2 3 Numerical Algorithms

1 23

Numerical Algorithms ISSN 1017-1398 Numer AlgorDOI 10.1007/s11075-013-9821-9

The Durrmeyer type modification ofthe q-Baskakov type operators with twoparameter α and β

Vishnu Narayan Mishra &Prashantkumar Patel

1 23

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Numer AlgorDOI 10.1007/s11075-013-9821-9

ORIGINAL PAPER

The Durrmeyer type modification of the q-Baskakovtype operators with two parameter α and β

Vishnu Narayan Mishra ·Prashantkumar Patel

Received: 7 February 2013 / Accepted: 18 December 2013© Springer Science+Business Media New York 2014

Abstract In this paper, we are dealing with q analogue of Durrmeyer type modi-fied the Baskakov operators with two parameter α and β, which introduces a newsequence of positive linear q-integral operators. We show that this sequence is anapproximation process in the polynomial weighted space of continuous functiondefined on the interval [0,∞). We study moments, weighted approximation proper-ties, the rate of convergence using a weighted modulus of smoothness, asymptoticformula and better error estimation for these operators.

Keywords Durrmeyer type operators · Weighted approximation ·Rate of convergence · q-integral · Stancu operators · Point-wise convergent

Mathematical Subject Classifications (2010) 41A25 · 41A35

V. N. Mishra (�) · P. PatelDepartment of Applied Mathematics and Humanities, Sardar Vallabhbhai National Instituteof Technology, Ichchhanath Mahadev Road, Surat, 395 007 (Gujarat), Indiae-mail: [email protected]

P. Patele-mail: [email protected]

V. N. MishraL. 1627 Awadh Puri Colony Beniganj, Phase -III, Opposite - I.T.I., Ayodhya Main Road, Faizabad,224 001, (Uttar Pradesh), India

P. PatelDepartment of Mathematics, St. Xavier College, Ahmedabad, Gujarat, 380 009, India

Author's personal copy

mailto:[email protected]

mailto:[email protected]

Numer Algor

1 Introduction

In last decade, the study of operators sequence base on q-integer has attracted moreand more attention. It has been shown that linear positive operators constructedby q-numbers are quite effective as far as the rate of convergence is concerned.Approximation properties of linear positive operators base q-integer different formthe classical case. We mention that the sequence of q-Bernstein polynomials for0 < q < 1 does not satisfy the conditions of Korovkin’s Theorem, and the limitof q-Bernstein operator is not the identity operator. They have been studied inten-sively, and their associations with different branches of analysis, such as convex andnumerical analysis, total positivity, probability distributions [1, 2]. The q-Bernsteinpolynomial were study by many authors [3–6]. The Durrmeyer type modification ofthe q-Bernstein operators was first introduced by Derriennic [7]. Recently Gupta [8]introduce, a simple q-analogue of well known Bernstein Durrmeyer operators andstudied some local results for these operators. Later Finta and Gupta [9] extendedthe studies and discussed some direct global approximation theorems for the q-Bernstein-Durrmeyer operators. Aral and Gupta introduce a Durrmeyer type modifi-cation of q-Baskakov operators and study their properties of approximation in [10].Baskakov-Durrmeyer operators and their different generalizations were studied byGupta et al. [11–14].

Aral and Gupta [10] introduce q analogues of Baskakov-Durrmeyer operators,which define as:

For any n ∈ N, q ∈ (0, 1),

Dn,q(f, x) = [n− 1]q∞∑

k=0

pqn,k(x)

∫ ∞/A

0pqn,k(t)f (t)dq t, (1.1)

where pq

n,k(x) =(n+ k − 1

k

)

q

qk22

xk

(1 + x)n+kq

for x ∈ [0,∞). For every real val-

ued continuous and bounded function f on [0,∞). These operators satisfy linearityproperty.

In 1983, Stancu [15] has gives the generalization of Bernstein polynomials withtwo parameter α, β satisfying the condition 0 ≤ α ≤ β. Several other researchershave studied in this direction and obtained different approximation properties ofmany operators, we mention some of them as [16–24] ect. Motivated by such typeoperators, the generalization of Baskakov-Durrmeyer type operators with parameterα and β is discuss in [25].

The aim of this paper is to study the approximation properties of the Durrmeyertype modification of the Baskakov type operators with two parameter α and β basedon q-integers. We investigate moments, asymptotic formula, weighted approximationproperties and better error estimations for the operators (2.2) in q-calculus. In thismanuscript we have use notations as given in [26, 27].

Consider q as a real number satisfying 0 < q < 1.

(1+x)nq = (−x; q)n ={(1 + x)(1 + qx)

(1 + q2x

)...

(1 + qn−1x

), n = 1, 2, ...,

1, n = 0.


Numer Algor

The q-improper integral define as

∫ ∞/A

0f (x)dqx = (1 − q)

∞∑

n=0

f

(qn

A

)qn

A, A > 0

provided sum converges absolutely.

