# Properties of q-analogue of Beta operator

Vijay Gupta1, Honey Sharma2, Taekyun Kim3* and Sang-Hun Lee4

Author Affiliations

1 School of Applied Science, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi-110078, India

2 Department of Mathematics, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar-144011 (Punjab), India

3 Department of Mathematics, Kwangwoon University, Seoul 139-701, South Korea

4 Division of General Education, Kwangwoon University, Seoul 139-701, South Korea

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Advances in Difference Equations 2012, 2012:86  doi:10.1186/1687-1847-2012-86

 Received: 20 April 2012 Accepted: 25 June 2012 Published: 25 June 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.

### Abstract

In this article, we introduce the q-variant of Beta operator. We find the recurrence formula for mth-order moments. Here, we establish some direct theorems in terms of modulus of continuity for these operators. We also propose conditions for better approximation. In the end, we also propose the Stancu-type generalization.

Mathematical Subject Classification: 41A25; 41A35.

##### Keywords:
q-integers; q-beta function; q-Beta operators; modulus of continuity; asymptotic formula

### 1. Introduction

In the last four decades after the integral modification of Bernstein polynomials by Durrmeyer, several new Durrmeyer-type operators were introduced and their approximation properties were discussed. In 2007, Gupta et al. [1]proposed a family of linear positive operators as

(1.1)

where

for f Cγ [0, ), where Cγ [0, ), γ > 0 be the class of all continuous functions defined on [0, ) satisfying the growth condition |f (t)| Ctγ, C > 0 and B(k, n + 1) is beta function. They [1] established the direct and inverse results for these operators.

In the recent years, q-calculus was used in approximation theory and several new operators were introduced and their approximation properties were discussed (see [2-6], etc.). Motivated by these operators, we now introduce the q-analogue of (1.1). For f C[0,∞) and 0 < q < 1, we propose the q-Beta operators as

(1.2)

where

and Bq(t, s) denote the q-Beta function [7] is given by

where . In particular, for any positive integer n, , K(x,0) = 1, and , n ∈ ℕ, a, b ∈ ℝ.

This article is the extension of the earlier work of [1]. Here, we consider the q variant of the operators discussed in [1] and obtain the recurrence relations for moments. We also obtain some direct results for the q operators, which also include the asymptotic formula. In the end, we establish the conditions for better approximation.

### 2. Preliminaries

To make the article self-content, here we mention certain basic definitions of q-calculus, details can be found in [8,9] and the other recent articles. For each nonnegative integer k, the q-integer [k]q and the q-factorial [k]q! are, respectively, defined by

and

For the integers n, k satisfying n k ≥ 0, the q-binomial coecients are defined by

The q-derivative of a function is defined by

and the q-improper integral (see [10]) is given by

The q-integral by parts is given by

Lemma 1. For n, k ≥ 0, we have

(2.1)

(2.2)

Proof. Using q-derivative operator, we can write

Equation (2.2) can be obtained directly by using q-quotient rule as follows:

Remark 1. By using (2.1) and Dqxk−1= [k − 1]qxk−2, we get

Hence, we obtain

(2.3)

Lemma 2. We have following equalities

(2.4)

(2.5)

Proof. Above equalities can be obtained by direct computations using definition of operator and (2.3). □

Theorem 1. If mth (m > 0, m ∈ ℕ)-order moment of operator (1.2) is defined as

then and for n > m, we have following recurrence relation,

(2.6)

Proof. By using (2.4), we have

by (2.5) and q-integration by parts, we get

by combining above two equations, we can write

Hence, the result follows. □

Corollary 2. We have

(2.7)

(2.8)

Corollary 3. If we denote central moments by , m = 1,2, then we have

(2.9)

(2.10)

Remark 2. As a special case when q → 1-, we have

The first two central moments for q → 1are

which are the moments obtained by Gupta et al. [1] for Beta operator.

### 3. Ordinary approximation

Let CB [0, ) be the space of all real valued continuous bounded function f on [0, ) endowed with the norm || f || = sup{|f(x)| : x ∈ [0, )}. Further let us consider the following K-functional:

where δ > 0 and

From [11], there exist an absolute constant C > 0 such that

(3.1)

where

is the second-order modulus of smoothness of f CB [0, ). By

we denote the usual modulus of continuity of f CB [0, ).

Theorem 4. Let 0 < q < 1, we have

for every x ∈ [0, ) and f CB [0, ), where C is a positive constant.

Proof. We consider modified operators defined by

(3.2)

x ∈ [0, ). The operators preserve the linear functions:

(3.3)

Let g W2 and t ∈ [0, ). Using Taylor's expansion, we have

and (3.3), we have

Therefore, from (3.2), we have

(3.4)

From Corollary 3, we get

(3.5)

By (3.2) and Corollary 1, we have

(3.6)

Now using (3.2), (3.5), and (3.6), we obtain

Thus taking infimum on the right-hand side over all g W2, we get

In the view of (3.1), we get

This completes the proof of the theorem. □

### 4. Weighted approximation

Here, we give weighted approximation theorem for the operator . Similar type of results are given in [3].

