We study some properties of q-BBH operators and establish the rate of convergence for BBH operators.. Also, we study convergence of the derivative of q-BBH operators.. There are several
Trang 1Volume 2008, Article ID 816367, 15 pages
doi:10.1155/2008/816367
Research Article
q-Parametric Bleimann Butzer and Hahn Operators
N I Mahmudov and P Sabancıgil
Eastern Mediterranean University, Gazimagusa, Turkish Republic of Northern Cyprus, Mersin 10, Turkey
Correspondence should be addressed to N I Mahmudov,nazim.mahmudov@emu.edu.tr
Received 4 June 2008; Accepted 20 August 2008
Recommended by Vijay Gupta
We introduce a new q-parametric generalization of Bleimann, Butzer, and Hahn operators in
C∗1x0, ∞ We study some properties of q-BBH operators and establish the rate of convergence for BBH operators We discuss Voronovskaja-type theorem and saturation of convergence for q-BBH operators for arbitrary fixed 0 < q < 1 We give explicit formulas of Voronovskaja-type for the
q-BBH operators for 0 < q < 1 Also, we study convergence of the derivative of q-BBH operators.
Copyrightq 2008 N I Mahmudov and P Sabancıgil This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
q-Bernstein polynomials
B n,q fx :n
k0
f
k
n
n k
x k
n−k−1
s0
were introduced by Phillips in 1 q-Bernstein polynomials form an area of an intensive
research in the approximation theory, see survey paper2 and references therein Nowadays,
there are new studies on the parametric operators Two parametric generalizations of
q-Bernstein polynomials have been considered by Lewanowicz and Wo´znycf 3, an analog
of the Bernstein-Durrmeyer operator and Bernstein-Chlodowsky operator related to the
q-Bernstein basis has been studied by Derriennic 4, Gupta 5 and Karsli and Gupta 6,
respectively, a q-version of the Szasz-Mirakjan operator has been investigated by Aral and
Gupta in7 Also, some results on q-parametric Meyer-K¨onig and Zeller operators can be
found in8 11
In12, Bleimann et al introduced the following operators:
H n fx 1
1 x n
n
k0
f
k
n − k 1
n k
Trang 2
There are several studies related to approximation properties of Bleimann, Butzer, and Hahn operatorsor, briefly, BBH, see, for example, 12–18 Recently, Aral and Do˘gru 19
introduced a q-analog of Bleimann, Butzer, and Hahn operators and they have established some approximation properties of their q-Bleimann, Butzer, and Hahn operators in the subspace of C B 0, ∞ Also, they showed that these operators are more flexible than classical BBH operators, that is, depending on the selection of q, rate of convergence of the q-BBH
operators is better than the classical one Voronovskaja-type asymptotic estimate and the
monotonicity properties for q-BBH operators are studied in20
In this paper, we propose a different q-analog of the Bleimann, Butzer, and Hahn
operators in C1x∗ 0, ∞ We use the connection between classical BBH and Bernstein
operators suggested in16 to define new q-BBH operators as follows:
H n,q fx : Φ−1B n 1,q Φfx, 1.3
where B n 1,q is a q-Bernstein operator, Φ and Φ−1 will be defined later Thanks to 1.3, different properties of Bn 1,q can be transferred to H n,q with a little extra effort Thus
the limiting behavior of H n,q can be immediately derived from 1.3 and the well-known
properties of B n 1,q It is natural that even in the classical case, when q 1, to define H nin the
space C∗1x0, ∞, the limit l f of f x/1 x as x→∞ has to appear in the definition of H n
Thus in C1x∗ 0, ∞ the classical BBH operator has to be modified as follows:
H n fx 1
1 x n
n
k0
f
k
n − k 1
n k
x k l f x n1
1 x n , x > 0, n ∈ N. 1.4 The paper is organized as follows In Section 2, we give construction of q-BBH operators
and study some elementary properties InSection 3, we investigate convergence properties
of q-BBH, Voronovskaja-type theorem and saturation of convergence for q-BBH operators for arbitrary fixed 0 < q < 1, and also we study convergence of the derivative of q-BBH operators.
2 Construction and some properties ofq-BBH operators
Before introducing the operators, we mention some basic definitions of q calculus.
