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q-analogue of a new sequence of linear positive operators
Journal of Inequalities and Applications volume 2012, Article number: 144 (2012)
Abstract
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 . An estimate for the rate of convergence and weighted approximation properties are also obtained.
MSC:41A25, 41A36.
1 Introduction
In the year 2003 Agrawal and Mohammad [1] introduced a new sequence of linear positive operators by modifying the well-known Baskakov operators having weight functions of Szasz basis function as
where
It is observed in [1] 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 [14].
A lot of works on q-calculus are available in literature of different branches of mathematics and physics. For systematic study, we refer to the work of Ernst [5], Kim [10, 11], and Kim and Rim [9]. The application of q-calculus in approximation theory was initiated by Phillips [13], who was the first to introduce q-Bernstein polynomials and study their approximation properties. Very recently the q-analogues of the Baskakov operators and their Kantorovich and Durrmeyer variants have been studied in [2, 3] and [7] respectively. We recall some notations and concepts of q-calculus. All of the results can be found in [5] and [8]. In what follows, q is a real number satisfying .
For ,
The q-binomial coefficients are given by
The q-Beta integral is defined by [12]
which satisfies the following functional equation:
For and each positive integer n, the q-Baskakov operators [2] are defined as
where
Remark 1 The first three moments of the q-Baskakov operators are given by
As the operators have mixed basis functions in summation and integration and have an interesting property of reproducing linear functions, we were motivated to study these operators further. Here we define the q-analogue of the operators as
where and
In case , the above operators reduce to the operators (1.1). In the present paper, we estimate a local approximation theorem and the rate of convergence of these new operators as well as their weighted approximation properties.
2 Moment estimation
Lemma 1 The following equalities hold:
-
(i)
,
-
(ii)
,
-
(iii)
.
Proof The operators are well defined on the function . Then for every , we obtain
Substituting and using (1.2), we have
where is the q-Baskakov operator defined by (1.3).
Next, we have
Again substituting and using (1.2), we have
Finally,
Again substituting , using (1.2) and , we have
 □
Remark 2 If we put , we get the moments of a new sequence considered in [1] as operators as
Lemma 2 Let, then forwe have
3 Direct theorems
By we denote the space of real valued continuous bounded functions f on the interval ; the norm- on the space is given by
The Peetre’s K-functional is defined by
where . By [4], pp.177], there exists a positive constant such that and the second order modulus of smoothness is given by
Also, for a usual modulus of continuity is given by
Theorem 1 Letand. Then for alland, there exists an absolute constantsuch that
Proof Let and . By Taylor’s expansion, we have
Applying Lemma 2, we obtain
Obviously, we have . Therefore,
Using Lemma 1, we have
Thus
Finally, taking the infimum over all and using the inequality , , we get the required result. This completes the proof of Theorem 1. □
We consider the following class of functions:
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 functions belonging to . Also, let be the subspace of all functions , for which is finite. The norm on is . We denote the modulus of continuity of f on closed interval , as by
We observe that for function , the modulus of continuity tends to zero.
Theorem 2 Let, andbe its modulus of continuity on the finite interval, where. Then for every,
Proof For and , since , we have
For and , we have
with .
From (3.1) and (3.2) we can write
for and . Thus
Hence, by using Schwarz inequality and Lemma 2, for every and
By taking we get the assertion of our theorem. □
4 Higher order moments and an asymptotic formula
Lemma 3 ([6])
Let, we have
Now, we present higher order moments for the operators (1.4).
Lemma 4 Let, we have
The proof of Lemma 4 can be obtained by using Lemma 3.
We consider the following classes of functions:
Theorem 3 Let, then the sequenceconverges to f uniformly onfor eachif and only if.
Theorem 4 Assume that, andas. For anysuch thatthe following equality holds
uniformly on any, .
Proof Let and be fixed. By using Taylor’s formula, we may write
where is the Peano form of the remainder, and . Applying to (4.1), we obtain
By the Cauchy-Schwarz inequality, we have
Observe that and . Then it follows from Theorem 3 and Lemma 4, that
uniformly with respect to . Now from (4.2), (4.3) and Remark 2, we get immediately
Then, we get the following
 □
References
Agrawal PN, Mohammad AJ: Linear combination of a new sequence of linear positive operators. Rev. Unión Mat. Argent. 2003, 44(1):33–41.
Aral A, Gupta V: Generalized q -Baskakov operators. Math. Slovaca 2011, 61(4):619–634. 10.2478/s12175-011-0032-3
Aral A, Gupta V: On the Durrmeyer type modification of the q -Baskakov type operators. Nonlinear Anal. 2010, 72(3–4):1171–1180. 10.1016/j.na.2009.07.052
DeVore RA, Lorentz GG: Constructive Approximation. Springer, Berlin; 1993.
Ernst, T: The history of q-calculus and a new method. U.U.D.M Report 2000, 16, ISSN 1101–3591, Department of Mathematics, Upsala University (2000)
Finta Z, Gupta V: Approximation properties of q -Baskakov operators. Cent. Eur. J. Math. 2010, 8(1):199–211. 10.2478/s11533-009-0061-0
Gupta V, Radu C: Statistical approximation properties of q -Baskakov-Kantorovich operators. Cent. Eur. J. Math. 2009, 7(4):809–818. 10.2478/s11533-009-0055-y
Kac VG, Cheung P: Quantum Calculus. Springer, New York; 2002.
Kim T, Rim S-H: A note on the q -integral and q -series. Adv. Stud. Contemp. Math. (Pusan) 2000, 2: 37–45.
Kim T: q -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys. 2008, 15: 51–57.
Kim T: q -Bernoulli numbers associated with q -Stirling numbers. Adv. Differ. Equ. 2008., 2008:
Koornwinder TH: q -special functions, a tutorial. In Deformation Theory and Quantum Groups with Applications to Mathematical Physics. Edited by: Gerstenhaber M, Stasheff J. Am. Math. Soc., Providence; 1992.
Phillips GM: Bernstein polynomials based on the q -integers. Ann. Numer. Math. 1997, 4: 511–518.
Wang X: The iterative approximation of a new sequence of linear positive operators. J. Jishou Univ. Nat. Sci. Ed. 2005, 26(2):72–78.
Acknowledgements
The work was done while the first author visited Division of General Education-mathematics, Kwangwoon University, Seoul, South Korea for collaborative research during June 15-25, 2010.
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Gupta, V., Kim, T. & Lee, SH. q-analogue of a new sequence of linear positive operators. J Inequal Appl 2012, 144 (2012). https://doi.org/10.1186/1029-242X-2012-144
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DOI: https://doi.org/10.1186/1029-242X-2012-144