Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 970659, 12 pages
doi:10.1155/2011/970659
Research Article
Approximation Order for Multivariate Durrmeyer
Operators with Jacobi Weights
Jianjun Wang,1 Chan-Yun Yang,2 and Shukai Duan3
1
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Department of Mechanical Engineering, Technology and Science Institute of Northern Taiwan,
No. 2 Xue-Yuan Road, Beitou, Taipei 112, Taiwan
3
School of Electronics and Information Engineering, Southwest University, Chongqing 400715, China
2
Correspondence should be addressed to Jianjun Wang, wjj@swu.edu.cn
Received 3 January 2011; Accepted 15 February 2011
Academic Editor: Pavel Drábek
Copyright q 2011 Jianjun Wang et al. 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.
Using the equivalence relation between K-functional and modulus of smoothness, we establish a
strong direct theorem and an inverse theorem of weak type for multivariate Bernstein-Durrmeyer
operators with Jacobi weights on a simplex in this paper. We also obtain a characterization
for multivariate Bernstein-Durrmeyer operators with Jacobi weights on a simplex. The obtained
results not only generalize the corresponding ones for Bernstein-Durrmeyer operators, but also
give approximation order of Bernstein-Durrmeyer operators.
1. Introduction
Let S Sd d 1, 2, . . . be a simplex in Rd defined by
S
x x1 , x2 , . . . , xd : xi ≥ 0, i 1, 2, . . . , d, |x| d
xi ≤ 1 .
1.1
i0
For p ≥ 1, we denote by Lp S the space of p-order Lebesgue integrable functions on S with
ωf p
⎧
1/p
p
⎪
⎪
ωxfx dx
< ∞ 1 ≤ p < ∞,
⎨
S
⎪
⎪
⎩maxωxfx
x∈S
p ∞,
1.2
2
Abstract and Applied Analysis
where L∞ S CS denote the space of continuous functions on S. For f ∈ LS, the
multivariate Bernstein-Durrmeyer Operators with d variables on S are given by
n d!
Mn,d f; x Pn·k x
Pn,k ufudu,
n!
S
|k|≤n
1.3
where Pn,k x n!/k!n − |k|!xk 1 − |x|n−|k| x ∈ S and x x1 , x2 , . . . , xd ∈ Rd , k k1 , k2 , . . . , kd ∈ N0d , with the convention
|x| d
xi ,
xk x1k1 x2k2 · · · xdkd ,
|k| i1
d
ki ,
k! k1 !k2 ! · · · kd !.
1.4
i1
For multivariate Jacobi weights ωx xα 1−|x|β , x ∈ S, α α1 , . . . , αd ∈ Rd , 0 < αi ,
β < 1, i 1, 2, . . . , d, xα x1α1 x2α2 · · · xdαd . We give some further notations, for x ∈ S, and we
√
write ϕi x ϕii x xi 1 − |x| 1 ≤ i ≤ d, ϕij x xi xj , 1 ≤ i < j ≤ d and
Di Dii Dijr
Dij Dijr−1 ,
∂
,
∂xi
1 ≤ i ≤ d,
Dij Di − Dj ,
1 ≤ i ≤ j ≤ d, r ∈ N,
D k
1 ≤ i < j ≤ d,
D1k1 D2k2
· · · Ddkd ,
k∈
1.5
N0d .
For f ∈ Lp S, the weighted Sobolev space is given by
r,p
Wφ S 0
f ∈ Lp S : ωf ∈ Lp S, Dk f ∈ Lloc S ,
ωϕrij Dijr f ∈ Lp S, |k| ≤ r, 1 ≤ i ≤ j ≤ d, r ∈ N ,
Wφr,∞ S 0
f ∈ CS : ωf ∈ CS, f ∈ Cr S , ωϕrij Dijr f ∈ CS, 1 ≤ i ≤ j ≤ d, r ∈ N ,
1.6
0
where S is the interior of S. To characterize the approximation capability of multivariate
Bernstein-Durrmeyer operators, we introduce the weighted K-functional
r r r
ωϕij Dij g Kϕr f, tr ω infr,p ωf − g p t
p
1≤i≤j≤d
g∈Wφ
1.7
and a measure of smoothness of f
ωϕr f, t ω sup
ωΔrhϕij eij f .
0<h≤t 1≤i≤j≤d
p
1.8
Abstract and Applied Analysis
3
Since 1967, Durrmeyer introduced Bernstein-Durrmeyer operators, and there are
many papers which studied theirproperties 1–7. In 1991, Zhang studied the characterization
of convergence for Mn,1 f; x with Jacobi weights. In 1992, Zhou 5 considered multivariate
Bernstein-Durrmeyer operators Mn,d f; x and obtained a characterization of convergence.
