulus of continuity and the Lipschitz-type function are alsoestablished for operators. Finally, we construct a bivariate generalization ofthe operator and also obtain the statistical approximation properties.
MSC: 41A25, 41A36.
1 Introduction
Gupta et al.[1] introduced discrete q-Beta operators as follows:
where
In the above paper, Gupta et al.[1] introduced and studied some approximation properties of these operators.They also obtained some global direct error estimates for the above operators usingthe second-order Ditzian-Totik modulus of smoothness and defined and studied thelimit discrete q-Beta operator. Also, they gave the following equalities:
In the recent years, applications of q-calculus in approximation theory isone of the interesting areas of research. Several authors have proposed theq analogues of Kantorovich-type modification of different linearpositive operators and studied their statistical approximation behaviors.
In 1974, Khan [2] studied approximation of functions in various classes using differenttypes of operators.
On the other hand, statistical convergence was first introduced by Fast [3], and it has become an area of active research. Also, statisticalconvergence was introduced by Gadjiev and Orhan [4], Doğru [5], Duman [6], Gupta and Radu [7], Ersan and Doğru [8] and Doğru and Örkcü [9].
In 2011, Örkcü and Doğru obtained weighted statistical approximationproperties of Kantorovich-type q-Szász-Mirakjan operators [10].
Recently, Doğru and Kanat [11] defined the Kantorovich-type modification of Lupas operators as follows:
In [11], Doğru and Kanat proved the following statistical Korovkin-typeapproximation theorem for operators (1.2).
Theorem 1 Let, , be a sequence satisfying the followingconditions:
then if f is any monotone increasing function defined on, for the positive linear operator, then
holds.
In [5], Doğru gave some example so that is statistically convergent to 1 in ordinary case.Throughout the present paper, we consider . Following [12, 13], for each non-negative integer n, we have
and
Further, we use the q-Pochhammer symbol, which is defined as
The q-derivative of a function f is defined by
The q-Jackson integral is defined as (see [14])
and over a general interval , one defines
Now, let us consider the following Kantorovich-type modification of discreteq-Beta operators for each positive integer n and:
where f is a continuous and non-decreasing function on the interval, .
It is seen that the operators are linear from the definition ofq-integral, and since f is a non-decreasing function,q-integral is positive, so are positive.
To obtain the statistical convergence of operators (1.6), we need the following basicresult.
2 Basic result
Lemma 1 The following equalities hold:
-
(i)
,
-
(ii)
,
-
(iii)
.
Proof By using (1.4), (1.5) and the equality , we have
Hence, by using and (2.2), we get
Similarly, using (2.2), and , we obtain
Finally, using (2.3), , and , we obtain
□
Remark 1 From Lemma 1, we have
Remark 2 If we put , we get the moments of Kantorovich-type modificationof discrete beta operators as
and
3 Korovkin-type statistical approximation properties
The study of Korovkin-type statistical approximation theory is a well-establishedarea of research, which deals with the problem of approximating a function with thehelp of a sequence of positive linear operators (see [9, 15, 16] for details). The usual Korovkin theorem is devoted to approximation bypositive linear operators on finite intervals. The main aim of this paper is toobtain the Korovkin-type statistical approximation properties of our operatorsdefined in (1.6), with the help of Theorem 1.
Let us recall the concept of a limit of a sequence extended to a statistical limit byusing the natural density δ of a set K of positive integers:
whenever the limit exists (see [17]). So, the sequence is said to be statistically convergent to a numberL, meaning that for every ,
and it is denoted by .
In this part, we will use the notation instead of for abbreviation.
Theorem 2 Letbe a sequence satisfying (1.3) forand a Kantorovich-type modification of discrete q-Beta operators given by (1.6). Then, for anyfunctionand, where, we have
wheredenotes the space of all real bounded functions f which are continuous in.
Proof Using , it is clear that
Now, by Lemma 1(ii), we have
For a given , we define the following sets:
and
From (3.1), one can see that . So, we get
By using (1.3), it is clear that
So,
then
Finally, by Lemma 1(iii), we have
where .