2 Construction of operators

We introduce q analogues of the Durrmeyer type modification of the Baskakov typeoperators with three parameter γ , α and β as follows:

For every n ∈ N, 0 < q < 1, γ > 0 and 0 ≤ α ≤ β,

D(α,β,γ )n,q (f, x) = [n− γ ]q

∞∑

k=0

pqn,k,γ (x)

∫ ∞/A

0pqn,k,γ (t)f

( [n]q t + α

[n]q + β

)dqt, (2.1)

where

pq

n,k,γ (x) = qk22

�q(n/γ + k)

�q(k + 1)�q(n/γ )· (γ x)k

(1 + γ x)(n/γ )+kq

for x ∈ [0,∞) and for every real valued continuous and bounded function f on[0,∞). These operators satisfy linearity property. By putting q = 1, α = 0, β = 0and γ = 1 the above operators reduce to the Baskakov-Durrmeyer operators discussin [28]. For q = 1 and γ = 1, the operators (2.1) convert to operators study byMishra and Patel [25]. For α = β = 0 and γ = 1, the operatorsD(α,β,γ )

n,q (f, x) reduceto operators discussed in [10]. We observe that the operators defined as above is notpreserved linear function it only preserved constant function. For γ = 1 the aboveoperators reduce in to following form

D(α,β)n,q (f, x) = [n− 1]q

∞∑

k=0

pq

n,k(x)

∫ ∞/A

0pq

n,k(t)f

( [n]q t + α

[n]q + β

)dqt, (2.2)

where pq

n,k(x) =(n+ k − 1

k

)

q

qk22

xk

(1 + x)n+kq

.

The present paper deals with some direct results for the operators D(α,β)n,q (f, x). Here

we estimate direct results in terms of modulus of continuity.

Lemma 1 [14] For n ∈ N, k ≥ 0, we have

Dq

[xk

(1 + x)n+kq

]= [k]qxk−1

(1 + x)n+kq

− qkxk[n+ k]q(1 + x)n+k+1

q

.


Numer Algor

Lemma 2 [10] The following equality holds:

Dn,q(1, x) = 1, Dn,q(t, x) =[

1 + [2]qq2[n− 2]q

]x + 1

q[n− 2]q , for n > 2,

Dn,q(t2, x) =

[1 + [3]q

q3[n− 3]q + [2]qq2[n− 2]q + q[2]q[3]q + [n]q

q6[n− 2]q [n− 3]q]x2

+[ [n]q + q(1 + [2]q)[n]q

q5[n− 2]q[n− 3]q]x + [2]q

q3[n− 2]q[n− 3]q , for n > 3.

Remark 1 By Lemma 1, it is observe that

x(1 + x)Dqpq

n,k(x) =( [k]qqk−1[n]q − qx

) [n]qq

pq

n,k(qx),

t

q

(1 + t

q

)Dqp

qn,k

(t

q

)=

( [k]qqk−1[n]q − t

) [n]qq2

pqn,k(t).

Lemma 3 If we define the moment as

Tn,m(x) = [n− 1]q∞∑

k=0

pqn,k(x)

∫ ∞/A

0pqn,k(t)t

mdqt.

Then, for n > m+ 2, we have the following recurrence relation:

([m+ 2]q − [n]q)Tn,m+1(qx) = q[m+ 1]qTn,m(qx)+ qx[n]qTn,m(qx)+ qx(1 + x)Dq [Tn,m(x)].

Proof Using Remark 1, we have

I = qx(1 + x)Dq [Tn,m(x)] = [n− 1]q∞∑

k=0

qx(1 + x)Dq

[pqn,k(x)

] ∫ ∞/A

0pqn,k(t)t

mdq t

= [n− 1]q [n]q∞∑

k=0

( [k]qqk−1[n]q − qx

)pqn,k(qx)

∫ ∞/A

0pqn,k(t)t

mdqt

= [n− 1]q [n]q∞∑

k=0

pqn,k(qx)

∫ ∞/A

0

( [k]qqk−1[n]q − t + t − qx

)pqn,k(t)t

mdq t.

I = qx(1 + x)Dq [Tn,m(x)] = [n− 1]q∞∑

k=0

pqn,k(qx)

∫ ∞/A

0q2 t

q

(1 + t

q

)Dqp

qn,k

(t

q

)tmdq t

+ [n]q [n− 1]q∞∑

k=0

pqn,k(x)

∫ ∞/A

0pqn,k(t)t

m+1dq t − qx[n]qTn,m(qx)

= [n− 1]q∞∑

k=0

pqn,k(qx)

∫ ∞/A

0

(qt + t2

)Dqp

qn,k

(t

q

)tmdq t

+ [n]qTn,m+1(qx)− qx[n]qTn,m(qx).


