Statistical approximation by Kantorovich-type discrete q-Betaoperators
© Mishra et al.; licensee Springer. 2013
Received: 2 July 2013
Accepted: 17 October 2013
Published: 29 November 2013
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.
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  studied approximation of functions in various classes using differenttypes of operators.
On the other hand, statistical convergence was first introduced by Fast , and it has become an area of active research. Also, statisticalconvergence was introduced by Gadjiev and Orhan , Doğru , Duman , Gupta and Radu , Ersan and Doğru  and Doğru and Örkcü .
In 2011, Örkcü and Doğru obtained weighted statistical approximationproperties of Kantorovich-type q-Szász-Mirakjan operators .
In , Doğru and Kanat proved the following statistical Korovkin-typeapproximation theorem for operators (1.2).
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
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.
and it is denoted by .
In this part, we will use the notation instead of for abbreviation.
wheredenotes the space of all real bounded functions f which are continuous in.
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 .
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.
Taking , a sequence satisfying (1.3), and using and then choosing as in (5.2), the theorem is proved. □
This gives us the pointwise rate of statistical convergence of the operators to .
We have the following theorem.
whereis given as in (5.2).
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.
In , 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  introduced a bivariate generalization of the q-MKZ operators andinvestigated its Korovkin-type approximation properties.
Recently, Ersan and Doğru  obtained the statistical Korovkin-type theorem and lemma for thebivariate linear positive operators defined in the space as follows.
= + + + .
In order to obtain the statistical convergence of operator (6.1), we need thefollowing lemma.
= + + + + + .
Proof By the help of the proofs for the bivariate operator in , the conditions may be easily obtained. So, the proof can be omitted.
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. □
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
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 .
Now, the rate of statistical convergence of bivariate operator (6.1) by means ofmodulus of continuity in will be given in the following theorem.
So, if it is substituted in the above equation, the proof iscompleted. □
We have the following theorem.
whereandare defined in (7.2), (7.3).
So, the proof is completed. □
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.
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.
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