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

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
Nội dung trích xuất từ tài liệu:
Báo cáo hóa học: " Research Article Lp Approximation by Multivariate Baskakov-Durrmeyer Operator"Hindawi Publishing CorporationJournal of Inequalities and ApplicationsVolume 2011, Article ID 158219, 7 pagesdoi:10.1155/2011/158219Research ArticleLp Approximation by MultivariateBaskakov-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 1Let Pn,k x x k ∞ k Bn,1 f, x Pn,k x f 1.1 n k0was 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 k0where f ∈ Lp 0, ∞ 1 ≤ p < ∞ . By now, the approximation behavior of the Baskakov-Durrmeyer operator is wellunderstood. 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 02 Journal of Inequalities and ApplicationsMore 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 pwhere 0 < α < 1. Let T ⊂ Rd d ∈ N , which is defined by x1 , x2 , . . . , xd : 0 ≤ xi < ∞, 1 ≤ i ≤ d}. T : Td : {x : 1.6Here and in the following, we will use the standard notations k1 , k2 , . . ...