In 2002, Xuan et al. studied the equivalent characterization of convergence for Mn,d f; x
with Jacobi weights and obtained the following result.
Theorem 1.1. For ωf ∈ Lp S, 0 < r < 1, the following results are equivalent:
i ωMn,d f − fp On−r ;
ii Kϕ2 f, tω Otr .
In this paper, using the Ditzian-Totik modulus of smoothness, we will give the upper
bound and lower bound of approximation function by Mn,d f; x on simplex. The main
results are as follows.
Theorem 1.2. If ωf ∈ Lp S, then
ωf 1
p
ωMn,d f − f ≤ C ωϕ2 f, √
.
p
n
n ω
1.9
And there exists a positive number δ 0 < δ < 1 such that the following inequality is satisfied:
ωϕ2
1
f, √
n
ω
n δ n C
ωMn,d f − fω .
≤
n k1 k
1.10
Throughout the paper, the letter C, appearing in various formulas, denotes a positive
constant independent of n, x, and f. Its value may be different at different occurrences, even
within the same formula.
From Theorem 1.2, we can easily obtain the following corollary.
Corollary 1.3. If ωf ∈ Lp S, 0 < r < 1, we has the following equivalent results:
i ωMn,d f − fp On−r ;
ii Kϕ2 f, tω Otr ;
iii ωϕ2 f, tω Ot2r .
2. Some Lemmas
To prove Theorem 1.2, we will show some lemmas in this section. For the simplex S, the
transformation T: S → S10 defined by
Tx1 , x2 , . . . , xd u1 , u2 , . . . , ud ,
ul ⎧
⎨xj
l j,
⎩1 − |x|
l/
j
2.1
4
Abstract and Applied Analysis
satisfies T 2 I, and I is the identity operator. So we have
∂
∂
∂
∂
∂
−
−
,
l
/j ,
∂ul ∂xl ∂xj
∂uj
∂xj
Mn,d f; x Mn,d fT ; Tx ;
Mn,d f; Tx Mn,d fT ; x ,
2.2
where fT u fTx.
Lemma 2.1. If ωf ∈ Lp S, then
ωMn,d f ≤ ωf ,
p
p
⎞
⎛
C
ωMn,d f − f ≤ ⎝ωf ωϕ2ij Dij2 f ⎠,
p
p
p
n
1≤i≤j≤d
r,p
f ∈ Wφ S.
2.3
Proof. Letting S {x : x1 , x ∈ Sd }, x x2 , x3 , . . . , xd , k k2 , k3 . . . , kd , k k1 , k,
Pn,k1 x1 n!/k1 !n − k1 !x1k1 1 − x1 n−k1 , then
n
Pn,k1 x1 Pn−k1 ,k
Mn,d f; x k≤n−k1
k1 0
×
1
Pn,k1 u1 0
n
Pn,k1 x1 k1 0
S
Pn−k1 ,k
n d!
n!
1
x
1 − x1
n d!
n!
u
fudu du1
1 − u1
Pn,k1 u1 1 − u1 d−1
0
k ≤n−k1
Pn−k1 ,k
x
1 − x1
×
Sd−1
n
k1 0
Pn−k1 ,k tfu1 , 1 − u1 tdt du1
Pn,k1 x1 n d
1
0
Pnd−1,k1 u1 Mn−k1 ,d−1
x
fu1 , 1 − u1 ·;
du1 .
1 − x1
2.4
Using the transformation T, 2.2, 2.4, the method of 7, we can easily get 2.3.
Lemma 2.2 see 8. If f ∈ Lp S, then
C−1 ωϕr f, t ω ≤ Kϕr f, tr ω ≤ Cωϕr f, t ω .
Proof. Lemma 2.2 is proved when f ∈ CS in 8. Similarly, we can prove f ∈ Lp S.
2.5
Abstract and Applied Analysis
5
Lemma 2.3. If 0 < a < 1, b > 0, x ∈ 0, 1, Pn,k x Cnk xk 1 −xn−k is basis function of the classical
Bernstein operators, then
a
n−1
n
Pn,k x
≤ Cx−a ,
k
k1
n−1
n
Pn,k x
n
−k
k1
b
2.6
≤ C1 − x−b .
