Báo cáo hóa học: "Research Article q-Parametric Bleimann Butzer and Hahn Operators" pptx

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 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 ∗ 1x 0, ∞. We study some properties of q-BBH operators and establish the rate of convergence for q-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. Copyright q 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 fx : n  k0 f  k n  n k  x k n−k−1  s0 1 − q s 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 paper 2 and references therein. Nowadays, there are new studies on the q-parametric operators. Two parametric generalizations of q- Bernstein polynomials have been considered by Lewanowicz and Wo ´ zny cf. 3,ananalog 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 in 7. Also, some results on q-parametric Meyer-K ¨ onig and Zeller operators can be found in 8–11. In 12, Bleimann et al. introduced the following operators: H n fx 1 1  x n n  k0 f  k n − k  1  n k  x k ,x>0,n∈ N. 1.2 2 Journal of Inequalities and Applications There are several studies related to approximation properties of Bleimann, Butzer, and Hahn operators or, b riefly, 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 in 20. In this paper, we propose a different q-analog of the Bleimann, Butzer, and Hahn operators in C ∗ 1x 0, ∞. We use the connection between classical BBH and Bernstein operators suggested in 16 to define new q-BBH operators as follows: H n,q fx :Φ −1 B n1,q Φfx, 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 B n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 n in the space C ∗ 1x 0, ∞, the limit l f of fx/1  x as x→∞ has to appear in the definition of H n . Thus in C ∗ 1x 0, ∞ the classical BBH operator has to be modified as follows: H n fx 1 1  x n n  k0 f  k n − k  1  n k  x k  l f x n1 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. In Section 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 of q-BBH operators Before introducing the operators, we mention some basic definitions of q calculus. Let q>0. For any n ∈ N ∪{0},theq-integer nn q is defined by n : 1  q  ··· q n−1 , 0 : 0; 2.1 and the q-factorial n! n q !by n!:12 ···n, 0!: 1. 2.2 For integers 0 ≤ k ≤ n,theq-binomial is defined by  n k  : n! k!n − k! . 2.3 Also, we use the following standard notations: z; q 0 : 1, z; q n : n−1  j0 1 − q j z, z; q ∞ : ∞  j0 1 − q j z, p n,k q; x :  n k  x k n−k−1  s0 1 − q s x,p ∞k q; x : x k 1 − q k k! ∞  s0 1 − q s x. 2.4 N. I. Mahmudov and P. Sabancıgil 3 It 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  k0 p nk q; x ∞  k0 p ∞k q; x1. 2.5 Introduce the following spaces. B ρ 0, ∞{f : 0, ∞→R |∃M f > 0 such that |fx|≤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 0 ρ 0, ∞  f ∈ C ρ 0, ∞ | lim x→∞ fx ρx  0  . 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 |fx| ρx . 2.7 We introduce the following auxiliary operators. Firstly, let us denote ψy y 1 − y ,y∈ 0, 1,ψ −1 x x 1  x ,x∈ 0, ∞. 2.8 Secondly, let Φ : C ∗ ρ 0, ∞→C0, 1 be defined by Φfy : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ fψy ρψy , if y ∈ 0, 1, l f  lim x→∞ fx ρx , if y  1. 2.9 Then Φ is a positive linear isomorphism, with positive inverse Φ −1 : C0, 1→C ∗ ρ 0, ∞ defined by Φ −1 gxρxgψ −1 x,g∈ C0, 1,x∈ 0, ∞. 