Let q > 0 For any n ∈ N ∪ {0}, the q-integer n n qis defined by
n : 1 q · · · q n−1, 0 : 0; 2.1
and the q-factorial n! n q! by
n! : 12 · · · n, 0! : 1. 2.2 For integers 0≤ k ≤ n, the q-binomial is defined by
n k
Also, we use the following standard notations:
z; q0: 1, z; q n:n−1
j0
1 − q j z , z; q∞:∞
j0
1 − q j z ,
p n,k q; x :
n k
x k
n−k−1
s0
1 − q s x , p ∞k q; x : x k
1 − q k k!
∞
s0
1 − q s x .
2.4
Trang 3It is agreed that an empty product denotes 1 It is clear that p nk q; x ≥ 0, p ∞k q; x ≥ 0 ∀x ∈
0, 1 and
n
k0
p nk q; x ∞
k0
Introduce the following spaces
B ρ 0, ∞ {f : 0, ∞→R | ∃M f > 0 such that |fx| ≤ M f ρ x ∀x ∈ 0, ∞},
C ρ 0, ∞ {f ∈ B ρ 0, ∞ | f is continuous},
C ρ∗0, ∞
f ∈ C ρ 0, ∞ | lim
x→∞
f x
ρ x l f exists and is finite
,
C ρ00, ∞
f ∈ C ρ 0, ∞ | lim
x→∞
f x
.
2.6
It is clear that C∗ρ 0, ∞ ⊂ C ρ 0, ∞ ⊂ B ρ 0, ∞ In each space, the norm is defined by
f ρ sup
x≥0
|fx|
We introduce the following auxiliary operators Firstly, let us denote
ψ y y
1− y , y ∈ 0, 1, ψ−1x
x
1 x , x ∈ 0, ∞. 2.8
Secondly, letΦ : C∗
ρ 0, ∞→C0, 1 be defined by
Φfy :
⎧
⎪
⎪
f ψy
ρ ψy , if y ∈ 0, 1,
l f lim
x→∞
f x
ρ x , if y 1.
2.9
ThenΦ is a positive linear isomorphism, with positive inverse Φ−1: C0, 1→C∗
ρ 0, ∞ defined
by
Φ−1gx ρxgψ−1x, g ∈ C0, 1, x ∈ 0, ∞. 2.10
For f ∈ C0, 1, t > 0, we define the modulus of continuity ωf; t as follows:
ω f; t : sup{|fx − fy| : |x − y| ≤ t, x, y ∈ 0, 1}. 2.11
We introduce new Bleimann-, Butzer-, and Hahn-BBH type operators based on q-integers
as follows
Definition 2.1 For f ∈ C∗
ρ 0, ∞, the q-Bleimann, Butzer, and Hahn operators are given by
H n,q fx : Φ−1B n 1,q Φfx
ρxn
k0
f ψk/n 1
ρ ψk/n 1 p n 1,k q; ψ−1x l f ρ xψ−1x n1, n ∈ N,
2.12 where
p n 1,k q; ψ−1x :
k
ψ−1x k−k
s0
1 − q s ψ−1x, k 0, 1, , n. 2.13
Trang 4Note that for q 1, ρ 1 x and l f 0, we recover the classical Bleimann, Butzer, and
Hahn operators If q 1, ρ 1 x but l f / 0, it is new Bleimann, Butzer, and Hahn operators with additional term l f x n1/ 1 x n Thus if f ∈ C0
1x0, ∞ then
H n,q fx :n
k0
f
q k n − k 1
n k
qx
1 x
k n−k
s1
1− q s x
1 x
. 2.14
To present an explicit form of the limit q-BBH operators, we consider
p ∞k q; ψ−1x : ψ−1x
k
1 − q k k!
∞
s0
1 − q s ψ−1x. 2.15
Definition 2.2 Let 0 < q < 1 The linear operator defined on C∗ρ 0, ∞ given by
H ∞,q fx : ρx∞
k0
f ψ1 − q k
ρ ψ1 − q kp ∞k q; ψ−1x 2.16
is called the limit q-BBH operator.
