q-Parametric Bleimann Butzer and Hahn Operators

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 Bn,qfx : n  k0 f _k n   n k  xk n_−k−1 s0 1 − qs_{x 1.1}

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:

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 CB0, ∞. 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 C_1x∗ 0, ∞. We use the connection between classical BBH and Bernstein operators suggested in16 to define new q-BBH operators as follows:

Hn,qfx : Φ−1Bn1,qΦfx, 1.3

where Bn1,q is a q-Bernstein operator, Φ and Φ−1 will be defined later. Thanks to 1.3,

different properties of Bn1,q can be transferred to Hn,q with a little extra effort. Thus

the limiting behavior of Hn,q can be immediately derived from 1.3 and the well-known

properties of Bn1,q. It is natural that even in the classical case, when q 1, to define Hnin the

space C∗_1x0, ∞, the limit lf of fx/1  x as x→∞ has to appear in the definition of Hn.

Thus in C_1x∗ 0, ∞ the classical BBH operator has to be modified as follows:

Hnfx  1 1  xn n  k0 f  k n− k  1   n k  xk lf x n1 1  xn, 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  · · ·  qn−1_{, 0 : 0; 2.1}

and the q-factorialn!  n_q! by

n! : 12 · · · n, 0! : 1. 2.2 For integers 0≤ k ≤ n, the q-binomial is defined by



n k



: _{k!n − k!}n! . 2.3

It is agreed that an empty product denotes 1. It is clear that pnkq; x ≥ 0, p∞kq; x ≥ 0 ∀x ∈ 0, 1 and n  k0 pnkq; x  ∞  k0 p∞kq; x  1. 2.5

Introduce the following spaces.

Bρ0, ∞  {f : 0, ∞→R | ∃Mf > 0 such that|fx| ≤ Mfρx ∀x ∈ 0, ∞}, Cρ0, ∞  {f ∈ Bρ0, ∞ | f is continuous}, C_ρ∗0, ∞   f ∈ Cρ0, ∞ | lim_x→∞ fx

ρx  lf exists and is finite

, C_ρ00, ∞   f ∈ Cρ0, ∞ | lim x→∞ fx ρx  0 . 2.6

It is clear that C∗_ρ0, ∞ ⊂ Cρ0, ∞ ⊂ Bρ0, ∞. In each space, the norm is defined by

f_ρ sup

x≥0

|fx|

ρx. 2.7

We introduce the following auxiliary operators. Firstly, let us denote

ψy  y

1− y, y∈ 0, 1, ψ

−1_{x } x

1 x, x∈ 0, ∞. 2.8 Secondly, letΦ : C∗_ρ0, ∞→C0, 1 be defined by

Φfy : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ fψy ρψy, if y∈ 0, 1, lf  lim x→∞ fx ρx, if y 1. 2.9 ThenΦ is a positive linear isomorphism, with positive inverse Φ−1: C0, 1→C∗_ρ0, ∞ defined by

Φ−1_{gx  ρxgψ}−1_{x, g ∈ C0, 1, x ∈ 0, ∞. 2.10}

For f∈ C0, 1, t > 0, we define the modulus of continuity ωf; t as follows:

ωf; 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∗_ρ0, ∞, the q-Bleimann, Butzer, and Hahn operators are given by

Note that for q 1, ρ  1  x and lf  0, we recover the classical Bleimann, Butzer, and

Hahn operators. If q 1, ρ  1  x but lf/ 0, it is new Bleimann, Butzer, and Hahn operators

with additional term lfxn1/1  xn. Thus if f ∈ C0_1x0, ∞ then Hn,qfx : n  k0 f  _k qk_{n − k  1}   n k   qx 1 x k n_−k s1  1− qs x 1 x  . 2.14

To present an explicit form of the limit q-BBH operators, we consider

p∞kq; ψ−1x : ψ −1_xk 1 − qk_k! ∞  s0 1 − qs_ψ−1_{x. 2.15} Definition 2.2. Let 0 < q < 1. The linear operator defined on C∗_ρ0, ∞ given by

H∞,qfx : ρx ∞  k0 fψ1 − qk_ ρψ1 − qk_p∞kq; ψ −1_{x 2.16}

is called the limit q-BBH operator.

