Approximation Results For q-Parametric BBH Operators

Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K.

Approximation Results For q-Parametric BBH Operators Nazim Mahmudov and Pembe Sabancıgil Abstract—The paper introduces q-parametric Bleimann, Butzer and Hahn (q-BBH) operators as a rational transformation of q-Bernstein-Lupa¸s operators. On their basis, a set of new results on q-BBH operators can be obtained easily from the corresponding properties of q-Bernstein-Lupa¸s operators. Furthermore, convergence properties of q-BBH operators are studied.

where Rn,q is the Lupa¸s q-Bernstein operator on C [0, 1] defined by ¸ µ ¶· n X [k] n Rn,q (f, x) = f k [n] k=0

k(k−1)

×

Keywords: Bernstein polynomials, BBH operators, Lupa¸ s operators.

1

The linear operator Un defined by n

k=0

µ

k n−k+1

¶µ

n k

q 2 xk (1 − x)n−k . (1 − x + qx)...(1 − x + q n−1 x)

Thanks to (1), different properties of Rn+1,q can be transferred to Un,q with some extra effort. Thus, the limiting behaviour of Un,q can be immediately derived from (1) and the well known properties of Rn+1,q .

Introduction

X 1 Un (f, x) := f n (1 + x)

∗

¶ xk ,

where f ∈ R[0,∞) was introduced by Bleimann, Butzer, and Hahn [5] to approximate continuous functions. In [5], [1] the authors pointed out some formal similarities and differences between Un and the classical Bernstein operator Bn . Connection suggested in [1] can be formulated by means of the following identity

2

Construction and some properties of the operators

Let q > 0. For any n ∈ N ∪ {0}, the q-integer [n] = [n]q is defined by

Un = Φ−1 ◦ Bn+1 ◦ Φ, Φ−1 and Φ are suitable positive linear operators which will be defined below. This idea was used in [8] to define new q-analogue of the Bleimann, Butzer, and Hahn operators as follows: ¡ ¢ Un,q (f ; x) := Φ−1 ◦ Bn+1,q ◦ Φ (f ; x) ,

[n] := 1 + q + ... + q n−1 , [0] := 0;

and the q-factorial [n]! = [n]q ! by

where Bn+1,q is a Philips q-analogue of the Bernstein operators.

[n]! := [1] [2] ... [n] , [0]! := 1.

Using the classical connection between Bernstein and BBH operators we propose the following q-analogue of the 0 Bleimann, Butzer and Hahn operators in C1+x [0, ∞) : ¡ ¢ Un,q (f ; x) := Φ−1 ◦ Rn+1,q ◦ Φ (f ; x) ,

(1)

∗ Eastern Mediterranean University Department of Mathematics Gazimagusa Mersin 10 Turkey Email: [email protected] ; [email protected]

ISBN:978-988-18210-1-0

For integers 0 ≤ k ≤ n, the q-binomial is defined by ·

n k

¸ :=

[n]! . [k]! [n − k]!

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Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K.

1. Φ−1 ◦ Φ is the identity operator on R[0,∞) .

Also, we use the following notations: n

(1 − x)q :=

n−1 Y

¡

¢ 1 − qj x ,

0 2. f ∈ C1+x [0, ∞) if and only if Φf ∈ C [0, 1].

j=0

∞ Y ¡

∞

(1 − x)q :=

0 3. If f ∈ C1+x [0, ∞), then f is convex if and only if f is convex and nonincreasing.

¢ 1 − qj x ,

j=0

·

n k

bn,k (q; x) := ×

k(k−1) 2

k

We introduce Bleimann, Butzer and Hahn type operators based on q-integers as follows.

¸

Definition 2 For f ∈ R[0,∞) the q-Bleimann, Butzer and Hahn operators are given by

n−k

q x (1 − x) , (1 − x + qx)...(1 − x + q n−1 x) ·

un,k (q; x) :=

q

b∞,k (q; x) :=

¸

n k

q k(k+1)/2 xk n , (1 + qx)q

k(k−1) 2

k

(1 − q) [k]!

k

(x/1 − x)

∞ Y

(1 +

qj

,

(x/1 − x))

j=0

u∞,k (q; x) :=

q k(k+1)/2 xk ∞

k

(1 + qx)q (1 − q) [k]!

,

C [0, ∞] = {f ∈ C [0, ∞) | f (x) has a finite limit at ∞} , 0 C1+x [0, ∞) = {f ∈ C [0, ∞) | f (x) = o (1 + x) , x → ∞} .