Lemma 2.3 H n,q , H ∞,q : C ρ∗0, ∞→C∗
ρ 0, ∞ are linear positive operators and
H n,q f ρ ≤ f ρ , H ∞,q f ρ ≤ f ρ 2.17
Proof We prove the first inequality, since the second one can be done in a like manner Thanks
to the definition, we have
|H n,q fx| ≤ ρx f ρn
k0
p n 1,k q; ψ−1x ρx|l f |ψ−1x n1
≤ ρx f ρn
k0
p n 1,k q; ψ−1x ρx f ρ ψ−1x n1
ρx f ρn1
k0
p n 1,k q; ψ−1x ρx f ρ
2.18
Lemma 2.4 The following recurrence formula holds:
H n,q
ρ t
t
1 t
m
x 1
n 1 m−1
x
1 x
m−1
j0
j
q j n j H n −1,q
ρ t
t
1 t
j
x.
2.19
In particular, we have
H n,q ρx ρx, H n,q
ρ t t
1 t
x ρx x
1 x , H n,q 1x 1,
H n,q
ρ t
t
1 t
2
x ρx
x
1 x
2
ρx x
1 x2
1
n 1 .
2.20
Trang 5Proof We prove only the recurrence formula, since the formulae2.20 can easily be obtained
by standard computations Since l f 1 for f ρtt/1 t m, we have
H n,q
ρ t
t
1 t
m
x
ρxn
k0
k
n 1
m
p n 1,k
q; ψ−1x
ρx
x
1 x
n1
ρxn
k0
k
n 1
m
k
x
1 x
k n−k
s0
1− q s x
1 x
ρx
x
1 x
n1
ρxn
k0
k m−1
n 1 m−1
n
x
1 x
k n−k
s0
1− q s x
1 x
ρx x
1 x
n1
ρxn
k1
m−1
j0
j
q j k − 1 j
n 1 m−1
×
n
x
1 x
k n−k
s0
1− q s x
1 x
ρx
x
1 x
n1
n 1 m−1
x
1 x
m−1
j0
j
q j n j
×
H n −1,q
ρ t
t
1 t
j
x − ρx
x
1 x
n
ρx
x
1 x
n1
n 1 m−1
x
1 x
m−1
j0
j
q j n j H n −1,q
ρ t
t
1 t
j
x
ρx
x
1 x
n1
n 1 m−1
m−1
j0
j
q j n j
n 1 m−1
x
1 x
m−1
j0
j
q j n j H n −1,q
ρ t
t
1 t
j
x.
2.21
Next theorem shows the monotonicity properties of q-BBH operators.
Theorem 2.5 If f ∈ C∗
1x0, ∞ is convex and
l f
f
n
q n
− f
n 1
q n1
then its q-BBH operators are nonincreasing, in the sense that
H n,q fx ≥ H n 1,q fx, n 1, 2, , q ∈ 0, 1, x ∈ 0, ∞. 2.23
Trang 6Proof We begin by writing
H n,q fx − H n 1,q fx
n
k0
f
q k n − k 1
n k
qx
1 x
k n−k
s1
1− q s x
1 x
−n1
k0
f
q k n − k 2
k
qx
1 x
kn−k1
s1
1− q s x
1 x
l f
x n1
1 x n1.
2.24
We now split the first of the above summations into two, writing
x
1 x
k n−k
s1
1− q s x
1 x
ψ k q n −k1 ψ k1, 2.25 where
ψ k
x
1 x
kn−k1
s1
1− q s x
1 x
The resulting three summations may be combined to give
H n,q fx − H n 1,q fx
n
k0
f
q k n − k 1
n k
q k ψ k q n −k1 ψ k1
−n1
k0
f
q k n − k 2
k
q k ψ k l f
x
1 x
n1
n
k0
f
q k n − k 1
n k
q k ψ kn1
k1
f
k − 1
q k−1n − k 2
n
q n1ψ k
−n1
k0
f
q k n − k 2
k
q k ψ k l f
x
1 x
n1
n
k1
k
a k q k ψ k
f
n
q n
− f
n 1
q n1
q n1
x
1 x
n1
l f
x
1 x
n1
,
2.27 where
a k n − k 1 n 1 f
q k n − k 1
q n n 1 −k1 k f
k − 1
q k−1n − k 2
− f
q k n − k 2
.
2.28
By assumption, the sum of the last three terms of 2.27 is positive Thus to show
monotonicity of H n,qit suffices to show nonnegativity of ak , 0 ≤ k ≤ n Let us write
α n − k 1 n 1 , x1 k
q k n − k 1 , x2 k − 1
q k n − k 2 . 2.29
Trang 7Then it follows that
1− α q n n 1 −k1 k ,
αx1 1 − αx2 k
q k n 1
1q n − k 2 n −k2 k − 1
k
q k n 1
1− q n −k2 q n −k2 1 − q k−1
1− q n −k2
q k n − k 2 ,
2.30
and we see immediately that
a k αfx1 1 − αfx2 − fαx1 1 − αx2 ≥ 0, 2.31
and so H n,q fx − H n 1,q fx ≥ 0.
