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q-analogue of a new sequence of linear positive operators

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9 pages
This paper deals with Durrmeyer type generalization of q -Baskakov type operators using the concept of q -integral, which introduces a new sequence of positive q -integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous functions defined on the interval [ 0 , ∞ ) . An estimate for the rate of convergence and weighted approximation properties are also obtained. MSC: 41A25, 41A36.
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Gupta et al.Journal of Inequalities and Applications2012,2012:144 http://www.journalofinequalitiesandapplications.com/content/2012/1/144
R E S E A R C H
q-analogue of a new sequence of linear positive operators 1 2* 3 Vijay Gupta , Taekyun Kim and Sang-Hun Lee
* Correspondence: tkkim@kw.ac.kr 2 Department of Mathematics, Kwangwoon University, Seoul 139-701, S. Korea Full list of author information is available at the end of the article
Open Access
Abstract This paper deals with Durrmeyer type generalization ofq-Baskakov type operators using the concept ofq-integral, which introduces a new sequence of positive q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous functions defined on the interval [0,). An estimate for the rate of convergence and weighted approximation properties are also obtained. MSC:Primary 41A25; 41A36 Keywords:Durrmeyer type operators; weighted approximation; rate of convergence;q-integral
1 Introduction In the year  Agrawal and Mohammad [] introduced a new sequence of linear positive operators by modifying the well-known Baskakov operators having weight functions of Szasz basis function as
Dn(f,x) =n pn,k(x)sn,k–(t)f(t)dt+pn,(x)f(), k=
where   k n+k– x pn,k(x) = , n+k k( +x)
k (nt) nt sn,k(t) =e. k!
x[,),
(.)
It is observed in [] that these operators reproduce constant as well as linear functions. Later, some direct approximation results for the iterative combinations of these operators were studied in []. A lot of works onq-calculus are available in literature of different branches of mathe-matics and physics. For systematic study, we refer to the work of Ernst [], Kim [, ], and Kim and Rim []. The application ofq-calculus in approximation theory was initiated by Phillips [], who was the first to introduceq-Bernstein polynomials and study their approximation properties. Very recently theq-analogues of the Baskakov operators and their Kantorovich and Durrmeyer variants have been studied in [, ] and [] respectively. We recall some notations and concepts ofq-calculus. All of the results can be found in [] and []. In what follows,qis a real number satisfying  <q< .
©2012 Gupta et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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