Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 101852, 13 pages
doi:10.1155/2011/101852
Research Article
Direct and Inverse Approximation Theorems for
Baskakov Operators with the Jacobi-Type Weight
Guo Feng1, 2
1
School of Mathematics and Information Engineering, Taizhou University, Zhejiang,
Taizhou 317000, China
2
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
Correspondence should be addressed to Guo Feng, gfeng20080576@163.com
Received 2 May 2011; Accepted 10 September 2011
Academic Editor: Ruediger Landes
Copyright q 2011 Guo Feng. 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.
We introduce a new norm and a new K-functional Kϕλ f; tw,λ . Using this K-functional, direct
and inverse approximation theorems for the Baskakov operators with the Jacobi-type weight are
obtained in this paper.
1. Introduction and Main Results
Let f be a function defined on the interval 0, ∞. The operators Vn f; x are defined as
follows:
∞ k
vn,k x,
f
Vn f; x n
k0
1.1
where
vn,k x nk−1
k
xk 1 x−n−k ,
1.2
which were introduced by Baskakov in 1957 1. Becker 2 and Ditzian 3 had studied these
operators and obtained direct and converse theorems. In 4, 5 Totik gave a result: if f ∈
CB 0, ∞, 0 < α < 1, then Vn f; x − fx∞ Onα if and only if xα 1 xα |Δ2h f; x| ≤
kh2α , where h > 0 and k is a positive constant. We may formulate the following question: do
2
Abstract and Applied Analysis
the Baskakov operators have similar property in the case of weighted approximation with the
Jacobi weights? It is well known that the weighted approximation is not a simple extension,
because the Baskakov operators are unbounded for the usual weighted norm fw wf∞ .
Xun and Zhou 6 introduced the norm
f wf f0
,
w
∞
f ∈ CB 0, ∞
1.3
and have discussed the rate of convergence for the Baskakov operators with the Jacobi
weights and obtained
wx
Vn f; x − fx
O n−α ⇐⇒ K f; t w Otα ,
1.4
where wx xa 1 x−b , 0 < a < 1, b > 0, 0 < α < 1, and CB 0, ∞ is the set of bounded
continuous functions on 0, ∞.
In this paper, we introduce a new norm and a new K-functional, using the K-functional,
and we get direct and inverse approximation theorems for the Baskakov operators with the
Jacobi-type weight.
First, we introduce some useful definitions and notations.
Definition 1.1. Let CB 0, ∞ denote the set of bounded continuous functions on the interval
0, ∞, and let
Ca,b,λ f | f ∈ CB 0, ∞, ϕ22−λ wf ∈ CB 0, ∞ ,
0
Ca,b,λ
f | f ∈ Ca,b,λ , f0 0 ,
1.5
where ϕx x1 x, wx xa 1 x−b , x ∈ 0, ∞, 0 ≤ a < λ ≤ 1, and b ≥ 0.
Moreover, the K-functional is given by
Kϕλ f; t w,λ inf ϕ21−λ f − g tϕ22−λ g ,
w
g∈D
w
1.6
0
where D {g | g ∈ Ca,b,λ
, g ∈ A.C.loc 0, ∞, ϕ22−λ g w < ∞}.
We are now in a position to state our main results.
0
Theorem 1.2. If f ∈ Ca,b,λ
, then
21−λ Vn f − f ≤ MK ϕλ f; n−1
ϕ
w
w,λ
1.7
.
0
Theorem 1.3. Suppose f ∈ Ca,b,λ
, 0 < α < 1. Then the following statements are equivalent:
1
2
ϕ21−λ xwx
Vn fx − fx O n−α ,
Kϕλ f; t w,λ Otα , 0 < t < 1.
n ≥ 2;
1.8
Abstract and Applied Analysis
3
Throughout this paper, M denotes a positive constant independent of x, n, and f
which may be different in different places. It is worth mentioning that for λ 1, we recover
the results of 6.
2. Auxiliary Lemmas
To prove the theorems, we need some lemmas. By simple computation, we have
∞
k2
k1
k
f; x nn 1 vn2,k x f
− 2f
f
n
n
n
k0
2.1
kk − 1 2kn k n kn k 1
k
.
vn,k xf
−
f; x n
x1 x
x2
1 x2
k0
2.2
Vn
or
Vn
∞
Lemma 2.1. Let c ≥ 0, d ∈ R. Then
−c k −d
k
vn,k x
≤ Mx−c 1 x−d ,
1
n
n
k1
∞
for x > 0.