Lemma 2.3. Hn,q, H∞,q: Cρ∗0, ∞→Cρ∗0, ∞ are linear positive operators and

Hn,qf_ρ≤ fρ, H∞,qf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

|Hn,qfx| ≤ ρxf_ρ n  k0 pn1,kq; ψ−1x  ρx|lf|ψ−1xn1 ≤ ρxf_ρn k0 pn1,kq; ψ−1x  ρxfρψ−1x n1  ρxf_ρn1 k0 pn1,kq; ψ−1x  ρxfρ. 2.18

Lemma 2.4. The following recurrence formula holds:

Proof. We prove only the recurrence formula, since the formulae2.20 can easily be obtained by standard computations. Since lf  1 for f  ρtt/1  tm, we have

Hn,q  ρt  t 1 t m x  ρxn k0  _k n  1 m pn1,k  q; ψ−1x   ρx  x 1 x n1  ρxn k0  _k n  1 m n 1 k   x 1 x k n_−k s0  1− qs x 1 x   ρx  x 1 x n1  ρxn k0 km−1 n  1m−1  n k− 1   x 1 x k n_−k s0  1− qs x 1 x   ρx x 1 x n1  ρxn k1 m−1 j0  m− 1 j  _qj_{k − 1}j n  1m−1 ×  n k− 1   x 1 x k n_−k s0  1− qs x 1 x   ρx  x 1 x n1  1 n  1m−1 x 1 x m_−1 j0  m− 1 j  qjnj ×  Hn−1,q  ρt  t 1 t j x − ρx  x 1 x n  ρx  x 1 x n1  1 n  1m−1 x 1 x m_−1 j0  m− 1 j  qj_nj_H n−1,q  ρt  t 1 t j x  ρx  x 1 x n1 1− 1 n  1m−1 m_−1 j0  m− 1 j  qjnj   1 n  1m−1 x 1 x m−1 j0  m− 1 j  qjnjHn−1,q  ρt  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 lf  f _n qn  − f _{n  1} qn1  qn1≥ 0, 2.22

then its q-BBH operators are nonincreasing, in the sense that

Proof. We begin by writing Hn,qfx − Hn1,qfx n k0 f  _k qk_{n − k  1}   n k   _qx 1 x k n_−k s1  1− qs x 1 x  −n1 k0 f  _k qk_{n − k  2}   n 1 k   _qx 1 x kn−k1 s1  1− qs x 1 x   lf xn1 1  xn1. 2.24

We now split the first of the above summations into two, writing  x 1 x k n_−k s1  1− qs x 1 x   ψk qn−k1ψk1, 2.25 where ψk  x 1 x kn_−k1 s1  1− qs x 1 x  . 2.26

The resulting three summations may be combined to give

Hn,qfx − Hn1,qfx n k0 f  _k qk_{n − k  1}   n k  qkψk qn−k1ψk1 −n1 k0 f  _k qkn − k  2   n 1 k  qk_ψ k lf  x 1 x n1 n k0 f  _k qk_{n − k  1}   n k  qkψk n1  k1 f  _{k − 1} qk−1_{n − k  2}   n k− 1  qn1ψk −n1 k0 f  _k qkn − k  2   n 1 k  qk_ψ k lf  x 1 x n1 n k1  n 1 k  akqkψk  f _n qn  − f _{n  1} qn1  qn1  x 1 x n1  lf  x 1 x n1 , 2.27 where ak n − k  1_{n  1} f  _k qk_{n − k  1}  qn_{n  1}−k1kf  _{k − 1} qk−1_{n − k  2}  − f  _k qk_{n − k  2}  . 2.28 By assumption, the sum of the last three terms of 2.27 is positive. Thus to show monotonicity of Hn,qit suffices to show nonnegativity of ak, 0≤ k ≤ n. Let us write