0 It is assumed that C1+x [0, ∞) is endowed with the norm

kf k1+x = sup x≥0

|f (x)| . 1+x

We consider the operators Φ : R[0,∞) → R[0,1] ,  ´ ³ t  , t ∈ [0, 1) ,  (1 − t) f 1−t Φ (f, t) :=   0, t = 1,

µ

x 1+x

¶

U∞,q (f ; x) µ ¶ ∞ X 1 − qk := f u∞,k (q; x) qk k=0 µ ¶ k(k+1)/2 ∞ X 1 − qk 1 q f xk = ∞ k k q (1 + qx)q (1 − q) [k]! k=0 is called the limit q-BBH operator. 0 0 [0, ∞) → C1+x [0, ∞) are Lemma 4 Un,q , U∞,q : C1+x linear positive operators and

kUn,q (f )k1+x ≤ kf k1+x , kU∞,q (f )k1+x ≤ kf k1+x . Lemma 5 We have Un,q (1; x) = 1, q n(n+1)/2 xn+1 . n (1 + qx)q

, x ∈ [0, ∞) .

Theorem 1 [10] We have the following relations:

ISBN:978-988-18210-1-0

Definition 3 Let 0 < q < 1. The linear operator defined on R[0,∞) given by

Un,q (t; x) = x −

and Φ−1 : R[0,1) → R[0,∞) , Φ−1 (g, x) := (1 + x) g

¡ ¢ Un,q (f ; x) := Φ−1 ◦ Rn+1,q ◦ Φ (f ; x) ¶ µ n X [k] un,k (q; x) = f q k [n − k + 1] k=0 µ ¶ n X 1 [k] = f n q k [n − k + 1] (1 + qx)q k=0 ¸ · n q k(k+1)/2 xk , n ∈ N. × k

0 Theorem 6 If f ∈ C1+x [0, ∞) is a convex function, then the sequence {Un,q (f ; x)} is nonincreasing in n for each q ∈ (0, 1] and x ∈ [0, ∞).

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Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K.

Proof. We start by writing

it suffices to show nonnegativity of ak , 0 ≤ k ≤ n. Let us write

Un,q (f ; x) − Un+1,q (f ; x) µ ¶ n X 1 [k] = f n+1 q k [n − k + 1] (1 + qx)q k=0 · ¸ ¡ ¢ n × q k(k+1)/2 xk 1 + q n+1 x k ¶ n+1 X µ 1 [k] f − n+1 q k [n − k + 2] (1 + qx)q k=0 · ¸ n+1 × q k(k+1)/2 xk k 1 (n+1)(n+2)/2 n+1 =− x n+1 q (1 + qx)q µ µ ¶ µ ¶¶ [n + 1] [n] × f −f q n+1 qn ¶ µ n−1 X 1 [k] + f n+1 q k [n − k + 1] (1 + qx)q k=0 · ¸ n × q (k+1)(k+2)/2 q n−k xk+1 k n−1 X µ [k + 1] ¶ 1 + f n+1 q k+1 [n − k] (1 + qx)q k=0 · ¸ n × q (k+1)(k+2)/2 xk+1 k+1 ¶ n−1 X µ 1 [k + 1] − f n+1 q k+1 [n − k + 1] (1 + qx)q k=0 · ¸ n+1 × q (k+1)(k+2)/2 xk+1 . k+1

q n−k [k + 1] [n − k] ,1 − α = , [n + 1] [n + 1] [k] [k + 1] x1 = k , x2 = k+1 . q [n − k + 1] q [n − k] α=

Then it follows that αx1 + (1 − α) x2 [k + 1] [k] [n + 1] q k [n − k + 1] [n − k] [k + 1] + [n + 1] q k+1 [n − k] [k + 1] q n−k+1 [k] + [n − k + 1] = k+1 q [n + 1] [n − k + 1] [k + 1] [n + 1] [k + 1] = k+1 = k+1 . q [n + 1] [n − k + 1] q [n − k + 1] =

We see immediately that ak = αf (x1 ) + (1 − α) f (x2 ) −f (αx1 + (1 − α) x2 ) ≥ 0 which proves the theorem.

3

|x−y|≤t

Un,q (f ; x) − Un+1,q (f ; x) 1 =− n+1 (1 + qx)q µ µ ¶ µ ¶¶ [n + 1] [n] ×q (n+1)(n+2)/2 xn+1 f − f q n+1 qn n−1 X · n+1 ¸ 1 + ak q (k+1)(k+2)/2 xk+1 , (2) n+1 k+1 (1 + qx) k=0

where µ ¶ [k] q n−k [k + 1] f ak = [n + 1] q k [n − k + 1] µ ¶ [n − k] [k + 1] + f [n + 1] q k+1 [n − k] ¶ µ [k + 1] . −f q k+1 [n − k + 1]

(3)

Now from Theorem 1 since f is nonincreasing the first term is nonnegative. Thus to show monotonicity of Un,q

ISBN:978-988-18210-1-0

Convergence properties of Un,q

For f ∈ C[0, 1], t > 0, the modulus of continuity is defined by ω(f, t) = sup |f (x) − f (y)| .