2.10 For f ∈ C0, 1,t>0, we define t he modulus of continuity ωf; t as follows: ωf; t : sup{|fx − fy| : |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 fx :Φ −1 B n1,q Φfx  ρx n  k0 fψk/n  1 ρψk/n  1 p n1,k q; ψ −1 x  l f ρxψ −1 x n1 ,n∈ N, 2.12 where p n1,k q; ψ −1 x :  n  1 k  ψ −1 x k n−k  s0 1 − q s ψ −1 x,k 0, 1, ,n. 2.13 4 Journal of Inequalities and Applications Note 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 n1 /1  x n .Thusiff ∈ C 0 1x 0, ∞ then H n,q fx : n  k0 f  k q k n − k  1  n k  qx 1  x  k n−k  s1  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 −1 x : ψ −1 x k 1 − q k k! ∞  s0 1 − q s ψ −1 x. 2.15 Definition 2.2. Let 0 <q<1. The linear operator defined on C ∗ ρ 0, ∞ given by H ∞,q fx : ρx ∞  k0 fψ1 − q k  ρψ1 − q k  p ∞k q; ψ −1 x 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 fx|≤ρxf ρ n  k0 p n1,k q; ψ −1 x  ρx|l f |ψ −1 x n1 ≤ ρxf ρ n  k0 p n1,k q; ψ −1 x  ρxf ρ ψ −1 x n1  ρxf ρ n1  k0 p n1,k q; ψ −1 x  ρx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  j0  m − 1 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 ρxρx,H n,q  ρt t 1  t  xρx x 1  x ,H n,q 1x1, H n,q  ρt  t 1  t  2  xρx  x 1  x  2  ρx x 1  x 2 1 n  1 . 2.20 N. I. Mahmudov and P. Sabancıgil 5 Proof. We prove only the recurrence formula, since the formulae 2.20 can easily be obtained by standard computations. Since l f  1forf  ρtt/1  t m , we have H n,q  ρt  t 1  t  m  x  ρx n  k0  k n  1  m p n1,k  q; ψ −1 x   ρx  x 1  x  n1  ρx n  k0  k n  1  m  n  1 k  x 1  x  k n−k  s0  1 − q s x 1  x   ρx  x 1  x  n1  ρx n  k0 k m−1 n  1 m−1  n k − 1  x 1  x  k n−k  s0  1 − q s x 1  x   ρx  x 1  x  n1  ρx n  k1 m−1  j0  m − 1 j  q j k − 1 j n  1 m−1 ×  n k − 1  x 1  x  k n−k  s0  1 − q s x 1  x   ρx  x 1  x  n1  1 n  1 m−1 x 1  x m−1  j0  m − 1 j  q j n j ×  H n−1,q  ρt  t 1  t  j  x − ρx  x 1  x  n   ρx  x 1  x  n1  1 n  1 m−1 x 1  x m−1  j0  m − 1 j  q j n j H n−1,q  ρt  t 1  t  j  x  ρx  x 1  x  n1  1 − 1 n  1 m−1 m−1  j0  m − 1 j  q j n j   1 n  1 m−1 x 1  x m−1  j0  m − 1 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 ∗ 1x 0, ∞ is convex and l f   f  n q n  − f  n  1 q n1  q n1 ≥ 0, 2.22 then its q-BBH operators are nonincreasing, in the sense that H n,q fx ≥ H n1,q fx,n 1, 2, , q∈ 0, 1,x∈ 0, ∞. 2.23 6 Journal of Inequalities and Applications Proof. We begin by writing H n,q fx − H n1,q fx  n  k0 f  k q k n − k  1  n k  qx 1  x  k n−k  s1  1 − q s x 1  x  − n1  k0 f  k q k n − k  2  n  1 k  qx 1  x  k n−k1  s1  1 − q s x 1  x   l f x n1 1  x n1 . 2.24 We now split the first of the above summations into two, writing  x 1  x  k n−k  s1  1 − q s x 1  x   ψ k  q n−k1 ψ k1 , 2.25 where ψ k   x 1  x  k n−k1  s1  1 − q s x 1  x  . 2.26 The resulting three summations may be combined to give H n,q fx − H n1,q fx  n  k0 f  k q k n − k  1  n k  q k ψ k  q n−k1 ψ k1  − n1  k0 f  k q k n − k  2  n  1 k  q k ψ k  l f  x 1  x  n1  n  k0 f  k q k n − k  1  n k  q k ψ k  n1  k1 f  k − 1 q k−1 n − k  2  n k − 1  q n1 ψ k − n1  k0 f  k q k n − k  2  n  1 k  q k ψ k  l f  x 1  x  n1  n  k1  n  1 k  a k q k ψ k   f  n q n  − f  n  1 q n1  q n1  x 1  x  n1  l f  x 1  x  n1 , 2.27 where a k  n − k  1 n  1 f  k q k n − k  1   q n−k1 k n  1 f  k − 1 q k−1 n − k  2  − f  k 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,q it suffices to show nonnegativity of a k , 0 ≤ k ≤ n. Let us write α  n − k  1 n  1 ,x 1  k q k n − k  1 ,x 2  k − 1 q k n − k  2 . 