If we choose , , , then one can write
by (1.3). Now, given , we define the following four sets:
It is obvious that . So, we get
So, the right-hand side of the inequalities is zero by (3.2), then
So, the proof is completed. □
4 Weighted statistical approximation
Let be the set of all functions f defined on satisfying the condition , where is a constant depending only on f. By, we denote the subspace of all continuous functionsbelonging to . Also, let be the subspace of all functions, for which is finite. The norm on is .
Theorem 3 Letbe a sequence satisfying (1.3) for. Then, for all nondecreasingfunctions, we have
Proof As a consequence of Lemma 1, since , where is a positive constant, is a sequence of linear positive operators actingfrom to . Using , it is clear that
Now, by Lemma 1(ii), we have
By using (1.3), it is clear that
then
Finally, by Lemma 1(iii), we have
If we choose , , , then one can write
by (1.3). Now, given , we define the following four sets:
It is obvious that . So, we get
So, the right-hand side of the inequalities is zero by (4.2), then
So, the proof is completed. □
5 Rates of statistical convergence
In this part, rates of statistical convergence of operator (1.6) by means of modulusof continuity and Lipschitz functions are introduced.
Lemma 2[15]
Letand, . The inequality
is satisfied.
Let , the space of all bounded and continuous functions on, and . Then, for , the modulus of continuity of f denoted by is defined to be
Then it is known that for , and also, for any and each , we have
Theorem 4 Letbe a sequence satisfying (1.3). For every non-decreasing, and, we have
where
Proof Let non-decreasing and . Using linearity and positivity of the operators and then applying (5.1), we get for and that
Taking into account and then applying Lemma 2 with and , we may write
By using the Cauchy-Schwarz inequality, we have
Taking , a sequence satisfying (1.3), and using and then choosing as in (5.2), the theorem is proved. □
Notice that by the conditions in (1.3), . By (5.1), we have
This gives us the pointwise rate of statistical convergence of the operators to .
Now we will study the rate of convergence of the operator with the help of functions of the Lipschitz class, where and . Recall that a function belongs to if the inequality
We have the following theorem.
Theorem 5 Let the sequencesatisfy the conditions given in (1.3), and let, with. Then
whereis given as in (5.2).
Proof Since are linear positive operators and, on with , we can write
If we take , , applying Lemma 2 and Hölder’s inequality,we obtain
Taking , as in (5.2), we get the desiredresult. □
6 The construct of the bivariate operators of Kantorovich type
The purpose of this part is to give a representation for the bivariate operators ofKantorovich type (1.6), introduce the statistical convergence of the operators tothe function f and show the rate of statistical convergence of theseoperators.
For and , let us define the bivariate case of operators (1.6)as follows:
where
In [18], Erkuş and Duman proved the statistical Korovkin-type approximationtheorem for the bivariate linear positive operators to the functions in space.
In 2006, Doğru and Gupta [19] introduced a bivariate generalization of the q-MKZ operators andinvestigated its Korovkin-type approximation properties.
Recently, Ersan and Doğru [8] obtained the statistical Korovkin-type theorem and lemma for thebivariate linear positive operators defined in the space as follows.
Theorem 6[8]
Letbe the sequence of linear positive operators acting frominto, where. Then, for any,
Lemma 3[8]
The bivariate operators defined in[8]satisfy the following items:
-
(i)
,
-
(ii)
,
-
(iii)
,
-
(iv)
= + + + .
In order to obtain the statistical convergence of operator (6.1), we need thefollowing lemma.
Lemma 4 The bivariate operators defined in (6.1) satisfy the followingequalities:
-
(i)
,
-
(ii)
,
-
(iii)
,
-
(iv)
= + + + + + .
Proof By the help of the proofs for the bivariate operator in [20], the conditions may be easily obtained. So, the proof can be omitted.