Consequently,

q

q

n−k

Theorem 7 Let q = qn satisfies 0 < qn < 1 and let qn → 1 as n → ∞. For any x ∈ [0, ∞) and for any 0 [0, ∞) the following inequality holds f ∈ C1+x 1 |Un,q (f ; x) − f (x)| 1+x ³ ´ p ≤ 2ω Φf, λn (x) , where λn (x) =

1 x . 2 (1 + x) [n + 1]qn

Proof. Positivity of Rn+1,qn implies that for any g ∈ C [0, 1] , |Rn+1,qn (g; x) − g (x)| ≤ Rn+1,qn (|g (t) − g (x)| ; x) .

(4)

On the other hand |(Φf ) (t) − (Φf ) (x)| ≤ ω (Φf, |t − x|) ¶ µ 1 ≤ ω (Φf, δ) 1 + |t − x| , δ > 0. δ

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Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K.

This inequality and (4) imply that |Rn+1,qn (Φf ; x) − (Φf ) (x)| µ ¶ 1 ≤ ω (Φf, δ) 1 + Rn+1,qn (|t − x| ; x) δ

[0, ∞) we have ¯ ¯ ¯ ¯ ¯bn,k (q; x ) − b∞,k (q; x )¯ ¯ 1+x 1+x ¯ ≤ bn,k (q;

x xq n x q n−k+1 ) + b∞,k (q; ) . 1+x 1−q 1+x 1−q

and |Un,q (f ; x) − f (x)| ¯ µ ¶ µ ¶¯ ¯ ¯ x x ¯ ¯ − (Φf ) = (1 + x) ¯Rn+1,qn Φf ; 1+x 1+x ¯ ≤ (1 + x) ω (Φf, δ) ¯ µ¯ ¶¶ ¯ 1 x ¯¯ x ¯ × 1 + Rn+1,qn ¯t − ; δ 1 + x¯ 1 + x µ 1 ≤ (1 + x) ω (Φf, δ) 1 + δ à ï !!1/2  ¯2 ¯ ¯ x ¯ x  × Rn+1,qn ¯¯t − ; 1 + x¯ 1 + x à à 1 x 1 = (1 + x) ω (Φf, δ) 1 + δ 1 + x [n + 1]qn  à ! µ ¶2 !1/2 x 1 x qn x  + 1− − 1 + x 1 + qn x [n + 1]qn 1+x à à 1 x 1 ≤ (1 + x) ω (Φf, δ) 1 + δ 1 + x [n + 1]qn !1/2  µ ¶2 x 1  − 1+x [n + 1]qn  à !1/2  1 x 1 , = (1 + x) ω (Φf, δ) 1 + δ (1 + x)2 [n + 1]qn µ

Using Lemma 9 we prove the following quantitative result for the rate of local convergence of Un,q (f ; x) in terms of the first modulus of continuity. 0 Theorem 10 Let 0 < q < 1 and f ∈ C1+x [0, ∞). Then for all 0 ≤ x < ∞ we have

|Un,q (f ; x) − U∞,q (f ; x)| µ ¶ ¡ ¢ 1+x ≤ 2 (1 + x) + 1 ω Φf, q n+1 . 1−q Proof. Consider ∆ (x) := Un,q (f ; x) − U∞,q (f ; x) ¡ ¢ = Φ−1 ◦ Rn+1,q ◦ Φ (f ; x) ¡ ¢ − Φ−1 ◦ R∞,q ◦ Φ (f ; x) ¡ ¢ = Φ−1 ◦ (Rn+1,q − R∞,q ) ◦ Φ (f ; x) ¡ ¢ = Φ−1 ◦ (Rn+1,q − R∞,q ) (Φf ; x) . Since Un,q (f ; x) and U∞,q (f ; x) possess the end point interpolation property, ∆ (0) = 0. For all x ∈ (0, ∞) we rewrite ∆ in the following form ∆ (x) =Φ

we have used¶ the explicit formula for µ ¯ ¯2 ¯ x x ¯ Rn+1,qn ¯t − 1+x , which can be found in ¯ ; 1+x p [9]. Now by choosing δ = λn (x), we obtain desired result.