Numer Algor

Using the q integral by parts∫ b

a

u(t)Dq(v(t))dq t = [u(t)v(t)]ba −∫ b

a

v(qt)

Dq [u(t)]dqt

I = −[n− 1]q∞∑

k=0

pq

n,k(x)

∫ ∞/A

0pq

n,k(t)(q[m+ 1]qtm + [m+ 2]q tm+1

)dqt

+ [n]qTn,m+1(qx)− qx[n]qTn,m(qx)=−q[m+ 1]qTn,m(qx)− [m+ 2]qTn,m+1(qx)+ [n]qTn,m+1(qx)

− qx[n]qTn,m(qx),which completes the proof of recurrence relation.

Lemma 4 The following equality holds:

D(α,β)n,q (1, x) = 1,D(α,β)

n,q (t, x) =[ [n]q[nq ] + β

[1 + [2]q

q2[n− 2]q]]

x + [n]q + qα[n− 2]qq[n− 2]q([n]q + β)

, for n > 2.

D(α,β)n,q (t2, x) =

[ [n]2q([n]q + β)2

[1 + [3]q

q3[n− 3]q + [2]qq2[n− 2]q + q[2]q [3]q + [n]q

q6[n− 2]q [n− 3]q]]

x2

+[ [n]2q([n]q + β)2

[ [n]q [2]2q + 2q3α[n− 3]qq5[n− 2]q [n− 3]q

]]x

+[2]q [n]2q + 2αq2[n]q [n− 3]q + q3α2[n− 2]q [n− 3]qq3[n− 2]q [n− 3]q([n]q + β)2

, for n > 3.

Proof By Lemma 2, we get

D(α,β)n,q (1, x) = Dn,q(1, x) = 1,

D(α,β)n,q (t, x) = [n]q

[n]q + βDn,q(t, x)+ α

[n]q + βDn,q(1, x)

=[ [n]q[n]q + β

[1 + [2]q

q2[n− 2]q]]

x + [n]q + qα[n− 2]qq[n− 2]q([n]q + β)

.

For n > 3, we have

D(α,β)n,q (t2, x) = 1

([n]q + β)2

[[n]2qDn,q(t

2, x)+ 2α[n]qDn,q(t, x)+ α2Dn,q(1, x)

]

=[ [n]2q([n]q + β)2

[(1 + [3]q

q3[n− 3]q + [2]qq2[n− 2]q + q[2]q [3]q + [n]q

q6[n− 2]q [n− 3]q]]

x2

+[ [n]2q([n]q + β)2

[ [n]q [2]2qq5[n− 2]q [n− 3]q + 2α

q2[n− 2]q]]

x

+ [2]q [n]2q + 2αq2[n]q [n− 3]qq3[n− 2]q [n− 3]q ([n]q + β)2

+ α2

([n]q + β)2,

we have the desired result.


Numer Algor

Remark 2 For all m ∈ N, 0 ≤ α ≤ β; we have the following recursive relationfor the images of the monomials tm under Dα,β

n,q (tm, x) in terms of Dq

n(tj , x), j =

0, 1, 2, ..., m as

D(α,β)n,q (tm, x) =

m∑

j=0

(m

j

) [n]jqαm−j

([n]q + β)mDq

n(tj , x).

From Lemma 3 and above recursive relation, we can write recurrence relation ofmoment of the Durrmeyer type modification of the q-Baskakov type operators withtwo parameter α and β.

Remark 3 We have

μn(q, x) = D(α,β)n,q (t − x, x) =

[ [n]q[n]q + β

[1 + [2]q

q2[n− 2]q]− 1

]x

+ [n]q + qα[n− 2]qq[n− 2]q([n]q + β)

, for n > 2,

αn(q, x) = D(α,β)n,q

((t − x)2, x

)= D(α,β)

n,q

(t2, x

)− 2xD(α,β)

n,q (t, x)+ x2D(α,β)n,q (1, x)

=[ [n]2q([n]q + β)2

[1 + [3]q

q3[n− 3]q + [2]qq2[n− 2]q + q[2]q [3]q + [n]q

q6[n− 2]q [n− 3]q]]

x2

−[ 2[n]2qq2[n− 2]q ([n]q + β)

− 1

]x2 −

[2([n]q + αq[n− 2]q )q[n− 2]q([n]q + β)

]x

+[ [n]2q([n]q + β)2

[ [n]q [2]2qq5[n− 2]q [n− 3]q + 2α

q2[n− 2]q]]

x

+ [2]q [n]2q + 2αq2[n]q [n− 3]q + q3α2[n− 2]q [n− 3]qq3[n− 2]q [n− 3]q([n]q + β)2

, for n > 3.

3 Local approximation

Let the space CB [0,∞) of all continuous and bounded functions be endowed withthe norm ‖f ‖ = sup{|f (x)| : x ∈ [0,∞)}. Further let us consider the followingK-functional:

K2(f, δ) = infg∈W 2

{‖f − g‖ + δ‖g′′‖}, (3.1)

where δ > 0 and W 2 = {g ∈ CB [0,∞) : g′, g′′ ∈ CB [0,∞)}. By the method asgiven in [29, p.177 Theorem 2.4], there exists an absolute constant C > 0 such that

K2(f, δ) ≤ Cω2

(f,

√δ), (3.2)

where

ω2

(f,

√δ)= sup

0<h≤√δ

supx∈[0,∞)

| f (x + 2h)− 2f (x + h)+ f (x) | (3.3)


Numer Algor

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

ω(f, δ) = sup0<h≤δ

supx∈[0,∞)

| f (x + h)− f (x) | (3.4)

we denote the usual modulus of continuity of f ∈ CB [0,∞).The first result is a direct local approximation theorem for the operators D(α,β)

n,q .