Proof. The first inequality can be inferred by Hölder inequality. In the following we prove the
second inequality.
i If 0 < b < 1, using Hölder inequality, we can easily obtain the result.
ii If b ≥ 1, let b m r, m ∈ N, 0 ≤ r < 1, then
b m r
n−1
n−1
n
n
n
Pn,k x
Pn,k x
n−k
n−k
n−k
k1
k1
≤ C1 − x−m
r
nm
Pnm,k x
nm−k
k1
n−1
2.7
≤ C1 − x−m−r C1 − x−b .
Lemma 2.3 is completed.
Lemma 2.4. If f ∈ Lp S, 1 ≤ p ≤ ∞, then
2 2
ωϕij Dij Mn,d f ≤ Cnωf p
p
1 ≤ i ≤ j ≤ d.
2.8
Proof. In the following we use the induction on the dimension number d to prove the result.
The case d 1 was proved by Lemma 4 of 6. Next, suppose that Lemma 2.4 is valid for
d r r ≥ 1; we prove it is also true for d r 1. To observe this, we use a decomposition
formula 2.4, and we have
2
Mn,d f; x
ωxϕ222 xD22
−
x1α1 1 − x1 |α|β
× 1 −
n
d
Pn,k1 x1 n dzα1 2 zα2 3 · · · zαd−1
k1 0
|z|β ϕ211 z
1
0
2
Pnd−1,k1 u1 D11
Mn−k1 ,d−1 fu1 , 1 − u1 ·; z du1 ,
2.9
6
Abstract and Applied Analysis
where z z1 , z2 , . . . , zd−1 x2 /1 − x1 , x3 /1 − x1 , . . . , xd /1 − x1 . Thus we have
2
Mn,d f; x ds
ωxϕ222 xD22
S
≤C
1
0
x1α1 1
− x1 −
|α|β
n
Pn,k1 x1 n d
1
Pnd−1,k1 u1 n − k1 0
k1 0
ωzfu1 , 1 − u1 zdz dx1 du1
×
z∈Sd−1
nd
≤C
n
n1
1 −
n k1 1 α1 n − k1 1 |α|β
Pnd−1,k1 u1 n1
0 k 0 n 1
2.10
1
ωzfu1 , 1 − u1 zdz du1
×
z∈Sd−1
≤ Cn
1
0
−
uα1 1 1 − u1 |α|β
1
u1
α1
−
1 − u1 −|α|−β
ωf u1 , 1 − u1 zdz du1
z∈Sd−1
Cnωf 1 .
In the above derivation, we have used the formula 6
1
0
x1α1 1
−
|α|β
− x1 −
k1 1 α1 n − k1 1 |α|β
1
Pn,k1 x1 dx1 ≤ C
n1 n1
n1
2.11
and the inequality
n k1 1 α1 n − k1 1 |α|β
−
k1 0
n1
n1
−
Pnd−1,k1 u1 ≤ Cuα1 1 1 − u1 |α|β .
2.12
When p ∞, we have
2
ωxϕ222 xD22
Mn,d f; x
−
x1α1 1 − x1 |α|β
× 1 −
n
k1 0
|z|β ϕ211 z
d
Pn,k1 x1 n dzα1 2 zα2 3 · · · zαd−1
1
0
2
Pnd−1,k1 u1 D11
Mn−k1 ,d−1 fu1 , 1 − u1 ·; z du1 ,
where z z1 , z2 , . . . , zd−1 x2 /1 − x1 , x3 /1 − x1 , . . . , xd /1 − x1 .
2.13
Abstract and Applied Analysis
7
From the Cauchy-Swartz inequality, Hölder inequality, and Lemma 2.3, we have
2
Mn,d f; x ωxϕ222 xD22
−
≤ Cx1α1 1 − x1 |α|β
n
Pn,k1 x1 n d
1
Pnd−1,k1 u1 n − k1 0
k1 0
d
× max zα1 2 zα2 3 · · · zαd−1
1 − |z|β fu1 , 1 − u1 zdu1
z∈Sd−1
n
−
≤ Cnωf ∞ x1α1 1 − x1 |α|β
Pn,k1 x1 n d
k1 0
1
Pnd−1,k1 u1 0
1
u1
α1
−
1 − u1 −|α|−β du1
1
1/2
n
−
α1
−2α1
|α|β
≤ Cn ωf ∞ x1 1 − x1 Pn,k1 x1 n d
Pnd−1,k1 u1 u1 du1
0
k1 0
×
1
1/2
−
−2|α|−2β
Pnd−1,k1 u1 1 − u1 du1
0
α1 /2
1
n
−
α1
α|β
|
−2
≤ Cnωf ∞ x1 1 − x1 Pn,k1 x1 n d
Pnd−1,k1 u1 u1 du1
0
k1 0
×
×
1
Pnd−1,k1 u1 du1
1−α1 /2 1
0
0
1
−
1/2−|α|β/2d
−
|α|β/2d
−2d
Pnd−1,k1 u1 1 − u1 du1
Pnd−1,k1 u1 du1
0
n
−
n d − 1 α1 /2
≤ Cnωf ∞ x1α1 1 − x1 |α|β
Pn,k1 x1 n d
k1 k1 − 1
k 2
1
×
n d − 1! n − d − k1 − 1!
n − d! n d − k1 − 1!