2.3
Proof. We notice 7
l
n
vn,k x
≤ Mx−l ,
k
k1
∞
for l ∈ N,
∞
k m
vn,k x 1 ≤ M1 xm ,
n
k0
2.4
for m ∈ Z.
For c 0, d 0, the result of 2.3 is obvious. For c > 0, d /
0, there exists m ∈ Z, such that
0 < −2d/m < 1. Using Hölder’s inequality, we have
−c −2c 1/2 1/2
∞
∞
∞
k
k
k −d
k −2d
vn,k x
≤
vn,k x
vn,k x 1 1
n
n
n
n
k1
k1
k1
2c1 c/2c1 −d/m
∞
∞
n
k m
≤
vn,k x
vn,k x 1 k
n
k1
k1
c/2c1 −d/m
≤ M x−2c1
1 xm
≤ Mx−c 1 x−d .
2.5
4
Abstract and Applied Analysis
For c > 0, d 0 or c 0, d /
0, the proof is similar to that of 2.5. Thus, this proof is
completed.
0
Lemma 2.2. Let f ∈ Ca,b,λ
, n ∈ N. Then
wxϕ21−λ xVn f; x ≤ Mϕ21−λ f .
w
2.6
Proof. By Lemma 2.1, we get
∞ k
vn,k x
wxϕ21−λ xVn f; x wxϕ21−λ x f
n
k1
∞
21−λ 2.7
21−λ
−1 k
2λ−1 k
≤ ϕ
f wxϕ
ϕ
x vn,k xw
w
n
n
k1
≤ Mϕ21−λ f .
w
0
Lemma 2.3. Let f ∈ Ca,b,λ
, n ∈ N. Then
22−λ Vn f ≤ Mnϕ21−λ f .
ϕ
w
w
2.8
0, n 1xx 1 ≤ 2n · 2x ≤ 4; using 2.1 and Lemma 2.1,
Proof. For x ∈ Enc 0, 1/n, x /
we have
wxϕ22−λ xVn f; x ≤ wxϕ21−λ xnn 1x1 x
∞
∞
k2 −21−λ k2
k1 −21−λ k1
×
vn2,k xw−1
ϕ
2 vn2,k xw−1
ϕ
n
n
n
n
k0
k0
∞
k
k
21−λ f
vn2,k xw−1
ϕ−21−λ
ϕ
w
n
n
k1
≤ Mnwxϕ21−λ xw−1 xϕ−21−λ xϕ21−λ f ≤ Mnϕ21−λ f .
w
w
2.9
Abstract and Applied Analysis
5
For x ∈ En 1/n, ∞, by 2.2, we get
wxϕ22−λ xVn f; x 2
∞
x1 x 1 2x k
k
k
2
−2λ
n wxϕ x vn,k xf
−x −
−x −
n
n
n
n
n
k1
∞
k
k
≤ n2 wxϕ−2λ xϕ21−λ f ϕ−21−λ
vn,k xw−1
w
n
n
k1
2
x1 x
1 2x k
k
·
−x − x
n
n n
n
: n2 wxϕ−2λ xϕ21−λ f I1 n, x I2 n, x I3 n, x.
2.10
w
Note that for x ∈ En , one has the following inequality 7
2m
n2m Vn t − x2m ; x ≤ Mnm ϕx ,
m ∈ N.
2.11
Applying Hölder’s inequality and Lemma 2.1, we have
I1 n, x 2
∞
k
k
k
−x
vn,k xw−1
ϕ−21−λ
n
n
n
k1
4 1/2
1/2 ∞
∞
k
−2 k
−41−λ k
−x
≤
vn,k xw
vn,k x
ϕ
n
n
n
k1
k1
≤ Mx−a−1λ 1 xbλ−1
2.12
x1 x
n
≤ Mn−1 w−1 xϕ2λ x,
∞
1 2x
k
k k
−
x
vn,k xw−1
ϕ−21−λ
I2 n, x n
n
n n
k1
1 2x
≤
n
−1
2 1/2
1/2 ∞
k
k
−41−λ k
ϕ
−x
vn,k xw
vn,k x
n
n
n
k1
k1
∞
−2
−3/2
≤ Mw xϕ xn
2λ
1 1/2
.
