Approximation by Multivariate Baskakov-Durrmeyer Operator
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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, [email protected] 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 Let Pn,k x
nk−1 k
xk 1 x−n−k , x ∈ 0, ∞, n ∈ N. The Baskakov operator defined by Bn,1
∞ k Pn,k xf f, x n k0
1.1
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 BaskakovDurrmeyer operator defined by see 3, 4 ∞ Pn,k xn − 1 Mn,1 f, x k0
∞
Pn,k tftdt,
x ∈ 0, ∞,
0
1.2
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 ωϕ2 f, t p sup f · 2hϕ· − 2f · hϕ· f·p , 0 0,
2.4
i1
r,p
where the infimum is taken over all g ∈ Wϕ T . For any vector e in Rd , we write the rth forward difference of a function f in the direction of e as ⎧ ⎛ ⎞ r r ⎪ ⎪ ⎨ ⎝ ⎠−1i fx ihe, x, x rhe ∈ T, r Δhe fx i0 i ⎪ ⎪ ⎩ 0, otherwise.
2.5
We then can define the modulus of smoothness of f ∈ Lp T 1 ≤ p < ∞, as d Δr ϕi ei f , ωϕr f, t p sup h p
2.6
0
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