α n − k  1

n  1 , x1 k

qk_{n − k  1}, x2

k − 1

Then it follows that 1− α  q n−k1_k n  1 , αx1 1 − αx2 k qk_{n  1}  1q n−k2_{k − 1} n − k  2   k qk_{n  1} _{1− q}n−k2_{ q}n−k2_{1 − q}k−1_ 1− qn−k2   k qk_{n − k  2}, 2.30

and we see immediately that

ak αfx1  1 − αfx2 − fαx1 1 − αx2 ≥ 0, 2.31

and so Hn,qfx − Hn1,qfx ≥ 0. Remark 2.6. It is easily seen that

lf  f _n qn  − f _{n  1} qn1  qn1  n  2  1 n  2Φf1  qn  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∗

ρ0, ∞. Then

Hn,qf − H∞,qf_ρ≤ CqωΦf, qn1, 3.1

where Cq  4/q1 − q ln1/1 − q  2.

First, we estimate I1, I3. By using the following inequalities: 0≤ k n  1 − 1 − qk  1− qk 1− qn1− 1 − q k_{ } qn11 − qk 1− qn1 ≤ q n1_, 0≤ 1 − 1 − qk  qk≤ qn1, k≥ n  2, 3.3 we get |I1| ≤ ρxωΦf, qn1 n1  k0 pn1,kq; ψ−1x  ρxωΦf, qn1, |I3| ≤ ρx ∞  kn2 ωΦf, qkp_∞kq; ψ−1x ≤ ρxωΦf, qn1. 3.4

Next, we estimate I2. Using the well-known property of modulus of continuity

ωg, λt ≤ 1  λωg, t, λ > 0, 3.5 we get |I2| ≤ ρx n1  k0 ωΦf, qk|pn1,kq; ψ−1x − p∞kq; ψ−1x| ≤ ρxωΦf, qn1_n1 k0 1  qk−n−1_|p n1,kq; ψ−1x − p∞kq; ψ−1x| ≤ 2ρxωΦf, qn1_ 1 qn1 n1  k0 qk|pn1,kq; ψ−1x − p∞kq; ψ−1x| : ρx 2 qn1ωΦf, qn1Jn1ψ−1x, 3.6 where Jn1ψ−1x  n1  k0 qk|pn1,kq; ψ−1x − p∞kq; ψ−1x|. 3.7

Now, using the estimation2.9 from 21, we have

Jn1ψ−1x ≤ q n1 q1 − qln 1 1− q n1  k0 pn1,kq; ψ−1x  p∞kq; ψ−1x ≤ 2qn1 q1 − qln 1 1− q. 3.8

From3.6 and 3.8, it follows that |I2| ≤ ρx 4 q1 − qln 1 1− qωΦf, q n1_{. 3.9}

Theorem 3.2. Let 0 < q < 1 be fixed and let f ∈ C∗

1x0, ∞. Then H∞,qfx  fx ∀x ∈ 0, ∞ if and only if f is linear.

Proof. By definition of H∞,qwe have

H_∞,qfx  Φ−1B_∞,qΦfx. 3.10

Assume that H∞,qfx  fx. Then B∞,qΦfx  Φfx. From 22, we know that B_∞,qg  g if and only if g is linear. So B_∞,qΦfx  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 {Hn,qfx}

does not approximate fx unless f is linear. It is completely in contrast to the classical case.

Theorem 3.4. Let q  qnsatisfies 0 < qn< 1 and let qn→1 as n→∞. For any x ∈ 0, ∞ and for any f∈ C∗_ρ0, ∞, the following inequality holds:

1 ρx|Hn,qnfx − fx| ≤ 2ω  Φf,λnx  , 3.11 where λnx  x/1  x21/n  1qn.