2.29 N. I. Mahmudov and P. Sabancıgil 7 Then it follows that 1 − α  q n−k1 k n  1 , αx 1 1 − αx 2  k q k n  1  1  q n−k2 k − 1 n − k  2   k q k n  1  1 − q n−k2  q n−k2 1 − q k−1  1 − q n−k2   k q k n − k  2 , 2.30 and we see immediately that a k  αfx 1 1 − αfx 2  − fαx 1 1 − αx 2  ≥ 0, 2.31 and so H n,q fx − H n1,q fx ≥ 0. Remark 2.6. It is easily seen that l f   f  n q n  − f  n  1 q n1  q n1 n  2  1 n  2 Φf1 qn  1 n  2 Φf  n n  1  − Φf  n  1 n  2  . 2.32 The condition 2.22 follows from convexity of Φf. On the other hand, Φf is convex if f is convex and nonincreasing, see 16. 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 ≤ CqωΦf, q n1 , 3.1 where Cq4/q1 − q ln1/1 − q  2. Proof. For all x ∈ 0, ∞, by the definitions of H n,q fx and H ∞,q fx, we have that H n,q f − H ∞,q fρx n  k0 fψk/n  1 ρψk/n  1 p n1,k q; ψ −1 x  l f ρx  x 1  x  n1 − ρx ∞  k0 fψ1 − q k  ρψ1 − q k  p ∞k q; ψ −1 x  ρx n1  k0  Φf  k n  1  − Φf1 − q k   p n1,k q; ψ −1 x  ρx n1  k0 Φf1 − q k  − Φf1p n1,k q; ψ −1 x − p ∞k q; ψ −1 x − ρx ∞  kn2 Φf1 − q k  − Φf1p ∞k q; ψ −1 x : I 1  I 2  I 3 . 3.2 8 Journal of Inequalities and Applications First, we estimate I 1 ,I 3 . By using the following inequalities: 0 ≤ k n  1 − 1 − q k  1 − q k 1 − q n1 − 1 − q k  q n1 1 − q k  1 − q n1 ≤ q n1 , 0 ≤ 1 − 1 − q k q k ≤ q n1 ,k≥ n  2, 3.3 we get |I 1 |≤ρxωΦf, q n1  n1  k0 p n1,k q; ψ −1 x  ρxωΦf, q n1 , |I 3 |≤ρx ∞  kn2 ωΦf, q k p ∞k q; ψ −1 x ≤ ρxωΦf, q n1 . 3.4 Next, we estimate I 2 . Using the well-known property of modulus of continuity ωg,λt ≤ 1  λωg,t,λ>0, 3.5 we get |I 2 |≤ρx n1  k0 ωΦf, q k |p n1,k q; ψ −1 x − p ∞k q; ψ −1 x| ≤ ρxωΦf, q n1  n1  k0 1  q k−n−1 |p n1,k q; ψ −1 x − p ∞k q; ψ −1 x| ≤ 2ρxωΦf, q n1  1 q n1 n1  k0 q k |p n1,k q; ψ −1 x − p ∞k q; ψ −1 x| : ρx 2 q n1 ωΦf, q n1 J n1 ψ −1 x, 3.6 where J n1 ψ −1 x  n1  k0 q k |p n1,k q; ψ −1 x − p ∞k q; ψ −1 x|. 3.7 Now, using the estimation 2.9 from 21, we have J n1 ψ −1 x ≤ q n1 q1 − q ln 1 1 − q n1  k0 p n1,k q; ψ −1 x  p ∞k q; ψ −1 x ≤ 2q n1 q1 − q ln 1 1 − q . 3.8 From 3.6 and 3.8, it follows that |I 2 |≤ρx 4 q1 − q ln 1 1 − q ωΦf, q n1 . 3.9 From 3.4,and3.9, we obtain the desired estimation. N. I. Mahmudov and P. Sabancıgil 9 Theorem 3.2. Let 0 <q<1 be fixed and let f ∈ C ∗ 1x 0, ∞.ThenH ∞,q fxfx ∀x ∈ 0, ∞ if and only if f is linear. Proof. By definition of H ∞,q we have H ∞,q fxΦ −1 B ∞,q Φfx. 3.10 Assume that H ∞,q fxfx. Then B ∞,q ΦfxΦfx.From22, we know that B ∞,q gg if and only if g is linear. So B ∞,q ΦfxΦfx if and only if Φfx 1 − xfx/1 − x  Ax  B. It follows that fx1  xAx/1  x  BA  Bx  B. The converse can be shown in a similar way. Remark 3.3. Let 0 <q<1 be fixed and let f ∈ C ∗ 1x 0, ∞. Then the sequence {H n,q fx} 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 fx − fx|≤2ω  Φf,  λ n x  , 3.11 where λ n xx/1  x 2 1/n  1 q n . Proof. Positivity of B n1,q n implies that for any g ∈ C0, 1 |B n1,q n gx − gx|≤B n1,q n |gt − gx|x. 3.12 On the other hand, |Φft − Φfx|≤ωΦf, |t − x| ≤ ωΦf, δ  1  1 δ |t − x|  ,δ>0. 3.13 This inequality and 3.12 imply that |B n1,q n Φfx − Φfx|≤ωΦf, δ  1  1 δ B n1,q n |t − x|x  , |Φ −1 B n1,q n Φfx − Φ −1 Φfx| ≤ ωΦf, δ  Φ −1 1 1 δ Φ −1 B n1,q n |t − x|x  ≤ ρxωΦf, δ  1  1 δ B n1,q n  |t − ψ −1 x| 2  ψ −1 x 1/2   ρxωΦf, δ  1  1 δ  x 1  x  2  x 1  x 2 1 n  1 q n −  x 1  x  2  1/2   ρxωΦf, δ  1  1 δ  x 1  x 2 1 n  1 q n  1/2  , 3.14 by choosing δ   λ n x, we obtain desired result. 