Volume 2011, Article ID 158219, 7 pagesdoi:10.1155/2011/158219 Research Article Baskakov-Durrmeyer Operator Feilong Cao and Yongfeng An Department of Mathematics, China Jiliang Universit
Trang 1Volume 2011, Article ID 158219, 7 pages
doi:10.1155/2011/158219
Research Article
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
Copyrightq 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 papproximation by the operator is established by using a decomposition technique
1 Introduction
Let P n,k x n k−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 xf
k n
1.1
was introduced by Baskakov1 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 bysee 3,4
M n,1
f, x
∞
k0
P n,k xn − 1
∞
0
P n,k tftdt, 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 modulussee 3
ω ϕ2
f, t
p sup
0<h≤t
f
· 2hϕ·− 2f· hϕ· f·
p , ϕ x x 1 x. 1.3
Trang 2More 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 ϕ2
f,√1
n
p
1
nf
p
, 1.4
ω2
ϕ
f, t
p O t 2α ⇐⇒M n,1 f − f
p On −α
, 1.5
where 0 < α < 1.
Let T ⊂ Rd d ∈ N, which is defined by
T : T d: {x : x1 , x2, , x d : 0 ≤ x i < ∞, 1 ≤ i ≤ d}. 1.6
Here and in the following, we will use the standard notations
x :x1 , x2, , x d , k :k1 , k2, , k d ∈ Nd
0,
x k : xk1
1 x k2
2 · · · x k d
d , k! k1!k2!· · · k d !, |x| :d
i1
x i , |k| :d
i1
k i , n
k
: n!
k!n − |k|! ,
∞
k0
: ∞
k1 0
∞
k2 0
· · ·∞
k d0
.
1.7
By means of the notations, for a function f defined on T the multivariate Baskakov operator
is defined assee 5
B n,d
f, x :∞
k0
f
k
n
P n,k x, 1.8
where
P n,kx 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 xφ n,k,d
f
, f ∈ L p T, 1.10
where
φ n,k,d
f
:
T P n,k ufudu
T P n,kudu n − 1n − 2 · · · n − d
T
P n,k ufudu. 1.11
Trang 3It 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 papproximation by the Baskakov-Durrmeyer operator given in1.10 By means of K-functional and modulus of smoothness defined in 5, we will extend1.4 to the case of higher dimension by using a decomposition technique
Fox x ∈ T, we define the weight functions
ϕ ix x i 1 |x|, 1 ≤ i ≤ d. 2.1 Let
D i r ∂ r
∂x r i
, r ∈ N, Dk D k1
1 D k2
2 · · · D k d
denote the differential operators For 1 ≤ p < ∞, we define the weighted Sobolev space as follows:
W ϕ r,p T f ∈ L p T : Dkf ∈ Lloc˙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 ϕ r
f, t r
p inf
f − g
p t rd
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 Rd , we write the rth forward difference of a function f in the
direction of e as
Δr
he fx
⎧
⎪
⎪
r
i0
⎛
⎝r
i
⎞
⎠−1i 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 ϕ r
f, t
p sup
0<h≤t
d
i1Δr
h ϕ iei f
where eidenotes the unit vector inRd , that is, its ith component is 1 and the others are 0.
In5, the following result has been proved
Trang 4Lemma 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 ω2
ϕ
f,√1
n
p
1
nf
p
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 onLemma 2.1and the following estimates:
M n,2 f − f
p ≤ const.