−
|α|β/2d
n d
−
1−|α|β/2d−α1 /2
−1
−
|α|β
α1 n−1
−
α1
n
n
|α|β
≤ Cn ωf ∞ x1 1 − x1 Pn,k1 x1 k1
n − k1
k 1
≤ Cnωf ∞ .
1
2.14
By Riesz interpolation theorem, we get
2 2
ωϕ22 D22 Mn,d f ≤ Cnωf p .
p
2.15
8
Abstract and Applied Analysis
Similarly, the other cases for i 1, 3, 4, . . . , dj can be proved. For i /
j, by the transformation
T, we have
2 2
2
Mn,d fT ≤ CnωT fT p Cnωf p .
ωϕij Dij Mn,d f ωT ϕ2jj Djj
p
p
2.16
Lemma 2.4 is completed.
r,p
Lemma 2.5. If f ∈ Wφ S ⊂ Lp S, 1 ≤ p ≤ ∞, then
2 2
ωϕij Dij Mn,d f ≤ Cωϕ2ij Dij2 f p
p
1 ≤ i ≤ j ≤ d.
2.17
Proof. We use the induction on the dimension number d to prove Lemma 2.5. The case d 1
was proved by Lemma 3 of 6, that is,
2 2
ωϕ D Mn,1 f ≤ Cωϕ2 D2 f .
p
p
2.18
Next, suppose that Lemma 2.5 is valid for d r r ≥ 1, and we prove it is also true for
d r 1. Noticing formula 2.4, we have
2
ωxϕ222 xD22
Mn,d f; x
−
x1α1 1 − x1 |α|β
× 1 −
n
k1 0
|z|β ϕ211 z
d
Pn,k1 x1 n dzα1 2 zα2 3 · · · zαd−1
1
0
2
Pnd−1,k1 u1 D11
Mn−k1 ,d−1 fu1 , 1 − u1 ·; z du1 ,
2.19
Abstract and Applied Analysis
9
where z z1 , z2 , . . . , zd−1 x2 /1 − x1 , x3 /1 − x1 , . . . , xd /1 − x1 . When p 1, from the
inductive assumption of p 1, we have
2
Mn,d f; x ds
ωxϕ222 xD22
S
≤C
1
0
x1α1 1
×
z∈Sd−1
nd
≤C
n1
0
Pn,k1 x1 n d
1
Pnd−1,k1 u1 0
k1 0
2
fu1 , 1 − u1 zdz dx1 du1
ωzϕ211 zD11
1
z∈Sd−1
≤C
− x1 n
1 −
n k1 1 α1 n − k1 1 |α|β
Pnd−1,k1 u1 n1
n1
0 k 0
×
1
−
|α|β
2
fu1 , 1 − u1 zdz du1
ωzϕ211 zD11
−
uα1 1 1 − u1 |α|β
2 ≤ Cωϕ222 D22
f .
1
u1
α1
−
1 − u1 −|α|−β
z∈Sd−1
2
f u1 , 1 − u1 zdz du1
ωϕ222 D22
1
2.20
When p ∞, we have
2
ωxϕ222 xD22
Mn,d f; x
−
x1α1 1 − x1 |α|β
n
k1 0
× 1 − |z|β ϕ211 z
d
Pn,k1 x1 n dzα1 2 zα2 3 · · · zαd−1
1
0
2.21
2
Pnd−1,k1 u1 D11
Mn−k1 ,d−1 fu1 , 1 − u1 ·; z du1 ,
where z z1 , z2 , . . . , zd−1 x2 /1 − x1 , x3 /1 − x1 , . . . , xd /1 − x1 . From the inductive
assumption, the Cauchy-Swartz inequality, Holder inequality, and Lemma 2.4, we get
2
Mn,d f; x ωxϕ222 xD22
≤
Cx1α1 1
−
|α|β
− x1 n
Pn,k1 x1 n d
k1 0
1
0
Pnd−1,k1 u1 d
× max zα1 2 zα2 3 · · · zαd−1
1 − |z|β ϕ2z Dz2 fu1 , 1 − u1 zdu1
z∈Sd−1
10
Abstract and Applied Analysis
−
≤ Cx1α1 1 − x1 |α|β
n
Pn,k1 x1 n d
1
0
k 0
1
2 2 × ωϕ22 D22 f du1
∞
2 2 ≤ Cωϕ22 D22 f .
