Báo cáo hóa học: " Research Article Lp Approximation by Multivariate Baskakov-Durrmeyer Operator"

Chia sẻ: Nguyen Minh Thang | Ngày: | Loại File: PDF | Số trang:7

Thêm vào BST

Báo xấu

36
lượt xem 6
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Lp Approximation by Multivariate Baskakov-Durrmeyer Operator

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Báo cáo hóa học: " Research Article Lp Approximation by Multivariate Baskakov-Durrmeyer Operator"

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 158219, 7 pages doi:10.1155/2011/158219 Research Article Lp Approximation by Multivariate Baskakov-Durrmeyer Operator Feilong Cao and Yongfeng An Department of Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang Province, China Correspondence should be addressed to Feilong Cao, feilongcao@gmail.com Received 14 November 2010; Accepted 17 January 2011 Academic Editor: Jewgeni Dshalalow Copyright q 2011 F. Cao and Y. An. 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. The main aim of this paper is to introduce and study multivariate Baskakov-Durrmeyer operator, which is nontensor product generalization of the one variable. As a main result, the strong direct inequality of Lp approximation by the operator is established by using a decomposition technique. 1. Introduction −n−k n k −1 , x ∈ 0, ∞ , n ∈ N. The Baskakov operator deﬁned by xk 1 Let Pn,k x x k ∞ k Bn,1 f, x Pn,k x f 1.1 n k0 was introduced by Baskakov 1 and can be used to approximate a function f deﬁned on 0, ∞ . It is the prototype of the Baskakov-Kantorovich operator see 2 and the Baskakov- Durrmeyer operator deﬁned by see 3, 4 ∞ ∞ Pn,k x n − 1 x ∈ 0, ∞ , Mn,1 f, x Pn,k t f t dt, 1.2 0 k0 where f ∈ Lp 0, ∞ 1 ≤ p < ∞ . By now, the approximation behavior of the Baskakov-Durrmeyer operator is well understood. It is characterized by the second-order Ditzian-Totik modulus see 3 sup f · 2hϕ · − 2f · hϕ · f· 2 ωϕ f, t , ϕx x1 x. 1.3 p p 0
2 Journal of Inequalities and Applications More precisely, for any function deﬁned on Lp 0, ∞ 1 ≤ p < ∞ , there is a constant such that 1 1 Mn,1 f − f ≤ const. ωϕ f, √ 2 f , 1.4 p p n n p O n−α , O t2α ⇐⇒ Mn,1 f − f 2 ωϕ f, t 1.5 p p where 0 < α < 1. Let T ⊂ Rd d ∈ N , which is deﬁned by x1 , x2 , . . . , xd : 0 ≤ xi < ∞, 1 ≤ i ≤ d}. T : Td : {x : 1.6 Here and in the following, we will use the standard notations k1 , k2 , . . . , kd ∈ Nd , x: x1 , x2 , . . . , xd , k: 0 d d k kk xk : x1 1 x2 2 · · · xdd , k1 !k2 ! · · · kd !, |x| : |k| : k! xi , ki , 1.7 i1 i1 ∞ ∞ ∞ ∞ n n! ··· : , : . k! n − |k| ! k k0 k1 0 k2 0 kd 0 By means of the notations, for a function f deﬁned on T the multivariate Baskakov operator is deﬁned as see 5 ∞ k Bn,d f, x : f Pn,k x , 1.8 n k0 where |k| − 1 n −n−|k| |x| xk 1 Pn,k x . 1.9 k Naturally, we can modify the multivariate Baskakov operator as multivariate Baskakov-Durrmeyer operator ∞ f ∈ Lp T , Mn,d f : Mn,d f, x : Pn,k x φn,k,d f , 1.10 k0 where Pn,k u f u du T n − 1 n − 2 ··· n − d φn,k,d f : Pn,k u f u du. 1.11 Pn,k u du T T
Journal of Inequalities and Applications 3 It is a multivariate generalization of the univariate Baskakov-Durrmeyer operators given in 1.2 and can be considered as a tool to approximate the function in Lp T . 2. Main Result We will show a direct inequality of Lp approximation by the Baskakov-Durrmeyer operator given in 1.10 . By means of K-functional and modulus of smoothness deﬁned in 5 , we will extend 1.4 to the case of higher dimension by using a decomposition technique. Fox x ∈ T , we deﬁne the weight functions 1 ≤ i ≤ d. |x| , 2.1 ϕi x xi 1 Let ∂r k k k r ∈ N, D1 1 D2 2 · · · Ddd , k ∈ Nd Dir Dk , 2.2 ∂xir 0 denote the diﬀerential operators. For 1 ≤ p < ∞, we deﬁne the weighted Sobolev space as follows: r,p f ∈ Lp T : Dk f ∈ Lloc T , ϕr Dir f ∈ Lp T ˙ Wϕ T , 2.3 i where |k| ≤ r , k ∈ Nd , and T denotes the interior of T . The Peetre K -functional on Lp T ˙ 0 1 ≤ p < ∞ , are deﬁned by d f −g Kϕ f, tr r tr ϕr Dir g 2.4 inf , t > 0, i p p p i1 r,p where the inﬁmum is taken over all g ∈ Wϕ T . For any vector e in Rd , we write the r th forward diﬀerence of a function f in the direction of e as ⎛⎞ ⎧ ⎪ r r ⎪ ⎨ ⎝ ⎠ −1 i f x r he ∈ T, ihe , x, x Δr e f x 2.5 ⎪i 0 i h ⎪ ⎩ 0, otherwise. We then can deﬁne the modulus of smoothness of f ∈ Lp T 1 ≤ p < ∞ , as d Δr ϕi ei f r 2.6 ωϕ f, t sup , h p p 0