Theorem 1 Let q ∈ (0, 1) and n > 3, we have∣∣∣D(α,β)

n,q (f, x)− f (x)

∣∣∣ ≤ Cω2

(f,

√(αn(q, x)+ μ2

n(q, x))

)+ ω(f, μn(q, x)),

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

Proof Let us define the auxiliary operator L(α,β)n,q by

L(α,β)n,q (f, x) = D(α,β)

n,q (f, x)+ f (x)− f (ζ(x, q)), (3.5)

where ζ(x, q) = [n]q[n]q + β

[[1+ [2]q[n]q

q2[n− 2]q]x+ [n]q + qα[n− 2]q

q[n]q [n− 2]q]

and for each

x ∈ [0,∞). The operator L(α,β)n,q are linear and preserve the linear functions:

L(α,β)n,q (t − x, x) = 0, t ∈ [0,∞). (3.6)

Let g ∈ W 2 and x, t ∈ [0,∞). By Taylor’s expansion, we have

g(t) = g(x)+ g′(x)(t − x)+∫ t

x

(t − u)g′′(u)du.

Applying L(α,β)n,q and by (3.6), we get

L(α,β)n,q (g, x) = g(x) + L(α,β)

n,q

(∫ t

x

(t − u)g′′(u)du, x).

Hence by (3.5), we have∣∣∣L(α,β)

n,q (g, x)− g(x)

∣∣∣ ≤∣∣∣D(α,β)

n,q

(∫ t

x

(t − u)g′′(u)du, x) ∣∣∣

+∣∣∣∣∫ ζ(x,q)

x

(ζ(q, x)− u)g′′(u)du∣∣∣∣

≤ D(α,β)n,q ((t − x)2‖g′′‖, x)+ [

ζ(x, q)− x]2‖g′′‖

=(D(α,β)

n,q

((t − x)2, x

)+ [

ζ(q, x)− x]2

)‖g′′‖.

From the ζ(q, x) and by remark (3), we have∣∣∣L(α,β)

n,q (g, x)− g(x)

∣∣∣ ≤ [αn(q, x)+ μ2n(q, x)]‖g′′‖. (3.7)

On the other hand, by (3.5), we obtain∣∣L(α,β)

n,q (f, x)∣∣ ≤ ∣∣D(α,β)

n,q (f, x)∣∣+ 2‖f ‖ ≤ ‖f ‖D(α,β)

n,q (1, x)+ 2‖f ‖ ≤ 3‖f ‖. (3.8)


Numer Algor

Now by (3.5), (3.7) and (3.8), we wield∣∣D(α,β)

n,q (f, x)− f (x)∣∣ ≤ ∣∣L(α,β)

n,q (f − g, x)− (f − g)(x)∣∣∣∣L(α,β)

n,q (g, x)− g(x)∣∣

+∣∣f (ζ(x, q)− f (x))∣∣

≤ 4‖f−g‖ +[αn(q, x)+ μ2

n(q, x)]+∣∣f (ζ(x, q)−f (x))

∣∣.

Hence taking infimum on the right hand side over all g ∈ W 2, we get∣∣D(α,β)

n,q (f, x)− f (x)∣∣ ≤ 4K2(f, [αn(q, x)+ μ2

n(q, x)])+ ω(f, μn(q, x)).

In view of (3.2), we find∣∣D(α,β)

n,q (f, x)− f (x)∣∣ ≤ Cω2

(f,

√[αn(q, x)+ μ2

n(q, x)])+ ω(f, μn(q, x)),

which complete the proof of Theorem 1.

4 Weighted approximation

The problem of the approximation of unbounded continuous functions by sequencesof linear positive operators have been investigated in many papers [30]. Korovkin-type theorems on weighted approximation of unbounded continuous functions onunbounded sets with single weight function were first proved by Gadzhiev [31, 32].Now, we give Gadzhiev’s results in weighted spaces. Let ρ(x) = 1 + φ2(x), whereφ(x) is a monotone increasing continuous function on the real axis and Bρ is the setof all functions f defined on the real axis satisfying the growth condition |f (x)| ≤Mf ρ(x), where Mf is a constant depending only on f . Then Bρ is a normed spacewith norm

‖f ‖ρ = sup

{ |f (x)|ρ(x)

: x ∈ R

},

for any f ∈ Bρ . Let Cρ denote the subspace of all continuous functions in Bρ , and

C∗ρ the subspace of all functions f ∈ Cρ for which lim|x|→∞

(f (x)

ρ(x)

)exists finitely.