Let and be the sequence that converges statistically to 1 butdoes not converge in ordinary sense, so for , it can be written as
Now, under the condition in (6.2), let us show the statistical convergence ofbivariate operator (6.1) with the help of the proof of Theorem 2. □
Theorem 7 Letandbe a sequence satisfying (6.2) for, and letbe a sequence of linear positive operators fromintogiven by (1.6). Then, for any functionand, where, , we have
Proof Using Lemma 4, the proof can be obtained similar to the proof ofTheorem 2. So, we shall omit this proof. □
7 Rates of convergence of bivariate operators
Let . Then the sup norm on is given by
We consider the modulus of continuity for bivariate case given by ,
It is clear that a necessary and sufficient condition for a function is
and satisfy the following condition:
for each . Then observe that any function in is continuous and bounded on K. Details ofthe modulus of continuity for bivariate case can be found in [21].
Now, the rate of statistical convergence of bivariate operator (6.1) by means ofmodulus of continuity in will be given in the following theorem.
Theorem 8 Letandbe a sequence satisfying the condition in (6.2). So, wehave
where
Proof By using the condition in (7.1), we get for and that
If the Cauchy-Schwarz inequality is applied, we have
So, if it is substituted in the above equation, the proof iscompleted. □
At last, the following theorem represents the rate of statistical convergence ofbivariate operator (6.1) by means of Lipschitz functions for the bivariate case, where and and , , then let us define as
We have the following theorem.
Theorem 9 Let the sequenceandsatisfy the conditions given in (6.2), and let, and, . Then
whereandare defined in (7.2), (7.3).
Proof Since are linear positive operators and, and , , we can write
If we take , , , , applying Hölder’s inequality, we obtain
So, the proof is completed. □
Authors’ information
Dr. VNM is an assistant professor at the Sardar Vallabhbhai National Institute ofTechnology, Ichchhanath Mahadev Road, Surat, Gujarat, India and he is a very activeresearcher in various fields of mathematics like Approximation theory, summabilitytheory, variational inequalities, fixed point theory and applications, operatoranalysis, nonlinear analysis etc. A Ph.D. in Mathematics, he is adouble gold medalist, ranking first in the order of merit in both B.Sc. and M.Sc.Examinations from the Dr. Ram Manohar Lohia Avadh University, Faizabad (UttarPradesh), India. Dr. VNM has undergone rigorous training from IIT, Roorkee, Mumbai,Kanpur; ISI Banglore in computer oriented mathematical methods and has experience ofteaching post graduate, graduate and engineering students. Dr. VNM has to his creditmany research publications in reputed journals including SCI/SCI(Exp.) accreditedjournals. Dr. VNM is referee of several international journals in the frame of pureand applied mathematics and Editor of reputed journals covering the subjectmathematics. The second author KK is a research scholar (R/S) in Applied Mathematicsand Humanities Department at the Sardar Vallabhbhai National Institute ofTechnology, Ichchhanath Mahadev Road, Surat (Gujarat), India under the guidance ofDr. VNM. Recently the third author LNM joined as a full-time research scholar at theDepartment of Mathematics, National Institute of Technology, Silchar-788010,District-Cachar, Assam, India and he is also very good active researcher inapproximation theory, summability analysis, integral equations, nonlinear analysis,optimization technique, fixed point theory and operator theory.
References
Gupta V, Agrawal PN, Verma DK: On discrete q -Beta operators. Ann. Univ. Ferrara 2011, 57: 39-66. 10.1007/s11565-011-0118-4
Khan, HH: Approximation of Classes of functions. Ph.D. thesis, AMU Aligarh(1974)
Fast H: Sur la convergence statistique. Colloq. Math. 1951, 2: 241-244.
Gadjiev AD, Orhan C: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 2002, 32: 129-138. 10.1216/rmjm/1030539612
Doğru O: On statistical approximation properties of Stancu type bivariategeneralization of q -Balázs-Szabados operators. Proc. Int. Conference on Numer. Anal. Approx. Th. Cluj-Napoca, Romania 2006, 179-194.
Duman O: A -statistical convergence of sequences of convolution operators. Taiwan. J. Math. 2008, 12(2):523-536.