Proof. Positivity of Bn1,qnimplies that for any g ∈ C0, 1

|Bn1,qngx − gx| ≤ Bn1,qn|gt − gx|x. 3.12

On the other hand,

|Φft − Φfx| ≤ ωΦf, |t − x| ≤ ωΦf, δ  11 δ|t − x|  , δ > 0. 3.13

This inequality and3.12 imply that |Bn1,qnΦfx − Φfx| ≤ ωΦf, δ  1 1 δBn1,qn|t − x|x  , |Φ−1_B n1,qnΦfx − Φ−1Φfx| ≤ ωΦf, δ  Φ−1_{1 }1 δΦ −1_B n1,qn|t − x|x  ≤ ρxωΦf, δ1 1 δBn1,qn  |t − ψ−1_x|2_ψ−1x1/2  ρxωΦf, δ  1 1 δ  x 1 x 2  x 1  x2 1 n  1qn −  x 1 x 21/2  ρxωΦf, δ  1 1 δ  x 1  x2 1 n  1qn 1/2 , 3.14

Corollary 3.5. Let q  qnsatisfies 0 < qn< 1 and let qn→1 as n→∞. For any f ∈ Cρ∗0, ∞ it holds that

lim

n→∞Hn,qnfx − 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 ∈ C1_{0, 1,}

lim n→∞ n qn Bn,qfx − B∞,qfx  Lqf, x 3.16 uniformly in x∈ 0, 1, where Lqf, x : ⎧ ⎪ ⎨ ⎪ ⎩ ∞  k0 k  f 1 − qk_{ −}f1 − qk − f1 − qk−1 1 − qk − 1 − qk−1  xk q; q_kx; q∞, 0≤ x < 1, 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 Lqf, x for q-BBH operators Vqf, x : Φ−1LqΦfx   x 1 x, q  ∞ ∞  k0 k ×  f _{1− q}k qk  1 qk − f _{1− q}k qk  −qkf1 − qk/qk − qk−1f1 − qk−1/qk−1 1 − qk_{ − 1 − q}k−1_  × _{q, q}1 k xk 1  xk−1  x 1 x; q  ∞ ∞  k0 kf _{1− q}k qk  1 qk − q k−1f1 − qk/qk − f1 − qk−1/qk−1 qk−1_{− q}k  × _{q; q}1 k xk 1  xk−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

qn1 Hn,qfx − H∞,qfx  Vqf, x, 3.19 in C∗_1x0, ∞.

Proof. We estimate the difference

SinceΦf is well defined on whole 0, 1, from 23, Theorem 1, we get that lim

n→∞Δ1x ≤ limn→∞_0≤u≤1sup



n  1_qn1 Bn1,q− B∞,q − Lq



Φfu  0. 3.21 Theorem is proved.

Remark 3.7. It is clear that Φf is differentiable in 0, 1 if f ∈ C1_{0, ∞. If Φf is not}

differentiable at x  1, then lim n→∞ n  1 qn1 Hn,qfx − H∞,qfx  Vqf, x, 3.22 uniformly on any0, A ⊂ 0, ∞. Theorem 3.8. If f ∈ C2_{0, ∞ and q} n→1 as n→∞, then lim n→∞n  1qn{Hn,qnfx − fx}  1 2f _{x1  x}2_{x 3.23} uniformly on any0, A ⊂ 0, ∞. Proof. By definition of Hn,qn, Hn,qnfx − fx  Φ−1Bn1,qnΦfx − Φ−1Φfx  Φ−1_B n1,qn− IΦfx  1  xBn1,qn− IΦfψ−1x, 3.24 and if L : 1/2f _{x1 − xx, then} 1 2f _{x1  x}2_{x Φ}−1_{LΦfx  1  xLΦfψ}−1_x  1 21  xΦf _ψ−1_xψ−1_{x1 − ψ}−1_x. 3.25

On the other hand, by24, Corollary 5.2 we have that lim n→∞_0≤u≤1sup  n  1qnBn1,qn− IΦfu − 1 2Φf _{uu1 − u}_{  0. 3.26}

From Theorem 3.6, we have the following saturation of convergence for the q-BBH operators for fixed q∈ 0, 1.

