Journal of Calculus of Variations
Volume 2013, Issue 1 489249
Research Article
Open Access

Some Approximation Properties of Modified Jain-Beta Operators

Vishnu Narayan Mishra, 

Corresponding Author

Vishnu Narayan Mishra

Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchanath Mahadev Road, Surat, Gujarat 395 007, India svnit.ac.in

L. 1627 Awadh Puri Colony Beniganj, Phase-III, Opp. I.T.I., Ayodhya Main Road Faizabad, Uttar Pradesh 224 001, India

Search for more papers by this author
Prashantkumar Patel, 

Prashantkumar Patel

Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchanath Mahadev Road, Surat, Gujarat 395 007, India svnit.ac.in

Department of Mathematics, St. Xavier College, Ahmedabad, Gujarat 380 009, India sxca.edu.in

Search for more papers by this author
First published: 26 December 2013
https://doi.org/10.1155/2013/489249
Citations: 2
Academic Editor: Jacob Engwerda

Abstract

Generalization of Szász-Mirakyan operators has been considered by Jain, 1972. Using these generalized operators, we introduce new sequences of positive linear operators which are the integral modification of the Jain operators having weight functions of some Beta basis function. Approximation properties, the rate of convergence, weighted approximation theorem, and better approximation are investigated for these new operators. At the end, we generalize Jain-Beta operator with three parameters α, β, and γ and discuss Voronovskaja asymptotic formula.

1. Introduction

For 0 < ϑ < ∞,  |κ | < 1, let
()
then
()
Equation (1) is a Poisson-type distribution which has been considered by Consul and Jain [1].
In 1970, Jain [2] introduced and studied the following class of positive linear operators:
()
where 0 ≤ κ < 1 and wκ(i, nx) has been defined in (1).
The parameter κ may depend on the natural number n. It is easy to see that κ = 0; (3) reduces to the well-known Szász-Mirakyan operators [3]. Different generalization of Szász-Mirakyan operator and its approximation properties is studied in [4, 5]. Kantorovich-type extension of was given in [6]. Integral version of Jain operators using Beta basis function is introduced by Tarabie [7], which is as follows:
()
In Gupta et al. [8] they considered integral modification of the Szász-Mirakyan operators by considering the weight function of Beta basis functions. Recently, Dubey and Jain [9] considered a parameter γ in the definition of [8]. Motivated by such types of operators we introduce new sequence of linear operators as follows:
For x ∈ [0, ∞) and γ > 0,
()
where wκ(i, nx) is defined in (1) and
()
The operators defined by (5) are the integral modification of the Jain operators having weight function of some Beta basis function. As special case, γ = 1 the operators (5) reduced to the operators recently studied in [7]. Also, if κ = 0 and γ = 1, then the operators (5) turn into the operators studied in [8].

In the present paper, we introduce the operators (5) and estimate moments for these operators. Also, we study local approximation theorem, rate of convergence, weighted approximation theorem, and better approximation for the operators . At the end, we propose Stancu-type generalization of (5) and discuss some local approximation properties and asymptotic formula for Stancu-type generalization of Jain-Beta operators.

2. Basic Results

Lemma 1 (see [2].)For ,  m = 0,1, 2, one has

()

Lemma 2. The operators , n > γ defined by (5) satisfy the following relations:

()

Proof. By simple computation, we get

()
()
()

Lemma 3. For x ∈ [0, ∞), n > γ, and with φx = t − x, one has

  • (i)

    ,

  • (ii)

     +((1 + (1 − κ) 2)/((1 − κ) 3(n − γ)))x.

Lemma 4. For x ∈ [0, ∞), n > γ, one has

()

Proof. Since max {x, x2} ≤ x + x2, (1 − κ) 2 ≤ 1, and (1 − κ) −2 ≤ (1 − κ) −3, we have

()
which is required.

3. Some Local Approximation

Let ,  Mf being a constant depending on f}. By , we denote the subspace of all continuous functions belonging to . Also, is subspace of all the function for which lim x→∞(f(x)/(1 + x2)) is finite. The norm on is .