Remark 2.6 It is easily seen that
l f
f
n
q n
− f
n 1
q n1
q n1
n 2
1
n 2 Φf1
q n 1
n 2 Φf
n
n 1
− Φf
n 1
n 2
.
2.32
The condition2.22 follows from convexity of Φf On the other hand, Φf is convex if f is
convex and nonincreasing, see16
3 Convergence properties
Theorem 3.1 Let q ∈ 0, 1, and let f ∈ C∗
ρ 0, ∞ Then
H n,q f − H ∞,q f ρ ≤ CqωΦf, q n1, 3.1
Proof For all x ∈ 0, ∞, by the definitions of H n,q fx and H ∞,q fx, we have that
H n,q f − H ∞,q f ρxn
k0
f ψk/n 1
ρ ψk/n 1 p n 1,k q; ψ−1x
l f ρ x
x
1 x
n1
− ρx∞
k0
f ψ1 − q k
ρ ψ1 − q kp ∞k q; ψ−1x
ρx n1
k0
Φf
k
n 1
− Φf1 − q k
p n 1,k q; ψ−1x
ρxn1
k0
Φf1 − q k − Φf1p n 1,k q; ψ−1x − p ∞k q; ψ−1x
− ρx∞
k n2
Φf1 − q k − Φf1p ∞k q; ψ−1x
: I1 I2 I3.
3.2
Trang 8First, we estimate I1, I3 By using the following inequalities:
0≤ n 1 k − 1 − q k 1− q k
1− q n1− 1 − q k q n11 − q k
1− q n1 ≤ q n1,
0≤ 1 − 1 − q k q k ≤ q n1, k ≥ n 2,
3.3
we get
|I1| ≤ ρxωΦf, q n1n1
k0
p n 1,k q; ψ−1x ρxωΦf, q n1,
|I3| ≤ ρx ∞
k n2
ω Φf, q k p ∞k q; ψ−1x ≤ ρxωΦf, q n1.
3.4
Next, we estimate I2 Using the well-known property of modulus of continuity
ω g, λt ≤ 1 λωg, t, λ > 0, 3.5
we get
|I2| ≤ ρx n1
k0
ω Φf, q k |p n 1,k q; ψ−1x − p ∞k q; ψ−1x|
≤ ρxωΦf, q n1n1
k0
1 q k −n−1 |p n 1,k q; ψ−1x − p ∞k q; ψ−1x|
≤ 2ρxωΦf, q n1 1
q n1
n1
k0
q k |p n 1,k q; ψ−1x − p ∞k q; ψ−1x|
: ρx 2
q n1ω Φf, q n1J n1ψ−1x,
3.6
where
J n1ψ−1x n1
k0
q k |p n 1,k q; ψ−1x − p ∞k q; ψ−1x|. 3.7 Now, using the estimation2.9 from 21, we have
J n1ψ−1x ≤ q n1
q 1 − qln
1
1− q
n1
k0
p n 1,k q; ψ−1x p ∞k q; ψ−1x
≤ 2q n1
q 1 − qln
1
1− q .
3.8
From3.6 and 3.8, it follows that
|I2| ≤ ρx 4
q 1 − qln
1
1− q ω Φf, q n1. 3.9
From3.4, and 3.9, we obtain the desired estimation
Trang 9Theorem 3.2 Let 0 < q < 1 be fixed and let f ∈ C∗
1x0, ∞ Then H ∞,q fx fx ∀x ∈ 0, ∞
if and only if f is linear.
H ∞,q fx Φ−1B ∞,q Φfx. 3.10
Assume that H ∞,q fx fx Then B ∞,q Φfx Φfx From 22, we know that
B ∞,q g g if and only if g is linear So B ∞,q Φfx Φfx if and only if Φfx
1 − xfx/1 − x Ax B It follows that fx 1 xAx/1 x B A Bx B.
The converse can be shown in a similar way
Remark 3.3 Let 0 < q < 1 be fixed and let f ∈ C∗
1x0, ∞ Then the sequence {H n,q fx} does not approximate f x unless f is linear It is completely in contrast to the classical case.