−1

◦

where

Corollary 8 Let q = qn satisfies 0 < qn < 1 and let 0 qn → 1 as n → ∞. For any f ∈ C1+x [0, ∞) it holds that lim kUn,q (f ; x) − f (x)k1+x = 0.

n+1 X·

(Φf )

k=0

µ

[k] [n + 1]

¶

¡

− (Φf ) 1 − q

k

¢

¸

× bn+1,k (q; x) −1

+Φ

◦

n+1 X

£ ¡ ¢ ¤ (Φf ) 1 − q k − (Φf ) (1)

k=0

× (bn+1,k (q; x) − b∞,k (q; x)) ∞ X ¢ ¤ £ ¡ − Φ−1 ◦ (Φf ) 1 − q k − (Φf ) (1) k=n+2

× b∞,k (q; x) =: I1 + I2 + I3 .

n→∞

We start with estimation of I1 and I3 . Since It is proved in [9] that, bn,k (q; x) → b∞,k (q; x) uniformly in x ∈ [0, 1) ¯as n → ∞. In the next lemma we give an ¯ ¯ x ¯ x ) − b∞,k (q; 1+x )¯ for x ∈ [0, ∞). estimate for ¯bn,k (q; 1+x Lemma 9 Let 0 < q < 1, k ≥ 0, n ≥ 1.For any x ∈

ISBN:978-988-18210-1-0

¡ ¢ ¡ ¢ [k] 1 − qk − 1 − qk = − 1 − qk n+1 [n + 1] 1−q 1 − qk = q n+1 ≤ q n+1 , 1 − q n+1 ¢ ¡ 0 ≤ 1 − 1 − q k = q k ≤ q n+1 , k > n + 1,

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Proceedings of the World Congress on Engineering 2009 Vol II WCE 2009, July 1 - 3, 2009, London, U.K.

we get |I1 |

(5)

¡ ¢ ≤ (1 + x) ω Φf, q n+1

n+1 X

bn+1,k (q;

k=0

x ) 1+x

¡ ¢ = (1 + x) ω Φf, q n+1 , |I3 |

(6) ∞ X

¡ ¢ ≤ (1 + x) ω Φf, q n+1 ¡

≤ (1 + x) ω Φf, q

n+1

¢

b∞,k (q;

k=n+2

x ) 1+x

.

ω (f, λt) ≤ (1 + λ) ω (f, t) , λ > 0,

[7] H. Oruc and G. M. Phillips, A generalization of the Bernstein polynomials, Proceedings of the Edinburgh Mathematical Society, 42, 403–413, 1999.

[9] S. Ostrovska, On the Lupa¸s q-analogue of the Bernstein operator, Rocky Mountain Journal of Mathematics, 36, 1615-1629, 2006.

and Lemma 9 we get |I2 | ≤ (1 + x)

[6] G.M. Phillips, Bernstein polynomials based on the qintegers, Annals of Numerical Mathematics, 4, 511– 518, 1997.

[8] N. I. Mahmudov and P. Sabancıgil, q-Parametric Bleimann Butzer and Hahn Operators, Journal of Inequalities and Applications, 2008 (2008), Article ID 816367, 15 pages doi:10.1155/2008/816367.

Finally we estimate I2 . Using the property,

n+1 X

[5] G. Bleimann, P. L. Butzer, and L. Hahn, A Bernstein-type operator approximating continuous functions on the semi-axis, Koninklijke Nederlandse Akademie vanWetenschappen. Indagationes Mathematicae, 42, 255–262, 1980.

¡ ¢ ω Φf, q k

[10] U. Abel and M. Ivan, Some identities for the operator of Bleimann, Butzer and Hahn involving divided differences, Calcolo, 36, 143-160, 1999.

k=0

¯ ¯ ¯ x ¯¯ x × ¯¯bn+1,k (q; ) − b∞,k (q; )¯ 1+x 1+x X¡ ¡ ¢ n+1 ¢ ≤ (1 + x) ω Φf, q n+1 1 + q k−n−1 k=0 ¯ ¯ ¯ x ¯¯ x ) − b∞,k (q; )¯ × ¯¯bn+1,k (q; 1+x 1+x

¡ ¢ ≤ 2 (1 + x) ω Φf, q n+1 ×

n+1 X

¯ ¯ q k ¯¯bn+1,k (q;

k=0 2

≤ 2 (1 + x)

1 q n+1

¯ x ¯¯ x ) − b∞,k (q; )¯ 1+x 1+x ¡ ¢ 1 ω Φf, q n+1 . 1−q

(7)

From (5), (6), and (7), we conclude the desired estimation.

References [1] A. Adell, Jose; F. Badia, German; de la Cal, Jesus On the iterates of some Bernstein-type operators. J. Math. Anal. Appl., 209, 529-541, 1997. [2] A. Lupa¸s, A q-analogue of the Bernstein operator, University of Cluj-Napoca, Seminar on numerical and statistical calculus, 9, 1987. [3] A. Aral, O. Do˘gru, Bleimann, Butzer, and Hahn operators based on the q-integers. J. Inequal. Appl. 2007, Art. ID 79410. [4] F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, De Gruyter Studies in Mathematics,Walter de Gruyter, Berlin, Germany, 1994.

ISBN:978-988-18210-1-0

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