⎧
⎪
⎪
f
1
n
2
i1
ϕ2
i D i2f
pf
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
ϕ2f
pf
p
, 2.10
which has been proved in3
Now we give the following decomposition formula:
M n,2
f, x
∞
k1 0
∞
k2 0
P n,k1x1P n k1,k2
x2
1 x1
n − 1n − 2
×
∞
0
P n,k1u1P n k1,k2
u2
1 u1
f u1 , u2du1du2
∞
k1 0
P n,k1x1n − 2
∞
0
P n −1,k1u1∞
k2 0
P n k1,k2
x2
1 x1
× n k1− 1
∞
0
P n k1,k2tfu1 , 1 u1tdt du1
∞
k0
P n,k1x1n − 2
∞
0
P n −1,k1u1M n k1,1
g u1, z
du1,
2.11
Trang 5g u1t fu1 , 1 u1t, 0 ≤ t < ∞, z x2
1 x1 , 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
− fx ∞
k1 0
P n,k1x1n − 2
×
∞
0
P n −1,k1u1M n k1,1
g u1, z
− g u1zdu1
M∗
n,1 h·, x1 − hx1
: J L,
2.13 where
h u1 : hu1 , x : f
u1, 1 u1 x2
1 x1
, 0≤ u1 < ∞,
M∗n,1
g, y
∞
l0
P n,l
y
n − 2
∞
0
P n −1,ltgtdt.
2.14
Then by the Jensen’s inequality, we have
J p
p≤
T
∞
k1 0
P n,k1x1
n − 2∞
0
P n −1,k1u1M n k1,1 g u1, z − g u1zdu1
p dx
≤
T
∞
k1 0
P n,k1x1n − 2
∞
0
P n −1,k1u1 M n k
1,1
g u1, z
− g u1z p
du1dx
∞
0
∞
k1 0
P n,k1x11 x1dx1n − 2
∞
0
P n −1,k1u1
× M n k1,1
g u1, z
− g u1z p
dzdu1
≤ ∞
k1 0
n k1− 1
n− 1
∞
0
P n −1,k1u1
∞
0
M n k
1,1
g u1, z
− g u1z p
dzdu1
≤ const.∞
k1 0
n k1− 1
n− 1
∞
0
P n −1,k1u1
1
n k1
p
ϕ2g u 1p
pg u
1p p
du1.
2.15
However, by definition, one also has
ϕ2tg
u1t t1 t1 u12D2
2f u1 , 1 u1t ϕ2
2D2
2f u1 , 1 u1t. 2.16
Trang 6J p p ≤ const.∞
k1 0
n k1− 1
n − 1n k1 p
∞
0
P n −1,k1u1
×
ϕ22D22f u1 , 1 u1t p
f u1 , 1 u1t p
dt du1
const.∞
k1 0
n k1− 1
n − 1n k1 p
∞
0
1
1 u1 P n −1,k1u1
×
∞
0
ϕ2
2u1 , u2D 2
2f u1 , u2 p
f u1 , u2 p
du1du2
≤ const.
n p
∞
k1 0
∞
0
P n,k1u1
∞
0
ϕ22u1 , u2D 2
2f u1 , u2 p f u1 , u2 p
du1du2
const.
n p
ϕ2
2D22fp
pfp p
.
2.17
To estimate the second term L, we use a similar method as to estimate2.10 see 3 and can get
L p≤ const.
n
ϕ2h
p h p
Denoting ϕ12x ϕ21x : √x1x2, D 2
12 : ∂2/ ∂x1 ∂x2, and D 2
21 : ∂2/ ∂x2 ∂x1, we have
ϕ2sh s
s 1 s D21f x2
1 x1 D212f x2
1 x1 D221f x22
1 x12D222 f
×
s, 1 s x2
1 x1
1 x11 x1 x2 ϕ21D21f ϕ2
12D212f ϕ2
21D221f s
1 s
x2
1 x1 x2 ϕ22D22f
s, 1 s x2
1 x1
.
2.19
Recalling that ϕ12x is no bigger than ϕ1x or ϕ2x, and the fact
D2
12fx ≤ sup D2
1fx , D2
2fx 2.20 proved in6 see 6, Lemma 2.1, we obtain
ϕ2h
p ≤ const.2
i1
ϕ2
i D2i f
Trang 7and hence
L p≤ const.
n
2
i1
ϕ2
i D2i f
pf
p
The second inequality of 2.9 has thus been established, and the proof of Theorem 2.2is 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.
... 1.11 Trang 3It is a multivariate generalization of the univariate Baskakov-Durrmeyer operators... the following result has been proved
Trang 4Lemma 2.1 There exists a positive constant, dependent...
du1,
2.11
Trang 5g u1t fu1 , 1