If we look at Lemma 2 and based on the famous Korovkin theorem [10], it is clear that does not form an approximation process. To do this approximation process, we replace constant κ by a number κn ∈ [0,1). If
()
then Lemma 2 gives
()
uniformly on any compact interval K ⊂ [0, ∞). Based on Korovkin’s criteria we state the following.

Theorem 5. Let with n > γ be defined in (5), where lim n→∞κn = 0. For any compact K ⊂ [0, ∞) and for each one has

()

Now, we establish a direct local approximation theorem for the modified operators in ordinary approximation. Let the space CB[0, ∞) of all continuous and bounded functions be endowed with the norm ∥f∥ = sup {|f(x)| : x ∈ [0, ∞)}. Further let us consider the following K-functional:
()
where δ > 0 and W2 = {g ∈ CB[0, ∞) : g′, g′′ ∈ CB[0, ∞)}. By the methods given in [11], there exists an absolute constant C > 0 such that
()
where
()
is the second order modulus of smoothness of f ∈ CB[0, ∞).

Theorem 6. For f ∈ CB[0, ∞) and n > γ, one has

()
where C is a positive constant.

Proof. We introduce the auxiliary operators as follows:

()
Let and x, t ∈ [0, ∞). By Taylor’s expansion we have
()
Applying , we get
()
Applying Lemma 2, we get
()
Since
()
()
Taking infimum overall g ∈ W2, we get
()
In view of (18)
()
which proves the theorem.

4. Rate of Convergence and Weighted Approximation

For any positive a, by
()
we denote the usual modulus of continuity of f on the closed interval [0, a]. We know that, for a function , the modulus of the continuity ωa(f, δ) tends to zero.

Now we give a rate of convergence theorem for the operator .

Theorem 7. Let and ωa+1(f, δ) be its modulus of continuity on the finite interval [0, a + 1]⊂[0, ∞), where a > 0. Then, for n > γ,

()

Proof. For x ∈ [0, a] and t > a + 1, since t − x > 1, we have

()
For x ∈ [0, a] and t ≤ a + 1, we have
()
with δ > 0.

From (31) and (32) we can write

()
for x ∈ [0, a] and t ≥ 0. Thus,
()
Hence, by Schwarz’s inequality and Lemma 3, for x ∈ [0, a],
()
By taking , we get
()
which proves the theorem.

Now we will discuss the weighted approximation theorem, where the approximation formula holds true on the interval [0, ∞).

Theorem 8. If , lim n→∞κn = 0, and n > γ, then,

()

Proof. Using the theorem in [12] we see that it is sufficient to verify the following three conditions:

()
Since , the first condition of (38) is fulfilled for r = 0. By Lemma 2 we have
()
and the second condition of (38) holds for r = 1 as n → ∞ with κn → 0.

Similarly, we can write, for n > γ,

()
which implies that
()
Thus, the proof is completed.

5. Better Error Approximation

In this section, we modified operator (5), in such way that the linear functions are preserved. The technique, which replaced x by appropriate function, was studied for many operators, for example, Bernstein, Szász, Szász-Beta operators, and so on [13–20].

We start by defining
()
We note that rκ(x)∈[0, ∞), for any 0 ≤ κ < 1. By replacing x by rκ(x) we give the following modification of the operators :
()
where
()
and x ∈ [0, ∞), n > γ; the term bn,i,γ(t) is given in (5).

Lemma 9. For x ∈ [0, ∞) and n > γ, one has

  • (i)

    ,

  • (ii)

    ,

  • (iii)

    .

Lemma 10. For x ∈ [0, ∞), n > γ, and with φx = t − x, one has

  • (i)

    ,

  • (ii)

    .

Lemma 11. For x ∈ [0, ∞), n > γ, one has

()

Proof. Since max {x, x2} ≤ x + x2 and (1 − κ) 2 ≤ 1, we have

()
which is required.

