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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 158219, 7 pages doi:10.1155/2011/158219 Research Article L p 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 L p approximation by the operator is established by using a decomposition technique. 1. Introduction Let P n,k x nk−1 k x k 1 x −n−k , x ∈ 0, ∞, n ∈ N. The Baskakov operator defined by B n,1 f, x ∞ k0 P n,k x f k n 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 Baskakov- Durrmeyer operator defined by see 3, 4 M n,1 f, x ∞ k0 P n,k x n − 1 ∞ 0 P n,k t f t dt, x ∈ 0, ∞ , 1.2 where f ∈ L p 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 ω 2 ϕ f, t p sup 0<h≤t f · 2hϕ · − 2f · hϕ · f · p ,ϕ x x 1 x . 1.3 2 Journal of Inequalities and Applications More precisely, for any function defined on L p 0, ∞1 ≤ p<∞, there is a constant such that M n,1 f − f p ≤ const. ω 2 ϕ f, 1 √ n p 1 n f p , 1.4 ω 2 ϕ f, t p O t 2α ⇐⇒ M n,1 f − f p O n −α , 1.5 where 0 <α<1. Let T ⊂ R d d ∈ N, which is defined by T : T d : { x : x 1 ,x 2 , ,x d :0≤ x i < ∞, 1 ≤ i ≤ d } . 1.6 Here and in the following, we will use the standard notations x : x 1 ,x 2 , ,x d , k : k 1 ,k 2 , ,k d ∈ N d 0 , x k : x k 1 1 x k 2 2 ···x k d d , k! k 1 !k 2 ! ···k d !, | x | : d i1 x i , | k | : d i1 k i , n k : n! k! n − | k | ! , ∞ k0 : ∞ k 1 0 ∞ k 2 0 ··· ∞ k d 0 . 1.7 By means of the notations, for a function f defined on T the multivariate Baskakov operator is defined as see 5 B n,d f, x : ∞ k0 f k n P n,k x , 1.8 where P n,k x n | k | − 1 k x k 1 | x | −n−|k| . 1.9 Naturally, we can modify the multivariate Baskakov operator as multivariate Baskakov-Durrmeyer operator M n,d f : M n,d f, x : ∞ k0 P n,k x φ n,k,d f ,f∈ L p T , 1.10 where φ n,k,d f : T P n,k u f u du T P n,k u du n − 1 n − 2 ··· n − d T P n,k u f u du. 1.11 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 L p T. 2. Main Result We will show a direct inequality of L p 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 ϕ i x x i 1 | x | , 1 ≤ i ≤ d. 2.1 Let D r i ∂ r ∂x r i ,r∈ N,D k D k 1 1 D k 2 2 ···D k d d , k ∈ N d 0 2.2 denote the differential operators. For 1 ≤ p<∞, we define the weighted Sobolev space as follows: W r,p ϕ T f ∈ L p T : D k f ∈ L loc ˙ T ,ϕ r i D r i f ∈ L p T , 2.3 where |k|≤r, k ∈ N d 0 ,and ˙ T denotes the interior of T. The Peetre K-functional on L p T 1 ≤ p<∞, are defined by K r ϕ f, t r p inf f − g p t r d i1 ϕ r i D r i g p ,t>0, 2.4 where the infimum is taken over all g ∈ W r,p ϕ T. For any vector e in R d , we write the rth forward difference of a function f in the direction of e as Δ r he f x ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r i0 ⎛ ⎝ r i ⎞ ⎠ −1 i f x ihe , x, x rhe ∈ T, 0, otherwise. 2.5 We then can define the modulus of smoothness of f ∈ L p T1 ≤ p<∞,as ω r ϕ f, t p sup 0<h≤t d i1 Δ r h ϕ i e i f p , 2.6 where e i denotes the unit vector in R d ,thatis,itsith component is 1 and the others are 0. In 5, the following result has been proved. 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 ∈ L p T, 1 ≤ p<∞ 1 const. ω r ϕ f, t p ≤ K r ϕ f, t r p ≤ const.ω r ϕ f, t p . 2.7 Now we state the main result of this paper. Theorem 2.2. If f ∈ L p T, 1 ≤ p<∞, then there is a positive constant independent of n and f such that M n,d f − f p ≤ const. ω 2 ϕ f, 1 √ n p 1 n f p . 2.8 Proof. Our proof is based on an induction argument for the dimension d. We will also use a decomposition method of the operator M n,d f. We report the detailed proof only for two dimensions. The higher dimensional cases are similar. Our proof depends on Lemma 2.1 and the following estimates: M n,2 f − f p ≤ const. ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ f p ,f∈ L p T , 1 n 2 i1 ϕ 2 i D 2 i f p f p ,f∈ W 2,p ϕ T . 2.9 The first estimate is evident as the M n,d f are positive and linear contractions on L p T1 ≤ p<∞. We can demonstrate the second estimate by reducing it to the one dimensional inequality M n,1 f − f p ≤ const. n ϕ 2 f p f p , 2.10 which has been proved in 3 Now we give the following decomposition formula: M n,2 f, x ∞ k 1 0 ∞ k 2 0 P n,k 1 x 1 P nk 1 ,k 2 x 2 1 x 1 n − 1 n − 2 × ∞ 0 P n,k 1 u 1 P nk 1 ,k 2 u 2 1 u 1 f u 1 ,u 2 du 1 du 2 ∞ k 1 0 P n,k 1 x 1 n − 2 ∞ 0 P n−1,k 1 u 1 ∞ k 2 0 P nk 1 ,k 2 x 2 1 x 1 × n k 1 − 1 ∞ 0 P nk 1 ,k 2 t f u 1 , 1 u 1 t dt du 1 ∞ k 1 0 P n,k 1 x 1 n − 2 ∞ 0 P n−1,k 1 u 1 M nk 1 ,1 g u 1 ,z du 1 , 2.11 Journal of Inequalities and Applications 5 where g u 1 t f u 1 , 1 u 1 t , 0 ≤ t<∞,z x 2 1 x 1 , 2.12 which can be checked directly and will play an important role in the following proof. From the decomposition formula, it follows that M n,2 f, x − f x ∞ k 1 0 P n,k 1 x 1 n − 2 × ∞ 0 P n−1,k 1 u 1 M nk 1 ,1 g u 1 ,z − g u 1 z du 1 M ∗ n,1 h · ,x 1 − h x 1 : J L, 2.13 where h u 1 : h u 1 , x : f u 1 , 1 u 1 x 2 1 x 1 , 0 ≤ u 1 < ∞, M ∗ n,1 g,y ∞ l0 P n,l y n − 2 ∞ 0 P n−1,l t g t dt. 2.14 Then by the Jensen’s inequality, we have J p p ≤ T ∞ k 1 0 P n,k 1 x 1 n − 2 ∞ 0 P n−1,k 1 u 1 M nk 1 ,1 g u 1 ,z − g u 1 zdu 1 p dx ≤ T ∞ k 1 0 P n,k 1 x 1 n − 2 ∞ 0 P n−1,k 1 u 1 M nk 1 ,1 g u 1 ,z − g u 1 z p du 1 dx ∞ 0 ∞ k 1 0 P n,k 1 x 1 1 x 1 dx 1 n − 2 ∞ 0 P n−1,k 1 u 1 × M nk 1 ,1 g u 1 ,z − g u 1 z p dzdu 1 ≤ ∞ k 1 0 n k 1 − 1 n − 1 ∞ 0 P n−1,k 1 u 1 ∞ 0 M nk 1 ,1 g u 1 ,z − g u 1 z p dzdu 1 ≤ const. ∞ k 1 0 n k 1 − 1 n − 1 ∞ 0 P n−1,k 1 u 1 1 n k 1 p ϕ 2 g u 1 p p g u 1 p p du 1 . 2.15 However, by definition, one also has ϕ 2 t g u 1 t t 1 t 1 u 1 2 D 2 2 f u 1 , 1 u 1 t ϕ 2 2 D 2 2 f u 1 , 1 u 1 t . 2.16 6 Journal of Inequalities and Applications Therefore, J p p ≤ const. ∞ k 1 0 n k 1 − 1 n − 1 n k 1 p ∞ 0 P n−1,k 1 u 1 × ϕ 2 2 D 2 2 f u 1 , 1 u 1 t p f u 1 , 1 u 1 t p dt du 1 const. ∞ k 1 0 n k 1 − 1 n − 1 n k 1 p ∞ 0 1 1 u 1 P n−1,k 1 u 1 × ∞ 0 ϕ 2 2 u 1 ,u 2 D 2 2 f u 1 ,u 2 p f u 1 ,u 2 p du 1 du 2 ≤ const. n p ∞ k 1 0 ∞ 0 P n,k 1 u 1 ∞ 0 ϕ 2 2 u 1 ,u 2 D 2 2 f u 1 ,u 2 p f u 1 ,u 2 p du 1 du 2 const. n p ϕ 2 2 D 2 2 f p p f p p . 2.17 To estimate the second term L, we use a similar method as to estimate 2.10see 3 and can get L p ≤ const. n ϕ 2 h p h p . 2.18 Denoting ϕ 12 xϕ 21 x : √ x 1 x 2 , D 2 12 : ∂ 2 /∂x 1 ∂x 2 ,andD 2 21 : ∂ 2 /∂x 2 ∂x 1 ,we have ϕ 2 s h s s 1 s D 2 1 f x 2 1 x 1 D 2 12 f x 2 1 x 1 D 2 21 f x 2 2 1 x 1 2 D 2 22 f × s, 1 s x 2 1 x 1 1 x 1 1 x 1 x 2 ϕ 2 1 D 2 1 f ϕ 2 12 D 2 12 f ϕ 2 21 D 2 21 f s 1 s x 2 1 x 1 x 2 ϕ 2 2 D 2 2 f s, 1 s x 2 1 x 1 . 2.19 Recalling that ϕ 12 x is no bigger than ϕ 1 x or ϕ 2 x, and the f act D 2 12 f x ≤ sup D 2 1 f x , D 2 2 f x 2.20 proved in 6see 6, Lemma 2.1,weobtain ϕ 2 h p ≤ const. 2 i1 ϕ 2 i D 2 i f p , 2.21 Journal of Inequalities and Applications 7 and hence L p ≤ const. n 2 i1 ϕ 2 i D 2 i f p f p . 2.22 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. . of Inequalities and Applications Volume 2011, Article ID 158219, 7 pages doi:10.1155/2011/158219 Research Article L p Approximation by Multivariate Baskakov-Durrmeyer Operator Feilong Cao and Yongfeng. L p 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. ∞ , 1.2 where f ∈ L p 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
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