Theorem 2 (See [31] and [32])

(a) There exists a sequence of linear positive operators An(Cρ → Bρ) such that

limn→∞‖An(φ

v)− φv‖ρ = 0, v = 0, 1, 2 (4.1)

and a function f ∗ ∈ Cρ\C∗ρ with limn→∞ ‖An(f

∗)− f ∗‖ρ ≥ 1.(b) If a sequence of linear positive operators An(Cρ → Bρ) such that satisfies

conditions (4.1) thenlimn→∞‖An(f )− f ‖ρ = 0,

for every f ∈ C∗ρ.

Throughout this paper we take the growth condition as ρ(x) = 1 + x2 andρε(x) = 1 + x2+ε , x ∈ [0,∞), ε > 0.


Numer Algor

Theorem 3 Let q = qn satisfies 0 < qn < 1 and let qn → 1 as n → ∞. For each

f ∈ C∗x2[0,∞), we have lim

n→∞∥∥∥D(α,β)

n,qn(f )− f

∥∥∥ρ= 0.

Proof Using the theorem in [32] we see that it is sufficient to verify the followingthree conditions

limn→∞

∥∥∥D(α,β)n,qn

(tr , x)− xr∥∥∥ρ= 0, r = 0, 1, 2. (4.2)

Since D(α,β)n,qn (1, x) = 1, the first condition of (4.2) is satisfied for r = 0.

Now,


(t, x)− x

∥∥∥ρ

= supx∈[0,∞)

| D(α,β)n,qn (t, x)− x |

1 + x2

≤ ([n]2qn − q2n[n− 2]qn([n]qn + β))

q2n[n− 2]qn([n]qn + β)

supx∈[0,∞)

x

1 + x2

+qn[n]qn + αq2n[n− 2]qn


≤ ([n]2qn − q2n[n− 2]qn([n]qn + β))


+qn[n]qn + αq2n[n− 2]qn


→ 0 as n → ∞.

and the second condition of (4.2) hold for r = 1.Similarly, we can write for n > 3


(t2, x

)− x2

∥∥∥ρ= sup

x∈[0,∞)


(t2, x

) − x2∥∥∥

1 + x2

≤[ [n]2qn([n]qn + β)2

[1 + [3]qn

q3n[n− 3]qn

+ [2]qnq2n[n− 2]qn

+ qn[2]qn [3]qn + [n]qnq6n[n− 2]qn [n− 3]qn

]− 1

]sup

x∈[0,∞)

x2

1 + x2

+ [n]qn([n]qn + β)2

[ [n]2qn [2]2qnq5n[n− 2]qn [n− 3]qn

+ 2α[n]qnq2n[n− 2]qn

]sup

x∈[0,∞)

x

1 + x2

+[n]2qn [2]qn + 2q2nα[n]qn [n− 3]qn + q3

nα2[n − 2]qn [n− 3]qn

q3n[n− 2]qn [n− 3]qn ([n]qn + β)2

≤[ [n]2qn([n]qn + β)2

[1 + [3]qn

q3n[n− 3]qn

+ [2]qnq2n[n− 2]qn

+ qn[2]qn [3]qn + [n]qnq6n[n− 2]qn [n− 3]qn

]− 1

]

+ [n]qn([n]qn + β)2

[ [n]2qn [2]2qnq5n[n− 2]qn [n− 3]qn

+ 2α[n]qnq2n[n− 2]qn

]

+[n]2qn [2]qn + 2q2nα[n]qn [n− 3]qn + q3

nα2[n − 2]qn [n− 3]qn

q3n[n− 2]qn [n− 3]qn ([n]qn + β)2 ,


Numer Algor

which implies that

‖D(α,β)n,qn

(t2, x

)− x2‖ρ → 0 as n → ∞.

Thus the proof is completed.We give the following theorem to approximate all functions in Cx2[0,∞). This typeof results are given in [33] for locally integrable functions.

Theorem 4 Let q = qn satisfies 0 < qn < 1 and let qn → 1 as n → ∞. For eachf ∈ Cx2[0,∞) and ε > 0, we have

limn→∞ sup

x∈[0,∞)

| D(α,β)n,qn (f, x)− f (x) |

(1 + x2

)1+ε= 0.

Proof For any fixed x0 > 0,

supx∈[0,∞)

|D(α,β)n,qn (f, x)− f (x)|

(1 + x2

)1+ε≤ sup

x≤x0

∣∣D(α,β)n,qn (f, x)− f (x)

∣∣(1 + x2

)1+ε

+ supx≥x0

∣∣D(α,β)n,qn (f, x)− f (x)

∣∣(1 + x2

)1+ε

≤ ‖D(α,β)n,qn

(f )− f ‖C[0,x0] + ‖f ‖x2 supx≥x0

∣∣D(α,β)n,qn

(1 + t2, x

) ∣∣(1 + x2

)1+ε

+ supx≥x0

∣∣f (x)∣∣

(1 + x2

)1+ε.