Theorem 12. Let f ∈ CB[0, ∞). Then for x ∈ [0, ∞) and n > γ, one has

()

Proof. Let and x ∈ [0, ∞). Using Taylor’s expansion

()
and Lemma 10, we have
()
Also, . Thus,
()
Since ,
()
Finally taking the infimum on right side over all , we get
()
In view of (18), we obtain
()
which proves the theorem.

Remark 13. We claim that the error estimation in Theorem 12 is better than that of (20), provided f ∈ C[0, ∞) and x > 0. Indeed, in order to get this better estimation we must show that τκ,n,γ(x) ≤ δκ,n,γ(x) + κx/(1 − κ). One can obtain that

()
Also,
()
which holds true, with n > γ > 0 and κ > 0. Thus, τκ,n,γ(x) ≤ δκ,n,γ(x) + κx/(1 − κ).

6. Stancu Approach

In 1968, Stancu introduced Bernstein-Stancu operator, which is a linear positive operator with two parameters α and β satisfying the condition 0 ≤ α ≤ β. Inspired by the Stancu-type generalization of Bernstein operator and the recent important work on several other operators are discuss in [21–27], we propose following modification of the operator as
()
where wκ(i, nx) and bn,i,γ(t) are defined in (5).

Lemma 14. For ,  s = 0,1, 2, the following inequalities holds:

  • (i)

    ,

  • (ii)

    ,

  • (iii)

      =  n3x2/((n + β) 2(n − γ)(1 − κ) 2) + (nx(2n + 2 (−n + 2α(n − γ))κ +(n − 4α(n − γ))κ2 +2α(n − γ)κ3)) / ((n + β) 2(n − γ)(1 − κ) 3) + α2/(n + β) 2.

The proof of the above lemma can be obtained by using linearity of operators and Lemma 2.

Lemma 15. If one denotes central moments by ,  m = 1,2, then one has

()

Theorem 16. Let with n > γ be defined in (56), where lim n→∞κn = 0. For any compact A ⊂ [0, ∞) and for each , one has

()

The proof is based on Korovkin’s criterion and Lemma 14.

Theorem 17. Let f ∈ CB[0, ∞) and n > γ, one has

()
for every 0 ≤ α ≤ β and x ∈ [0, ∞), where B is a positive constant.

The proof of Theorem 17 is just similar to Theorem 6.

Now, we establish the Voronovskaja-type asymptotic formula for the operators .

Theorem 18. Let f be bounded and integrable on [0, ∞), first and second derivatives of f exist at a fixed point x ∈ [0, ∞), and κn ∈ (0,1) such that κn → 0 as n → ∞; then

()

Proof. Let f, f′, and x ∈ [0, ∞) be fixed. By Taylor’s expansion we can write

()
where r(t, x) is Peano form of the remainder, , and lim t→xr(t, x) = 0.

Applying to the previous, we obtain

()
By Cauchy-Schwarz’s inequality, we have
()
Observe that r2(x, x) = 0 and . Then it follows that
()
uniformly with respect to x ∈ [0, A].

Now, from (63) and (64), we obtain .

Using κn → 0 as n → ∞, we obtain

()
Using above limits, we have
()
which proves the theorem.

Remark 19. In particular, if α = β = 0 and γ = 1, then the operators , κn ∈ (0,1) such that κn → 0 as n → ∞, reduce to the Jain-Beta operators recently introduced by Tarabie [7]. We obtain the following conclusion of the above asymptotic formula for the Jain-Beta operator in the ordinary approximation as follows:

()

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

    Acknowledgments

    The authors would like to express their deep gratitude to the anonymous learned referees for their valuable suggestions and comments. The second author is thankful to the Department of Mathematics, St. Xavier College, Ahmedabad, Gujarat, for carrying out his research work under the supervision of Dr. Vishnu Narayan Mishra at SVNIT, Surat, Gujarat, India. Special thanks are due to Professor Jacob C. Engwerda for kind cooperation and smooth behavior during communication and for the efforts to send the reports of the paper timely.

        The full text of this article hosted at iucr.org is unavailable due to technical difficulties.