1
x
2.13
6
Abstract and Applied Analysis
Note that for x > 1/n, one has 1 1/x < 2n. Hence,
I2 n, x ≤ Mn−1 w−1 xϕ2λ x,
∞
k
k x1 x
I3 n, x ϕ−21−λ
vn,k xw−1
n
n
n
k1
2.14
2.15
≤ Mn−1 x1 xx−a−1λ 1 xb−1λ
Mn−1 w−1 xϕ2λ x.
Combining 2.9–2.14, we get
wxϕ22−λ xVn f; x ≤ Mnϕ21−λ f .
2.16
22−λ Vn f ≤ Mnϕ21−λ f .
ϕ
2.17
w
w
Thus,
w
w
The proof is completed.
Lemma 2.4. Let f ∈ D, n ∈ N, and n ≥ 2. Then
22−λ Vn f ≤ Mϕ22−λ f .
ϕ
w
w
2.18
Proof. 1 For the case λ / 1 or a / 0, if λ − 2 b ≥ 0, using 2.1 and Lemma 2.1, we have
wxϕ22−λ xVn f; x 1/n 1/n ∞
k
wxϕ22−λ xnn 1 vn2,k x
u v dudv
f n
0
0
k0
∞
≤ ϕ22−λ f wxϕ22−λ xnn 1 vn2,k x
w
k0
1/n 1/n λ−a−2 λb−2
k
k
·
uv
dudv
1 uv
n
n
0
0
Abstract and Applied Analysis
7
1/n 1/n λ−a−2 ∞
k
k2 λb−2
22−λ 22−λ
≤ ϕ
f wxϕ
dudv
1
xnn1 vn2,k x
w
n
n
0
0
k1
1/n 1/n
ϕ22−λ f wxϕ22−λ xnn1vn2,0 x
uvλ−a−2 1u vλb−2 dudv
w
0
0
λ−a−2 ∞
k
k 2 λb−2
≤ ϕ22−λ f wxϕ22−λ xnn 1 vn2,k xn−2
1
w
n
n
k1
1/n
1
uλ−a−1 du
3λb−2 ϕ22−λ f wxϕ22−λ xnn 1vn2,0 x
w
1a−λ
0
λ−a−2 ∞
k λb−2
k
≤ 2 · 3λb−2 ϕ22−λ f wxϕ22−λ x vn2,k x
1
w
n
n
k1
λ−a
1
1
2 · 3λb−2 ϕ22−λ f wxϕ22−λ xnn 1vn2,0 x
w
1 a − λλ − a n
λ−a x2a−λ nn 1
1
λb−2 22−λ 1
.
≤2·3
f ϕ
n
w
1 a − λλ − a1 x n
2.19
i If x ∈ Enc ,
x2a−λ nn 1
1 a − λλ − a1 xn
2
λ−a
1
nn 1
2
1
.
≤
≤
n
1 a − λλ − a n
1 a − λλ − a
2.20
ii If x ∈ En , n ≥ 2,
x2a−λ nn 1
1 a − λλ − a1 xn
λ−a
λ−a
nλ−a x2 nn 1
1
1
≤
n
1 a − λλ − a1 xn n
≤
x2 nn 1
1 a − λλ − ann − 1x2
≤
3
.
1 a − λλ − a
2.21
Combining 2.19–2.21, we have
wxϕ22−λ xVn f; x ≤ Mϕ22−λ f .
2.22
22−λ
xVn f ≤ Mϕ22−λ f .
ϕ
2.23
w
Thus,
w
w
8
Abstract and Applied Analysis
If λ − 2 b < 0, we have
wxϕ22−λ xVn f; x 1/n 1/n λ−a−2 ∞
k
k λb−2
22−λ 22−λ
≤ ϕ
f wxϕ
dudv
1
xnn 1 vn2,k x
w
n
n
0
0
k1
1/n 1/n
22−λ λ−a−2
22−λ
ϕ
f wxϕ
dudv
.
xnn 1vn2,0 x
u v
w
0
0
2.24
By using the method similar to that of 2.19–2.23, it is not difficult to obtain the same
inequality as 2.23.
2 For the case λ 1, a 0, the proof is similar to that of case 1 and even simpler.
Therefore the proof is completed.
