q-analogue of a new sequence of linear positive operators

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.

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.

Lemma 1The 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 2Let, then forwe have

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 1Letand. 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 2Let, 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. □

Lemma 3 ([6])

Let, we have

Now, we present higher order moments for the operators (1.4).

Lemma 4Let, we have

The proof of Lemma 4 can be obtained by using Lemma 3.

We consider the following classes of functions:

Theorem 3Let, then the sequenceconverges tofuniformly onfor eachif and only if.

Theorem 4Assume 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

 □

The authors declare that they have no competing interests.

All authors contributed in preparing the manuscript equally.

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.

1. Agrawal, PN, Mohammad, AJ: Linear combination of a new sequence of linear positive operators. Rev. Unión Mat. Argent.. 44(1), 33–41 (2003). PubMed Abstract | Publisher Full Text

2. Aral, A, Gupta, V: Generalized q-Baskakov operators. Math. Slovaca. 61(4), 619–634 (2011). Publisher Full Text

3. Aral, A, Gupta, V: On the Durrmeyer type modification of the q-Baskakov type operators. Nonlinear Anal.. 72(3-4), 1171–1180 (2010). Publisher Full Text

4. DeVore, RA, Lorentz, GG: Constructive Approximation, Springer, Berlin (1993)

5. 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)

6. Finta, Z, Gupta, V: Approximation properties of q-Baskakov operators. Cent. Eur. J. Math.. 8(1), 199–211 (2010). Publisher Full Text

7. Gupta, V, Radu, C: Statistical approximation properties of q-Baskakov-Kantorovich operators. Cent. Eur. J. Math.. 7(4), 809–818 (2009). Publisher Full Text

8. Kac, VG, Cheung, P: Quantum Calculus, Springer, New York (2002)

9. Kim, T, Rim, S-H: A note on the q-integral and q-series. Adv. Stud. Contemp. Math. (Pusan). 2, 37–45 (2000)

10. Kim, T: q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys.. 15, 51–57 (2008)

11. Kim, T: q-Bernoulli numbers associated with q-Stirling numbers. Adv. Differ. Equ.. 2008, (2008)

12. Koornwinder, TH: q-special functions, a tutorial. In: Gerstenhaber M, Stasheff J (eds.) Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Am. Math. Soc., Providence (1992)

13. Phillips, GM: Bernstein polynomials based on the q-integers. Ann. Numer. Math.. 4, 511–518 (1997)

14. Wang, X: The iterative approximation of a new sequence of linear positive operators. J. Jishou Univ. Nat. Sci. Ed.. 26(2), 72–78 (2005)