Gupta V, Radu C: Statistical approximation properties of q -Baskokov-Kantorovichoperators. Cent. Eur. J. Math. 2009, 7(4):809-818. 10.2478/s11533-009-0055-y
Ersan S, Doğru O: Statistical approximation properties of q -Bleimann, Butzer and HahnOperators. Math. Comput. Model. 2009, 49: 1595-1606. 10.1016/j.mcm.2008.10.008
Doğru O, Örkcü M: Statistical approximation by a modification of q -Meyer-Königand Zeller operators. Appl. Math. Lett. 2010, 23(3):261-266. 10.1016/j.aml.2009.09.018
Örkcü M, Doğru O: Weighted statistical approximation by Kantorovich type q -Szász-Mirakjan operators. Appl. Math. Comput. 2011, 217(20):7913-7919. 10.1016/j.amc.2011.03.009
Doğru O, Kanat K: On Statistical approximation properties of the Kantorovich type Lupasoperators. Math. Comput. Model. 2012, 55: 1610-1621. 10.1016/j.mcm.2011.10.059
Gupta V: Some approximation properties of q -Durrmeyer operators. Appl. Math. Comput. 2008, 197(1):172-178. 10.1016/j.amc.2007.07.056
Kim T: New approach to q -Euler polynomials of higher order. Russ. J. Math. Phys. 2010, 17(2):218-225. 10.1134/S1061920810020068
Jackson FH: On a q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193-203.
Örkcü M, Doğru O: q -Szász-Mirakjan Kantorovich type operators preserving sometest. Appl. Math. Lett. 2011, 24: 1588-1593. 10.1016/j.aml.2011.04.001
Liu S: Rate of A -Statistical approximation of a modified Q Bernstein Operators. Bull. Math. Anal. Appl. 2012, 4(3):129-137.
Niven I, Zuckerman HS, Montgomery H: An Introduction to the Theory of Numbers. 5th edition. Wiley, New York; 1991.
Erkuş E, Duman O: A -Statistical extension of the Korovkin type approximationtheorem. Proc. Indian Acad. Sci. Math. Sci. 2003, 115(4):499-507.
Doğru O, Gupta V: Korovkin-type approximation properties of bivariate q -Meyer-König and Zeller operators. Calcolo 2006, 43: 51-63. 10.1007/s10092-006-0114-8
Ersan S: Approximation properties of bivariate generalization of Bleimann, Butzer andHahn operators based on the q -integers. Proc. of the 12th WSEAS Int. Conference on Applied Mathematics, Cairo, Egypt 2007, 122-127.
Anastassiou GA, Gal SG: Approximation Theory: Moduli of Continuity and Global SmoothnessPreservation. Birkhäuser, Boston; 2000.
Acknowledgements
This research work is supported by CPDA, SVNIT, Surat, India. The authors wouldlike to thank the anonymous learned referees for their valuable suggestionswhich improved the paper considerably. The authors are also thankful to all theeditorial board members and reviewers of prestigious journal Advances inDifference Equations. Special thanks are due to our great master and friendacademician Prof. Ravi P. Agarwal, Texas A and M University-Kingsville, TX, USA,for kind cooperation, smooth behavior during communication and for his effortsto send the reports of the manuscript timely. The authors are also grateful toall the editorial board members and reviewers of prestigious Science CitationIndex (SCI) journal i.e. Advances in Difference Equations (ADE). Thanksare also Prof. Christopher D. Rualizo, Journal Editorial Office of SpringerOpen. This research work is totally supported by CPDA, SVNIT, Surat (Gujarat),India.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
VNM, KK and LNM computed the moments of modified operators and established theasymptotic formula. VNM conceived of the study and participated in its design andcoordination. VNM, KK and LNM contributed equally and significantly in writing thispaper. All the authors drafted the manuscript, read and approved the finalmanuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Mishra, V.N., Khatri, K. & Mishra, L.N. Statistical approximation by Kantorovich-type discrete q-Betaoperators. Adv Differ Equ 2013, 345 (2013). https://doi.org/10.1186/1687-1847-2013-345
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2013-345
Keywords
- discrete q-Beta operators
- Kantorovich-type operators
- Korovkin-type approximation theorem
- rate of statistical convergence
- modulus of continuity
- Lipschitz-type functions