Corollary 3.9. Let 0 < q < 1 and f ∈ C∗

1x0, ∞ ∩ C10, ∞. Then

Hn,qfx − H∞,qfx_1x  oqn1 3.28 if and only if Vqf, x ≡ 0, and this is equivalent to

f _{1− q}k qk  1 qk − 1 qk−1   f1 − qk qk  − f1 − qk−1 qk−1  , k 1, 2, . . . . 3.29

Theorem 3.10. Let 0 < q < 1 and f ∈ C∗

1x0, ∞ ∩ C10, ∞. If f is a convex function, then

Hn,qfx − H∞,qfx_1x  oqn1 if and only if f is a linear function. Proof. IfHn,qf − H∞,qf1x  oqn1, then byCorollary 3.9

f _{1− q}k qk _qk−1_{− q}k q2k−1  f 1 − qk_ qk  − f1 − qk−1 qk−1  , k 1, 2, . . . . 3.30 Hence for k 1, 2, . . . _1−qk_/qk 1−qk−1_/qk−1  f  1− qk qk  − f _t_{dt 0. 3.31}

Since f is convex and f is continuous on0, ∞, we get f t  f 1 − qk_/qk_{ ∀t ∈ 1 −} qk−1_/qk−1_{,1 − q}k_/qk_{. Hence f t ≡ f 0, and therefore ft  At  B. Conversely, if f is}

linear, thenHn,qfx − H∞,qfx_1x 0.

One of the remarkable properties of the q-Bernstein approximation is that derivatives of Bnf of any order converge to corresponding derivatives of f, see 25. Next theorem

shows the same property for Hnqfor the first derivative. Theorem 3.11. Let f ∈ C∗

1x0, ∞∩C10, ∞ and let {qn} be a sequence chosen so that the sequence

εn n

1 qn q2n · · ·  qnn−1

− 1 3.32

converges to zero from above faster than{1/3n_{}. Then}

Since Hn,qnfx is a composition of differentiable functions, it is differentiable at any x ∈ 0, A and d dxHn,qnfx  d dx  1  xBn1,qnΦf  x 1 x   Bn1,qnΦf  x 1 x   1 1 x d dxBn1,qnΦf  x 1 x  . 3.35 By24, Theorem 4.1  Bn1,qnΦf  x 1 x  − Φf  x 1 x   ≤ 2ωΦf,  Bn1,qn  t− x 1 x 2 _x 1 x  , 3.36 and by25, Theorem 3 lim n→∞_0≤x≤Asup  _dxd Bn1,qnΦf  x 1 x  − Φf  x 1 x    0. 3.37 Thus the desired limit follows from the following inequality:

 _dxd Hn,qnfx − d dxfx    d dxHn,qnfx − d dx1  xΦf  x 1 x   ≤Bn1,qnΦf  x 1 x  − Φf  x 1 x   _{1 x}1 _dxd Bn1,qnΦf  x 1 x  − Φf  x 1 x   ≤ 2ω  Φf,  Bn1,qn  t− x 1 x 2 _x 1 x   d dxBn1,qnΦf  x 1 x  − Φf  x 1 x    2ω  Φf,  x 1  x2 1 n  1qn   d dxBn1,qnΦf  x 1 x  − Φf  x 1 x   ≤ 2ω  Φf,  A n  1qn   d dxBn1,qnΦf  x 1 x  − Φf  x 1 x  . 3.38

Remark 3.12. In1, it is shown that

Immediately from the definition of Hn,q, we get an analog of3.39 for Hn,q: Hn,qfx  Φ−1Bn1,qΦfx  Φ−1n1 k0  n 1 k  Δk_Φf 0xk n1 k0  n 1 k  Δk_Φf 0 xk 1  xk−1. 3.41 Acknowledgment

The research is supported by the Research Advisory Board of Eastern Mediterranean University under project BAP-A-08-04.

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