Lemma 2.5 see 8, page 200. Let Ωt be an increasing positive function on 0, a, the inequality
r > α
r
h
α
Ωt
Ωh ≤ M t t
2.25
holds true for h, t ∈ 0, a. Then one has
Ωt Otα .
2.26
3. Proofs of Theorems
3.1. Proof of Theorem 1.2
Proof. First, we prove it as follows.
i If x ∈ Enc , then
wxϕ21−λ x
x
∞
vn,k x
k0
k
− u
w−1 uϕ−22−λ udu
≤ Mn−1 .
n
k/n
3.1
ii If x ∈ En , then
Vn t − x2 1 tb−2λ ; x ≤ Mn−1 ϕ2 x1 xb−2λ .
3.2
Abstract and Applied Analysis
9
The Proof of 3.1
In fact, i for k 0, since x ∈ Enc , we have
wxϕ21−λ xvn,0 x
x
uw−1 uϕ−22−λ udu
0
wxϕ
21−λ
x1 x
−n
3.3
x
u
λ−a−1
1 u
b−2λ
du.
0
If b − 2 λ ≤ 0, we get
wxϕ21−λ x1 x−n
x
uλ−a−1 1 ub−2λ du
0
≤ Mwxϕ
21−λ
3.4
x1 x−n xλ−a
≤ Mn−1 .
If b − 2 λ > 0, we have
wxϕ21−λ x1 x−n
x
uλ−a−1 1 ub−2λ du
0
≤ Mwxϕ21−λ x1 x−nb−2λ
x
uλ−a−1 du
3.5
0
−1
≤ Mn .
ii If k ≥ 1, since x ∈ Enc , we have
21−λ
wxϕ
x
∞
x vn,k x
k
− u
w−1 uϕ−22−λ udu
n
k/n
k1
∞
k
k b x
−a
− x ϕ−22−λ x 1 vn,k x
u
du
n
n
k/n
k1
1−a
∞
k b
k
k
−2
1−a
≤ Mwxϕ x vn,k x
−x
1
−x
n
n
n
k1
≤ Mwxϕ21−λ x
≤ Mwxϕ−2 x
2−a ∞
k b
k
−x
1
vn,k x
≤ Mn−1 .
n
n
k1
Combining 3.4, 3.5 and 3.6, we obtain 3.1.
3.6
10
Abstract and Applied Analysis
The proof of 3.2
If b − 2 λ ≤ 0, by 9.5.10 and 9.6.3 of 7, using the Cauchy-Schwarz inequality and the
Hölder inequality, we obtain
1/2 1/2
Vn 1 t2b−2λ ; x
Vn t − x2 1 tb−2λ ; x ≤ Vn t − x4 ; x
1/2 2−b−λ/2
Vn 1 t−2 ; x
≤ Vn t − x4 ; x
3.7
≤ Mn−1 ϕ2 x1 xb−2λ .
If b − 2 λ > 0, by 2.3, we get Vn 1 tb−2λ ; x ≤ M1 xb−2λ , and using the
Cauchy-Schwarz inequality and the Hölder inequality, we have
1/2 1/2
Vn 1 t2b−2λ ; x
Vn t − x2 1 tb−2λ ; x ≤ Vn t − x4 ; x
3.8
≤ Mn−1 ϕ2 x1 x
b−2λ
.
Combining 3.7 and 3.8, we obtain 3.2.
Next, we prove Theorem 1.2. For g ∈ D, if x ∈ Enc , by 3.1, we have
wxϕ21−λ x Vn g; x − gx t
wxϕ21−λ xVn
t − ug udu; x x
≤ wxϕ
21−λ
t
22−λ −1
−22−λ
g Vn |t − u|w uϕ
xϕ
udu
; x
x
w
x
∞
22−λ 21−λ
≤ Mϕ
g wxϕ
x vn,k x
w
k0
≤ Mn−1 ϕ22−λ g .
w
k
− u
w−1 uϕ−22−λ udu
n
k/n
3.9
Abstract and Applied Analysis
11
If x ∈ En , by 3.2, we get
wxϕ21−λ x Vn g; x − gx t
21−λ
wxϕ
xVn
t − ug udu; x x
t
22−λ 21−λ
−1
−22−λ
≤ Mϕ
g wxϕ
xVn |t − u|w uϕ
udu
; x
x
w
≤ Mϕ22−λ g ϕ−2 xVn t − x2 ; x
w
3.10
x−2−aλ wxϕ21−λ xVn t − x2 1 tb−2λ ; x ≤ Mn−1 ϕ22−λ g .