The first term of the above inequality tends to zero from Theorem 5. For some cal-

culation, it is easily seen that for any fixed x0 > 0, supx≥x0

∣∣D(α,β)n,qn

(1 + t2, x

) ∣∣(1 + x2

)1+εtends

to zero as n → ∞. We can choose x0 > 0 so large that the last part of the aboveinequality can be made small enough.Thus the proof is completed.

5 Rate of convergence

In this section, we want to estimate the rate of convergence for the sequence of theoperators D(α,β)

n,qn For any positive a, by

ωa(f, δ) = sup|t−x|≤δ

supx,t∈[0,a]

|f (t) − f (x)|

we denote the usual modulus of continuity of f on the closed interval [0, a]. Weknow that for a function f ∈ Cx2[0,∞), the modulus of continuity ωa(f, δ) tends tozero.

Now we give a rate of convergence theorem for the operator D(α,β)n,q .


Numer Algor

Theorem 5 Let f ∈ Cx2 [0,∞), q = qn ∈ (0, 1) such that qn → 1 as n → ∞ andωa+1(f, δ) be its modulus of continuity on the finite interval [0, a + 1] ⊂ [0,∞),where a > 0. Then for every n > 3,

|D(α,β)n,qn

(f )− f | ≤ 6Mf

(1 + a2

)αn(qn, x)+ 2ωa+1

(f,

√αn(qn, x)

).

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

|f (t)− f (x)| ≤ Mf

(2 + x2 + t2

)

≤ Mf

(2 + 3x2 + 2(t − x)2

)

≤ 3Mf

(1 + x2 + (t − x)2

)

|f (t) − f (x)| ≤ 6Mf

(1 + a2

)(t − x)2. (5.1)

For x ∈ [0, a] and t ≤ a + 1, we have

|f (t)− f (x)| ≤ ωa+1(f, |t − x|) ≤[

1 + |t − x|δ

]ωa+1(f, δ) (5.2)

with δ > 0.From (5.1) and (5.2), we get

|f (t)− f (x)| ≤ 6Mf (1 + a2)(t − x)2 +[

1 + |t − x|δ

]ωa+1(f, δ).

For x ∈ [0, a] and t ≥ 0

|D(α,β)n,qn

(f, x)− f (x)| ≤ D(α,β)n,qn

(|f (x)− f (t)|, x)≤ 6Mf

(1 + a2

)D(α,β)

n,qn

((t − x)2, x

)

+ωa+1(f, δ)

[1 + 1

δ

[D(α,β)

n,qn

((t − x)2, x

)] 12].

Hence, by Schwarz’s inequality and remark (3), for every qn ∈ (0, 1) and x ∈ [0, a]

|D(α,β)n,qn

(f, x)− f (x)| ≤ 6Mf

(1 + a2

)αn(qn, x)+ ωa+1(f, δ)

[1 + 1

δ

√αn(qn, x)

].

By taking δ = √αn(qn, x), we get the assertion of our theorem. We can give esti-

mates of the errors |D(α,β)n,qn (f, ·) − f |, n ∈ N, for unbounded functions by using a

weighted modulus of smoothness associated to the space Bρε [0,∞). We would liketo take a weighted modulus of continuity �ρε(f ; δ) which tends to zero as δ → 0. let

�ρε(f ; δ) := sup0<h≤δ,x≥0

|f (x + h)− f (x)|(1 + (x + h)2+ε)

, (5.3)

for every f ∈ Bρε [0,∞). The weighed modulus of continuity �ρε(f ; δ) was definedby Lopez-Moreno in [34]. It is known that �ρε(f ; δ) has the following properties.


Numer Algor

Lemma 5 [34] Let f ∈ Bρε [0,∞). Then

(i) �ρε(f ; δ) ≤ 2‖f ‖ρε , δ > 0 ε ≥ 0.

(ii) limδ→0+

�ρε(f ; δ) = 0.

(iii) For each m ∈ N, �ρε(f ;mδ) ≤ m�ρε(f ; δ), δ > 0.

(iv) For each λ ∈ R+, �ρε(f ; λδ) ≤ (1 + λ)�ρε(f ; δ), δ > 0.

Theorem 6 Let 0 < q < 1 and ε ≥ 0. For all non-decreasing f ∈ Bρε [0,∞). Thenwe have the inequality

|D(α,β)n,qn

(f ; x)− f | ≤√D(α,β)

n,qn

(μ2x,ε; x

)(1 + 1

δ

√D(α,β)

n,qn

(�2

x ; x))

�ρε(f ; δ),

x ≥ 0, δ > 0, n ∈ N, where μx,ε(t) := 1 + (x + |t − x|)2+ε, �x(t) := |t − x|,t ≥ 0.

Proof From the definition of �ρε(f ; δ), and lemma 5(iv), we can write

|f (t)− f (x)| ≤ (1 + (x + |t − x|)2+ε)

(1 + |t − x|

δ

)�ρε(f ; δ)

= μx,ε(t)

(1 + 1

δ�x(t)

)�ρε(f ; δ).