Theorem 3.4 Let q q n satisfies 0 < q n < 1 and let q n →1 as n→∞ For any x ∈ 0, ∞ and for any
f ∈ C∗
ρ 0, ∞, the following inequality holds:
1
ρ x |H n,q n fx − fx| ≤ 2ω
Φf,λ n x
where λ n x x/1 x21/n 1 q n .
Proof Positivity of B n 1,q n implies that for any g ∈ C0, 1
|B n 1,q n gx − gx| ≤ B n 1,q n |gt − gx|x. 3.12
On the other hand,
|Φft − Φfx| ≤ ωΦf, |t − x|
≤ ωΦf, δ
11
δ |t − x|
This inequality and3.12 imply that
|B n 1,q n Φfx − Φfx| ≤ ωΦf, δ
1 1
δ B n 1,q n |t − x|x
,
|Φ−1B n 1,q n Φfx − Φ−1Φfx|
≤ ωΦf, δ
Φ−11 1
δΦ−1B n 1,q n |t − x|x
≤ ρxωΦf, δ1 1
δ B n 1,q n
|t − ψ−1x|2
ψ−1x 1/2
ρxωΦf, δ
1 1
δ
x
1 x
2
1 x2
1
n 1 q n −
x
1 x
21/2
ρxωΦf, δ
1 1
δ
x
1 x2
1
n 1 q n
1/2
,
3.14
by choosing δλ n x, we obtain desired result.
Trang 10Corollary 3.5 Let q q n satisfies 0 < q n < 1 and let q n →1 as n→∞ For any f ∈ C∗
ρ 0, ∞ it holds
that
lim
n→∞ H n,q n fx − fx ρ 0. 3.15
Next, we study Voronovskaja-type formulas for the BBH operators For the
q-Bernstein operators, it is proved in23 that for any f ∈ C10, 1,
lim
n→∞
n
q n B n,q fx − B ∞,q fx L q f, x 3.16
uniformly in x ∈ 0, 1, where
L q f, x :
⎧
⎪
⎪
∞
k0
k
f 1 − q k −f 1 − q k − f1 − q k−1
1 − q k − 1 − q k−1
x k
q; q k x; q∞, 0 ≤ x < 1,
3.17
Similarly, we have the following Voronovskaja-type theorem for the q-BBH operators for fixed
q ∈ 0, 1 Before stating the theorem we introduce an analog of L q f, x for q-BBH operators
V q f, x : Φ−1L q Φfx
x
1 x , q
∞
∞
k0
k
×
f
1− q k
q k
1
q k − f
1− q k
q k
−q k f 1 − q k /q k − q k−1f 1 − q k−1/q k−1
1 − q k − 1 − q k−1
× q, q1
k
x k
1 x k−1
x
1 x ; q
∞
∞
k0
kf
1− q k
q k
1
q k − q k−1f 1 − q k /q k − f1 − q k−1/q k−1
q k−1− q k
× q; q1
k
x k
1 x k−1.
3.18
Theorem 3.6 Let 0 < q < 1, f ∈ C∗
1x0, ∞ ∩ C10, ∞, and Φf is differentiable at x 1 Then
lim
n→∞
n 1
q n1 H n,q fx − H ∞,q fx V q f, x, 3.19
in C∗1x0, ∞.
Δx :
n 1 q n1 H n,q fx − H ∞,q fx − V q f, x
n 1 q n1 Φ−1B n 1,q Φfx − Φ−1B ∞,q Φfx − Φ−1L q Φfx
Φ−1n 1
q n1 B n 1,q − B ∞,q − L q
Φ
fx
1 x
n 1 q n1 B n 1,q − B ∞,q − L q
Φfψ−1x
.
3.20
... x and l f 0, we recover the classical Bleimann, Butzer, and< /p>Hahn operators If q 1, ρ x but l f / 0, it is new Bleimann, Butzer, and Hahn. ..
Definition 2.1 For f ∈ C∗
ρ 0, ∞, the q -Bleimann, Butzer, and Hahn operators are given by
H n,q fx : Φ−1B... sup{|fx − fy| : |x − y| ≤ t, x, y ∈ 0, 1}. 2.11
We introduce new Bleimann- , Butzer- , and Hahn- BBH type operators based on q-integers
as follows
Definition