10 Journal of Inequalities and Applications Corollary 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 fx − fx ρ  0. 3.15 Next, we study Voronovskaja-type formulas for the q-BBH operators. For the q- Bernstein operators, it is proved in 23  that for any f ∈ C 1 0, 1, lim n→∞ n q n B n,q fx − B ∞,q fx  L q f, x3.16 uniformly in x ∈ 0, 1, where L q f, x : ⎧ ⎪ ⎨ ⎪ ⎩ ∞  k0 k  f  1 − q k  − f1 − q k  − f1 − q k−1  1 − q k  − 1 − q k−1   x k q; q k x; 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 L q f, x for q-BBH operators V q f, x :Φ −1 L q Φfx  x 1  x ,q  ∞ ∞  k0 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−1 f1 − q k−1 /q k−1  1 − q k  − 1 − q k−1   × 1 q, q k x k 1  x k−1   x 1  x ; q  ∞ ∞  k0 k  f   1 − q k q k  1 q k − q k−1 f1 − q k /q k  − f1 − q k−1 /q k−1  q k−1 − q k  × 1 q; q k x k 1  x k−1 . 3.18 Theorem 3.6. Let 0 <q<1,f∈ C ∗ 1x 0, ∞ ∩ C 1 0, ∞, and Φf is differentiable at x  1.Then lim n→∞ n  1 q n1 H n,q fx − H ∞,q fx  V q f, x, 3.19 in C ∗ 1x 0, ∞. Proof. We estimate the difference Δx :     n  1 q n1 H n,q fx − H ∞,q fx − V q f, x          n  1 q n1 Φ −1 B n1,q Φfx − Φ −1 B ∞,q Φfx − Φ −1 L q Φfx           Φ −1  n  1 q n1 B n1,q − B ∞,q  − L q  Φ  fx     1  x      n  1 q n1 B n1,q − B ∞,q  − L q  Φfψ −1 x     . 3.20 [...]... operators of Bleimann, Butzer, and Hahn, ” Journal of Approximation Theory, vol 97, no 1, pp 181–198, 1999 18 R A Khan, “A note on a Bernstein-type operator of Bleimann, Butzer, and Hahn, ” Journal of Approximation Theory, vol 53, no 3, pp 295–303, 1988 19 A Aral and O Dogru, Bleimann, Butzer, and Hahn operators based on the q-integers,” Journal of ˘ Inequalities and Applications, vol 2007, Article ID... generalization of Bleimann, Butzer and Hahn operators,” Mathematica Pannonica, vol 9, no 2, pp 165–171, 1998 15 O Agratini, “A class of Bleimann, Butzer and Hahn type operators,” Analele Universit˘ tii Din a¸ Timisoara, vol 34, no 2, pp 173–180, 1996 ¸ 16 J A Adell, F G Bad´a, and J de la Cal, “On the iterates of some Bernstein-type operators,” Journal of ı Mathematical Analysis and Applications, vol... 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Inequalities and Applications 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 ∈ C1 0, ∞ ∩ C1 0, ∞ Then x Hn,q f x − H∞,q f x o qn 1 x 1 3.28 if and only if Vq f, x ≡ 0, and this is equivalent to f 1 − qk qk 1 1 − k−1 k q q 1 − qk qk f 1 − qk−1 qk−1 −f , k 1, 2, 3.29 ∗ Theorem 3.10 Let 0 < q < 1 and f ∈... 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C1 0, ∞ and let {qn } be a sequence chosen so that the sequence n εn 1 qn 2 qn ··· n−1 qn −1 3.32 converges to zero from above faster than {1/3n } Then lim Hn,qn f x n→∞ f x 3.33 uniformly on any 0, A ⊂ 0, ∞ Proof By definition Hn,qn f x 1 x Bn 1,qn Φ f x 1 x 3.34 N I Mahmudov and P Sabancıgil 13 Since Hn,qn f x is a composition of differentiable functions, it is differentiable at any x ∈ 0, A and d Hn,qn . Inequalities and Applications Note 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. 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. of Inequalities and Applications Volume 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