Then, we have the inequality

∣∣D(α,β)n,qn

(f ; x)− f (x)∣∣ ≤ �ρε(f ; δ)D(α,β)

n,qn

(μx,ε

(1 + 1

δ�x

); x

)

≤ �ρε(f ; δ){D(α,β)

n,qn(μx,ε; x)+D(α,β)

n,qn

(μx,ε�x

δ; x

)}.

(5.4)

Applying the Cauchy-Schwarz inequality to the second term, we get

D(α,β)n,qn

(μx,ε�x

δ; x

)≤

{D(α,β)

n,qn

((μx,ε)

2; x)}1/2

{D(α,β)

n,qn

((�x

δ

)2

; x)}1/2

= 1

δ

{D(α,β)

n,qn

(μ2x,ε; x

)}1/2 {D(α,β)

n,qn

(�2

x ; x)}1/2

.

Now, by (5.4), we obtain

∣∣D(α,β)n,qn

(f ; x)− f∣∣ ≤

√D(α,β)

n,qn

(μ2x,ε; x

)(1 + 1

δ

√D(α,β)

n,qn

(�2

x ; x))

�ρε(f ; δ).


Numer Algor

6 Voronovskaja type theorem

In this section we establish the asymptotic formula for the operators D(α,β)n,q :

Lemma 6 Assume that qn ∈ (0, 1), qn → 1 as n → ∞. Then, for every x ∈ [0,∞),we have

limn→∞[n]qnD(α,β)

n,qn(t − x, x) = (2 − β)x + 1 + α, (6.1)


n,qn

((t − x)2, x

)= 2x(1 + x). (6.2)

Theorem 7 For qn ∈ (0, 1), the sequence D(α,β)n,qn converges to f uniformly on [0, A]

for each f ∈ C∗x2[0,∞) if and only if lim

n→∞ qn = 1.

Proof The proof is similar to the Theorem 2 in [35].

Theorem 8 Assume that qn ∈ (0, 1), qn → 1 as n → ∞. Then for any f ∈C∗x2[0,∞) such that f ′, f ′′ ∈ C∗

x2[0,∞) and x ∈ [0,∞), we have

limn→∞[n]qn(D(α,β)

n,qn(f, x)− f (x)) = (α + 1 + (2 − β)x)f ′(x)+ x(1 + x)f ′′(x),

for every x ≥ 0.

Proof Let f, f ′, f ′′ ∈ C∗x2 [0,∞) and x ∈ [0,∞) be fixed. By Taylor’s expansion

we can write

f (t) = f (x)+ f ′(x)(t − x)+ 1

2f ′′(x)(t − x)2 + r(x, t)(t − x)2, (6.3)

where r(t, x) is Peano form of the remainder, r(·, x) ∈C∗x2 [0,∞) and lim

t→xr(t,x)=0.

Applying D(α,β)n,qn to above, we obtain

[n]qn[D(α,β)n,qn

(f, x)− f (x)] = f ′(x)[n]qnD(α,β)n,qn

(t − x, x)

+1

2f ′′(x)[n]qnD(α,β)

n,qn

((t − x)2, x

)

+ [n]qnD(α,β)n,qn

(r(t, x)(t − x)2, x

).

By Cauchy-Schwarz inequality, we have

D(α,β)n,qn

(r(t, x)(t − x)2, x

)≤

√D(α,β)

n,qn

(r(t, x)2, x

)√D(α,β)n,qn

((t − x)4, x

). (6.4)


Numer Algor

Observe that r2(x, x) = 0 and r2(·, x) ∈ C∗x2[0,∞). Then it follows from Theorem

that


n,qn

(r(t, x)2, x

)= r2(x, x) = 0, (6.5)

uniformly with respect to x ∈ [0, A]. Now from (6.4) and (6.5) and Lemma 6, weobtain


n,qn

(r(t, x)(t − x)2, x

)= 0.

limn→∞[n]qn[D(α,β)

n,qn(f, x)− f (x)] = lim

n→∞[n]qn[f ′(x)D(α,β)

n,qn(t − x, x)

+ 1

2f ′′(x)D(α,β)

n,qn

((t − x)2, x

)]

+ limn→∞[n]qn

[D(α,β)

n,qn

(r(t, x)(t − x)2, x

)]

= ((2 − β)x + 1 + α)f ′(x)+(x2 + x

)f ′′(x),

7 Better Estimation

In this section we give better estimates for the operators D(α,β)n,q as follows:

To make the convergence faster King [36] proposed an approach to modify theclassical Bernstein polynomial, so that the sequence preserve test function e0 and e2.After that this approach was applied to some well known operators. As the operatorD(α,β)

n,q introduction in (2.2) preserve only the constant so further modification ofthese operators is proposed to be made so that the modified operators preserve theconstant as well as linear functions, for this purpose the modification of D(α,β)

n,q asfollows:

D∗(α,β)n,q (f, x) = [n− 1]q

∞∑

k=0

pq

n,k(rn,q(x))

∫ ∞/A

0pq

n,k(t)f

( [n]q t + α

[n]q + β

)dqt,

(7.1)

where rn,q(x) = q2[n−2]q([n]q+β)x−q[n]q−αq2[n− 2]q[n]2q

and 0< rn,q(x)≤∞.