−
Pnd−1,k1 u1 x1−α1 1 − x1 −|α|−β
∞
2.22
By Riesz interpolation theorem, we get
2 2
2 f .
ωϕ22 D22 Mn,d f ≤ Cωϕ222 D22
p
2.23
p
Similarly, the other cases for i 1, 3, 4, . . . , dj can be proved. For i /
j, by the transformation
T, we have
2 2
2
2
Mn,d fT ≤ CωT ϕ2jj Djj
fT ≤ Cωϕ2ij Dij2 f ωϕij Dij Mn,d f ωT ϕ2jj Djj
p
p
p
p
2.24
Lemma 2.5 is completed.
Lemma 2.6 see 9. Let {σn }, {φn } be nonnegative sequences (σ1 0, n ∈ N. For l > 0, if the
sequences {σn }, {φn } satisfy
l
k
σn ≤ Q
σk φk
n
Q ≥ 1, 1 ≤ k ≤ n, n ∈ N,
2.25
one has
σn ≤ Mn−s
n
ks−1 φk .
k1
If Q 1, then l s. If Q > 1, then 0 < s < l.
2.26
Abstract and Applied Analysis
11
3. The Proof of Theorems
r,p
Now we prove 1.9 of Theorem 1.2. By using Lemma 2.1, for arbitrary g ∈ Wφ S ⊂ Lp S,
we have
ωMn,d f − f ≤ C ωMn,d f − g ωMn,d g − ωg ω f − g p
p
p
p
⎛
⎞⎞
⎛
1
2 2 ⎠⎠
≤ C⎝ω f − g p ⎝
ωϕij Dij g ωg p
p
n 1≤i≤j≤d
3.1
⎛
⎞
1
1
≤ C⎝ω f − g p ωϕ2ij Dij2 g ωf p ⎠.
p
n 1≤i≤j≤d
n
Hence, from Lemma 2.2, we obtain
1
ωMn,d f − f ≤ C Kϕ2 f, 1
ωf p
p
n ω n
1
2
ωf p .
≤ C ωϕ f, t ω n
3.2
Next, we prove 1.10 of Theorem 1.2. Leting σn C1/nωϕ2ij Dij2 Mn,d f 1 ≤ i ≤ j ≤ d,
φn CωMn,d f − fp , then σ1 0. By Lemmas 2.4 and 2.5, we have
p
1
1
σn ≤ C ωϕ2ij Dij2 Mn,d f − Mk,d f C ωϕ2ij Dij2 Mn,d Mk,d f p
p
n
n
1
≤ Cω f − Mk,d f p C ωϕ2ij Dij2 Mk,d f p
n
k
C σk φk
n
3.3
C > 1.
Using Lemma 2.6, we get σn ≤ C1/n
n
k1 n/k
δ
φk 0 < δ < 1. That is,
n δ n 2 2
ω Mk,d f − f p .
ωϕij Dij Mn,d f ≤ C
p
k
k1
3.4
When n ≥ 2, there exists m ∈ N such that n/2 ≤ m ≤ n and satisfies the equation
ωMm,d f − f min ωMk,d f − f .
p
p
n/2≤k≤n
3.5
Thus,
ωMm,d f − f ≤ 2
ωMk,d f − f .
p
p
n n/2≤k≤n
3.6
12
Abstract and Applied Analysis
Using Lemma 2.2, we have
1
1
ωϕ2 f, √
≤ CKϕ2 f,
n
n ω
⎛
⎞
1
≤ C⎝ω Mm,d f − f p ωϕ2ij Dij2 Mm,d f ⎠
p
n 1≤i≤j≤d
3.7
n δ n 1
ωMk,d f − fp .
≤C
n k1 k
Theorem 1.2 is completed.
Acknowledgments
This paper was supported by the Natural Science Foundation of China nos. 11001227,
60972155, Natural Science Foundation Project of CQ CSTC no. CSTC, 2009BB2306,
CSTC2009BB2305, and the Fundamental Research Funds for the Central Universities no.
XDJK2010B005, XDJK2010C023.
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Journal of
Complex Analysis
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Volume 2014
International
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Stochastic Analysis
Abstract and
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International Journal of
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Discrete Dynamics in
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Volume 2014
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
Optimization
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Volume 2014
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Volume 2014