Abstract
In this article, we introduce the qvariant of Beta operator. We find the recurrence formula for mthorder 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 Stancutype generalization.
Mathematical Subject Classification: 41A25; 41A35.
Keywords:
qintegers; qbeta function; qBeta operators; modulus of continuity; asymptotic formula1. Introduction
In the last four decades after the integral modification of Bernstein polynomials by Durrmeyer, several new Durrmeyertype operators were introduced and their approximation properties were discussed. In 2007, Gupta et al. [1]proposed a family of linear positive operators as
where
for f ∈ C_{γ }[0, ∞), where C_{γ }[0, ∞), γ > 0 be the class of all continuous functions defined on [0, ∞) satisfying the growth condition f (t) ≤ Ct^{γ}, 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, qcalculus was used in approximation theory and several new operators were introduced and their approximation properties were discussed (see [26], etc.). Motivated by these operators, we now introduce the qanalogue of (1.1). For f ∈ C[0,∞) and 0 < q < 1, we propose the qBeta operators as
where
and B_{q}(t, s) denote the qBeta 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 selfcontent, here we mention certain basic definitions of qcalculus, details can be found in [8,9] and the other recent articles. For each nonnegative integer k, the qinteger [k]_{q }and the qfactorial [k]_{q}! are, respectively, defined by
and
For the integers n, k satisfying n ≥ k ≥ 0, the qbinomial coecients are defined by
The qderivative of a function is defined by
and the qimproper integral (see [10]) is given by
The qintegral by parts is given by
Lemma 1. For n, k ≥ 0, we have
Proof. Using qderivative operator, we can write
Equation (2.2) can be obtained directly by using qquotient rule as follows:
□
Remark 1. By using (2.1) and D_{q}x^{k−1}= [k − 1]_{q}x^{k−2}, we get
Hence, we obtain
Lemma 2. We have following equalities
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,
Proof. By using (2.4), we have
by (2.5) and qintegration by parts, we get
by combining above two equations, we can write
Hence, the result follows. □
Corollary 2. We have
Corollary 3. If we denote central moments by , m = 1,2, then we have
Remark 2. As a special case when q → 1^{}, we have
The first two central moments for q → 1^{− }are
which are the moments obtained by Gupta et al. [1] for Beta operator.
3. Ordinary approximation
Let C_{B }[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 Kfunctional:
where δ > 0 and
From [11], there exist an absolute constant C > 0 such that
where
is the secondorder modulus of smoothness of f ∈ C_{B }[0, ∞). By
we denote the usual modulus of continuity of f ∈ C_{B }[0, ∞).
Theorem 4. Let 0 < q < 1, we have
for every x ∈ [0, ∞) and f ∈ C_{B }[0, ∞), where C is a positive constant.
Proof. We consider modified operators defined by
x ∈ [0, ∞). The operators preserve the linear functions:
Let g ∈ W^{2 }and t ∈ [0, ∞). Using Taylor's expansion, we have
and (3.3), we have
Therefore, from (3.2), we have
From Corollary 3, we get
By (3.2) and Corollary 1, we have
Now using (3.2), (3.5), and (3.6), we obtain
Thus taking infimum on the righthand side over all g ∈ W^{2}, 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 M_{f }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 = q_{n }∈ (0, 1) such that q_{n }→ 1 as n → ∞, then for each , we have
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 → ∞.
By Corollary 2, for n > 1,
as n → ∞, we get
Similarly for n > 1, we have
as n → ∞, we get
By (4.2), (4.3), (4.4), and Korovkin's theorem, we get the desired result. □
Theorem 6. Let q = q_{n }∈ (0, 1) such that q_{n }→ 1 as n → ∞, then for each and α> 0, we have
Proof. For any fixed x_{0 }> 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 x_{0 }> 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 (q_{n}) denote a sequence such that 0 < q_{n }< 1 and q_{n }→ 1 as n → ∞. Then we have for a point x ∈ (0, ∞)
Proof. By qTaylor's formula [7] on f, we have
for 0 < q < 1, where
We know that for n large enough
That is for any ε > 0, A > 0, there exists a δ > 0 such that
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 χ_{x }(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 (q_{n}) denote a sequence such that 0 < q_{n }< 1 and q_{n }→ 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 e_{0 }and e_{1}. As the qBeta 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 {u_{n,q}(x)} of real valued continues functions on [0, ∞) with 0 ≤ u_{n,q}(x) < ∞, as
We replace x in definition of operator (1.2) with u_{n,q}(x). Therefore, modified operator is given as
Remark 3. By simple computation we can write
Therefore,
Theorem 9. Let f ∈ C_{B}(I), then for every x ∈ I and for n > 1, C > 0, we have
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 = q_{n }satisfies 0 < q_{n }< 1 and let q_{n }→ 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
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 BernsteinStancu operators in [15] a linear positive operator depending on two nonnegative 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 qBeta Stancu operators as
where 0 ≤ α ≤ β.
In each of following theorems, we assume that q = q_{n}, where q_{n }is a sequence of real numbers such that 0 < q_{n }< 1 for all n and lim_{n→∞ }q_{n }= 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 ∈ C_{B }[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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