Thus x ≥ [n]q + qα[n− 2]qq[n− 2]q([n]q + β)

. We consider x ∈ In,q =[

[n]q+qα[n−2]qq[n−2]q ([n]q+β)

,∞).


Numer Algor

Lemma 7 For x ∈ In,q , we have

D∗(α,β)n,q (1, x) = 1,D∗(α,β)

n,q (t, x) = x,

D∗(α,β)n,q

(t2, x

)= q6[n−2]q [n−3]q + q3[n− 2]q [3]q + q4[2]q [n− 3]q + q[2]q [3]q + [n]q

q6[n− 2]q [n− 3]q([n]q + β)2

×[q2[n− 2]q([n]q + β)x − q[n]q − αq2[n− 2]q

[n]q]2

+[ [n]2q([n]q + β)2

[ [n]q [2]2qq5[n− 2]q [n− 3]q + 2α

q2[n− 2]q]]

rn,q (x)

+ [2]q [n]2q + 2αq2[n]q [n− 3]qq3[n− 2]q [n− 3]q([n]q + β)2

+ α2

([n]q + β)2

Remark 4 For every x ∈ In,q , it is easy to verify that D∗(α,β)n,q (t − x, x) = 0.

We denote D∗(α,β)n,q

((t − x)2, x

)by α∗

n(q, x).

Theorem 9 Let f ∈ CB(In,q) and 0 < q < 1. Then for all x ∈ In,q and n > 3,there exist an absolute constant M > 0 such that

| D∗(α,β)n,q (f, x)− f (x) |≤ Mω2(f,

√α∗n(q, x)).

Proof Let g ∈ CB(In,q) and x, t ∈ In,q . Using the Taylor’s formula, we get

g(t) − g(x) = (t − x)g′(x)+∫ t

x

(t − u)g′′(u)du.

Applying D∗(α,β)n,q and by remark (4), we get

D∗(α,β)n,q (g, x)− g(x) = D∗(α,β)

n,q

(∫ t

x

(t − u)g′′(u)du, x).

Obviously, we have∣∣∣∣∫ t

x

(t − u)g′′(u)du∣∣∣∣ ≤ ‖g′′‖(t − x)2.

|D∗(α,β)n,q (g, x)− g(x)| ≤ ‖g′′‖D∗(α,β)

n,q ((t − x)2, x) = ‖g′′‖α∗n(q, x).

Since ‖D∗(α,β)n,q (f, x)‖ ≤ ‖f ‖, we have

|D∗(α,β)n,q (f, x)−f (x)| ≤ |D∗(α,β)

n,q (f−g, x)−(f − g)(x)| + |D∗(α,β)n,q (g, x)− g(x)|

≤ 2‖f − g‖ + α∗n(q, x)‖g′′‖.

Now, taking infimum on right hand side over all g ∈ C2B(0,∞) and from (3.2), we

get,

|D∗(α,β)n,q (f, x)− f (x)| ≤ K2(f, α

∗n(q, x)) ≤ Mω2(f,

√α∗n(q, x)).

Thus the proof is completed.


Numer Algor

Theorem 10 Assume that qn ∈ (0, 1), qn → 1 as n → ∞. Then for any f ∈C∗x2(In,qn) such that f ′, f ′′ ∈ C∗

x2(In,qn), we have

limn→∞[n]qn[D∗(α,β)

n,qn(f, x)− f (x)] = x(1 + x)f ′′(x).

The proof follows from along the lines of Theorem 6.

Remark 5 Recently, Aral and Gupta [14], gives q-generalization of the classicalBaskakov operators. Motivated by this operators we introduce following modified q-Baskakov operators:For f ∈ C[0,∞), q > 0 and each positive integer n,

Bn,q,γ (f, x) =∞∑

k=0

bqn,k,γ (x)f

( [k]qqk−1[n]q

), (7.2)

where bqn,k,γ (x) = q

k(k−1)2

�q(n/γ + k)

�q(k + 1)�q(n/γ )· (γ x)k

(1 + γ x)(n/γ )+kq

.

One can study its moments, local approximation properties and Voronovskajatype asymptotic result based on q-integers. Using moments of Bn,q,γ (f, x) one can

discuss approximation properties of the operators D(α,β,γ )n,q as defined in (2.1).

Acknowledgments The authors are thankful to the anonymous referee for making valuable commentsleading to the better presentation of the paper. Special thanks are due to Prof. Dr. Claude Brezinski, Editor-in-Chief, Numerical Algorithm, for kind cooperation and useful suggestion for improvement of the paper.

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Numer Algor

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