4 Journal of Inequalities and Applications Lemma 2.1. There exists a positive constant, dependent only on p and r , such that for any f ∈ Lp T , 1≤p
Journal of Inequalities and Applications 5 where x2 0 ≤ t < ∞, gu1 t f u1 , 1 u1 t , z , 2.12 1 x1 which can be checked directly and will play an important role in the following proof. From the decomposition formula, it follows that ∞ Mn,2 f, x − f x Pn,k1 x1 n − 2 k1 0 ∞ ∗ × gu1 , z − gu1 z du1 Mn,1 h · , x1 − h x1 Pn−1,k1 u1 Mn k1 ,1 0 :J L, 2.13 where x2 0 ≤ u1 < ∞ , h u1 : h u1 , x : f u1 , 1 u1 , 1 x1 2.14 ∞ ∞ ∗ Pn,l y n − 2 Mn,1 g, y Pn−1,l t g t dt. 0 l0 Then by the Jensen’s inequality, we have ∞ ∞ p p ≤ n−2 gu1 , z − gu1 z du1 dx J Pn,k1 x1 Pn−1,k1 u1 Mn k1 ,1 p T k1 0 0 ∞ ∞ p ≤ Pn,k1 x1 n − 2 gu1 , z − gu1 z Pn−1,k1 u1 Mn du1 dx k1 ,1 T k1 0 0 ∞∞ ∞ x1 dx1 n − 2 Pn,k1 x1 1 Pn−1,k1 u1 0 k1 0 0 2.15 p × gu1 , z − gu1 z Mn dzdu1 k1 ,1 ∞ ∞ ∞ k1 − 1 n p ≤ gu1 , z − gu1 z Pn−1,k1 u1 Mn dzdu1 k1 ,1 n−1 0 0 k1 0 ∞ ∞ p k1 − 1 n 1 p p ≤ const. ϕ2 gu1 Pn−1,k1 u1 gu1 du1 . n−1 p n k1 p 0 k1 0 However, by deﬁnition, one also has u 1 2 D2 f u 1 , 1 ϕ2 t gu1 t 2 ϕ2 D2 f 2 t1 t1 u1 t u1 , 1 u1 t . 2.16 2
6 Journal of Inequalities and Applications Therefore, ∞ ∞ n k1 − 1 p ≤ const. J Pn−1,k1 u1 p p n − 1 n k1 0 k1 0 p p × ϕ2 D2 f 2 u1 , 1 u1 t f u1 , 1 u1 t dt du1 2 ∞ ∞ n k1 − 1 1 const. Pn−1,k1 u1 p n − 1 n k1 1 u1 0 k1 0 2.17 ∞ p p × ϕ2 2 u1 , u2 D2 f u1 , u2 f u1 , u2 du1 du2 2 0 ∞ ∞ const. ∞ p p ≤ ϕ2 u1 , u2 D2 f u1 , u2 2 Pn,k1 u1 f u1 , u2 du1 du2 2 np k 0 0 0 1 const. p p ϕ2 D2 f 2 f . 2 p np p To estimate the second term L, we use a similar method as to estimate 2.10 see 3 and can get const. ≤ ϕ2 h L h . 2.18 p p n p √ 2 2 x1 x2 , D12 : ∂2 / ∂x1 ∂x2 , and D21 : ∂2 / ∂x2 ∂x1 , we Denoting ϕ12 x ϕ21 x : have ϕ2 s h s 2 x2 x2 x2 x2 × s, 1 2 D2 f D2 f 2 s1 s D1 f D22 f s 1 x1 12 1 x1 21 2 1 x1 1 x1 1 x1 s x2 x2 ϕ2 D1 f 2 ϕ2 D12 f 2 ϕ2 D21 f 2 ϕ2 D2 f s, 1 s . x1 x2 2 2 1 12 21 1 x1 x2 1 s1 1 x1 2.19 Recalling that ϕ12 x is no bigger than ϕ1 x or ϕ2 x , and the fact ≤ sup D1 f x , D2 f x 2 2 2 D12 f x 2.20 proved in 6 see 6, Lemma 2.1 , we obtain 2 ≤ const. ϕ2 h ϕ2 Di2 f 2.21 , i p p i1
Journal of Inequalities and Applications 7 and hence 2 const. ≤ ϕ2 Di2 f 2.22 L f . p i p n p i1 The second inequality of 2.9 has thus been established, and the proof of Theorem 2.2 is ﬁnished. Acknowledgment The research was supported by the National Natural Science Foundation of China no. 90818020 . References 1 V. A. Baskakov, “An instance of a sequence of linear positive operators in the space of continuous functions,” Doklady Akademii Nauk SSSR, vol. 113, pp. 249–251, 1957. 2 Z. Ditzian and V. Totik, Moduli of Smoothness, vol. 9 of Springer Series in Computational Mathematics, Springer, New York, NY, USA, 1987. 3 M. Heilmann, “Direct and converse results for operators of Baskakov-Durrmeyer type,” Approximation Theory and its Applications, vol. 5, no. 1, pp. 105–127, 1989. 4 A. Sahai and G. Prasad, “On simultaneous approximation by modiﬁed Lupas operators,” Journal of Approximation Theory, vol. 45, no. 2, pp. 122–128, 1985. 5 F. Cao, C. Ding, and Z. Xu, “On multivariate Baskakov operator,” Journal of Mathematical Analysis and Applications, vol. 307, no. 1, pp. 274–291, 2005. 6 W. Chen and Z. Ditzian, “Mixed and directional derivatives,” Proceedings of the American Mathematical Society, vol. 108, no. 1, pp. 177–185, 1990.