Let be the set of all functions f defined on the interval [0, ) satisfying the condition

where Mf is a constant depending on f. is a normed space with the norm

denotes the subspace of all continuous functions in and denotes the subspace of all functions with .

Theorem 5. Let q = qn ∈ (0, 1) such that qn 1 as n → ∞, then for each , we have

(4.1)

Proof. By the Korovkin's theorem (see [12]), converges to f uniformly as n → ∞ for if it satisfies for i = 0, 1, 2 uniformly as n → ∞.

As, ,

(4.2)

By Corollary 2, for n > 1,

as n → ∞, we get

(4.3)

Similarly for n > 1, we have

as n → ∞, we get

(4.4)

By (4.2), (4.3), (4.4), and Korovkin's theorem, we get the desired result. □

Theorem 6. Let q = qn ∈ (0, 1) such that qn 1 as n → ∞, then for each and α> 0, we have

(4.5)

Proof. For any fixed x0 > 0, we have

By Theorem 5 and Corollary 2 first two terms of above inequality tends to 0 as n → ∞. Last term of inequality can be made small enough for large x0 > 0. This completes the proof.

### 5. Central moments and asymptotic formula

In this section, we observe that it is not possible to estimate recurrence formula in q calculus, there may be some techniques, but at the moment it can be considered as an open problem. Here we establish the recurrence relation for the central moments and obtain asymptotic formula.

Lemma 3. If we de ne the central moments as

then

and for n > m, we have the following recurrence relation:

proof. Using the identity

and q derivatives of product rule, we have

Thus

Using the identities

and

we obtain the following identity after simple computation

Using the above identity and q integral by parts

we have

Finally, using

and

we get

This completes the proof of recurrence relation.

Theorem 7. Let f C [0, ) be a bounded function and (qn) denote a sequence such that 0 < qn < 1 and qn 1 as n → ∞. Then we have for a point x ∈ (0, )

Proof. By q-Taylor's formula [7] on f, we have

for 0 < q < 1, where

(5.1)

We know that for n large enough

(5.2)

That is for any ε > 0, A > 0, there exists a δ > 0 such that

(5.3)

for |t − x| < δ and n sufficiently large. Using (5.1), we can write

where

We can easily see that

In order to complete the proof of the theorem, it is suffficient to show that . We proceed as follows:

Let

and

so that

where χx (t) is the characteristic function of the interval {t : |t − x| < δ}.

It follows from (5.1)

if |t − x| δ, then , where M > 0 is a constant. Since

we have

and

Using Lemma 3, we have

Thus, for n suciently large . This completes the proof of theorem. □

Corollary 8. Let f C [0, ) be a bounded function and (qn) denote a sequence such that 0 < qn < 1 and qn 1 as n → ∞. Suppose that the first and second derivative f '(x)

and f '' (x) exist at a point x ∈ (0, ), we have

### 6. Better error approximation

King [13] in 2003 proposed a new approach to modify the Bernstein polynomials to improve rate of convergence, by making operator to preserve test functions e0 and e1. As the q-Beta operators reproduce only constant functions, this motivated us to propose the modification of (1.2), so that they reproduce constant as well as linear functions.

Define sequence {un,q(x)} of real valued continues functions on [0, ) with 0 ≤ un,q(x) < ∞, as

We replace x in definition of operator (1.2) with un,q(x). Therefore, modified operator is given as

(6.1)

Remark 3. By simple computation we can write

Therefore,

Theorem 9. Let f CB(I), then for every x I and for n > 1, C > 0, we have

(6.2)

where .

Proof. Let , by Taylor's series

therefore, by linearity and Remark 3, we get

Also

Therefore,

on choosing , taking infimum over , we get the desired result.

Theorem 10. Let q = qn satisfies 0 < qn < 1 and let qn 1 as n → ∞. For each we have

Proof. To prove theorem (Using the Theorem in [14]), it is sufficient to verify the following three conditions

(6.3)

By Remark 3, we have and , the first and second condition of (6.3) is fulfilled for ν = 0 and ν = 1.

And

which implies that as n → ∞

Thus, the proof is completed. □

### 7. Stancu approach

In 1968, Stancu introduced Bernstein-Stancu operators in [15] a linear positive operator depending on two non-negative parameters α and β satisfying the condition 0 ≤ α β. Recently, many researcher applied this approach to many operators, for detail see [16,17,2], etc.

For f C[0, ) and 0 < q < 1, we define the q-Beta Stancu operators as

(7.1)

where 0 ≤ α β.

In each of following theorems, we assume that q = qn, where qn is a sequence of real numbers such that 0 < qn < 1 for all n and limn→∞ qn = 1.

Theorem 11. For , s = 0, 1, 2 the following identities hold:

Proof of the theorem can be obtained directly by using linearity of operator and Corollary 2.

Corollary 12. If we denote central moments by , m = 1, 2, then we have

Theorem 13. Let 0 < q < 1, we have

for every α, β ≥ 0, x ∈ [0, ) and f CB [0, ), where C is a positive constant.

Proof of theorem is just similar to Theorem 4.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

### Acknowledgements

The authors would like to express their deep gratitude to the referees for their valuable suggestions and comments.

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