w
0
Therefore, for f ∈ Ca,b,λ
, g ∈ D, by Lemma 2.2 and 3.9, 3.10, and the definition of
−1
Kϕλ f; n w,λ , we obtain
wxϕ21−λ x Vn f; x − fx ≤ wxϕ21−λ x Vn f − g; x wxϕ21−λ x fx − gx wxϕ21−λ x Vn g; x − gx 3.11
≤ Mϕ21−λ f − g wxϕ21−λ x fx − gx w
wxϕ21−λ x Vn g; x − gx ≤ M ϕ21−λ f − g n−1 ϕ22−λ g .
w
w
Taking the infimum on the right-hand side over all g ∈ D, we get
wxϕ21−λ x Vn f; x − fx ≤ MKϕλ f; n−1
w,λ
.
3.12
This completes the proof of Theorem 1.2.
3.2. Proof of Theorem 1.3
Proof. By Theorem 1.2, we know 2 ⇒ 1. Now,we will prove 1 ⇒ 2. In view of 1, we
get
21−λ Vn f − f ≤ Mn−α .
ϕ
w
3.13
12
Abstract and Applied Analysis
By the definition of K-functional, we may choose g ∈ D to satisfy
21−λ f − g n−1 ϕ22−λ g ≤ 2Kϕλ f; n−1
ϕ
w
w
w,λ
3.14
.
Using Lemma 2.2 and Lemma 2.3, we have
Kϕλ f; t w,λ ≤ ϕ21−λ Vn f − f tϕ22−λ Vn f w
w
≤ Mn−α t ϕ22−λ Vn f − g ϕ22−λ Vn g w
w
≤ Mn−α t nMϕ21−λ f − g Mϕ22−λ g w
3.15
w
≤ Mn−α tnM ϕ21−λ f − g n−1 ϕ22−λ g .
w
w
Taking the infimum on the right-hand side over all g ∈ D, we get
Kϕλ f; t
w,λ
−α
≤M n
t
−1
−1 Kϕλ f; n
.
w,λ
n
3.16
By Lemma 2.4, we get
Kϕλ f; n−1
w,λ
≤ M n−α .
3.17
Leting n 1−1 < t ≤ n−1 , we get
Kϕλ f; t w,λ ≤ MKϕλ f; n−1
w,λ
≤M
n
n1
−α
n 1−α ≤ Mn 1−α ≤ Mtα .
3.18
This completes the proof of Theorem 1.3.
Acknowledgments
The author would like to thank Professor Ruediger Landes and the anonymous referees for
their valuable comments, remarks, and suggestions which greatly help us to improve the
presentation of this paper and make it more readable. The project is supported by the Natural
Science Foundation of China Grant no. 10671019.
References
1 V. A. Baskakov, “An example of a sequence of linear positive operators in the spaces of continuous
functions,” Doklady Akademii Nauk SSSR, vol. 112, pp. 249–251, 1957.
2 M. Becker, “Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial
weight spaces,” Indiana University Mathematics Journal, vol. 27, no. 1, pp. 127–142, 1978.
Abstract and Applied Analysis
13
3 Z. Ditzian, “On global inverse theorems of Szász and Baskakov operators,” Canadian Journal of
Mathematics, vol. 31, no. 2, pp. 255–263, 1979.
4 V. Totik, “Uniform approximation by Baskakov and Meyer-König and Zeller operators,” Periodica
Mathematica Hungarica, vol. 14, no. 3-4, pp. 209–228, 1983.
5 V. Totik, “Uniform approximation by exponential-type operators,” Journal of Mathematical Analysis and
Applications, vol. 132, no. 1, pp. 238–246, 1988.
6 P. C. Xun and D. X. Zhou, “Rate of Convergence for Baskakov Operators with Jacobi-Weights,” Acta
Mathematics Applicatae Sinica, vol. 18, pp. 127–139, 1995 Chinese.
7 Z. Ditzian and V. Totik, Moduli of Smoothness, vol. 9, Springer, New York, NY, USA, 1987.
8 W. Chen, Approximation Theory of Operators, Xiamen University Publishing House, Xiamen, China, 1989.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014