Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 205298, 16 pages
doi:10.1155/2009/205298
Research Article
The Direct and Converse Inequalities for
Jackson-Type Operators on Spherical Cap
Yuguang Wang and Feilong Cao
Institute of Metrology and Computational Sciences, China Jiliang University, Hangzhou 310018, China
Correspondence should be addressed to Feilong Cao, feilongcao@gmail.com
Received 10 July 2009; Accepted 23 October 2009
Recommended by Marta A D Garcı́a-Huidobro
Approximation on the spherical cap is different from that on the sphere which requires us to
construct new operators. This paper discusses the approximation on the spherical cap. That is, the
m ∞
} is constructed to approximate the function defined on the
so-called Jackson-type operator {Jk,s
k1
spherical cap Dx0 , γ. We thus establish the direct and inverse inequalities and obtain saturation
m ∞
} on the cap Dx0 , γ. Using methods of K-functional and multiplier, we obtain
theorems for {Jk,s
k1
m
m
f − fD,p ≤ ω2 f, k −1 D,p ≤ C2 maxv≥k Jv,s
f − fD,p and that the saturation
the inequality C1 Jk,s
−2
2
order of these operators is Ok , where ω f, tD,p is the modulus of smoothness of degree 2, the
constants C1 and C2 are independent of k and f.
Copyright q 2009 Y. Wang and F. Cao. 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.
1. Introduction
In the past decades, many mathematicians dedicated to establish the Jackson and Bernsteintype theorems on the sphere see 1–9. Early works, such as Butzer and Johnen 3,
Nikol’skii and Lizorkin 8, 9, and Lizorkin and Nikol’skii 5 had successfully established
the direct and inverse theorems on the sphere. In 1991, Li and Yang 4 constructed Jackson
operators on the sphere and obtained the Jackson and Bernstein-type theorems for the Jackson
operators.
Jackson operator on the sphere is defined by see 4
Jk,s f x :
1
|Sn−2 |
Sn−1
f y Dk,s arccos x · y f y dω y ,
1.1
where k and s are positive integers,
Dk,s θ sinkθ/2
sinθ/2
2s
1.2
2
Journal of Inequalities and Applications
is the classical Jackson kernel, f is measurable function of degree p on the sphere Sn−1 in Rn ,
dωy is the elementary surface piece, |Sn−1 | is the measurement of Sn−1 . For f ∈ Lp Sn−1 ,
1 ≤ p ≤ ∞ L∞ Sn−1 is the collection of continuous functions on Sn−1 , Li and Yang 4
proved that
k C1 Jk,s f − f Sn−1 ,p ≤ ω2 f, k−1 ≤ C2 k−2 vJv,s f − f Sn−1 ,p ,
1.3
v1
and the saturation order for Jk,s is k−2 , where C1 and C2 are independent of positive integer k
and f, and ω2 f, t is the modulus of smoothness of degree 2 on the unit sphere Sn−1 .
Naturally, we desire to obtain the similar results on the spherical caps. To achieve the
goal, a key issue is to establish the inverse inequality on the cap.
when discussing
Recently, Belinsky et al. 2 constructed mth translation operator Sm
θ
the averages of functions on the sphere. This inspires us to construct the mth Jackson-type
m
m
on the spherical cap. We then prove a strong-type converse inequality for Jk,s
,
operator Jk,s
which helps us get the direct and inverse theorems of approximation on the spherical cap.
Also, we obtain that the saturation order for the constructed Jackson-type operator is k−2 , the
same to that of the Jackson operator on the sphere.
2. Definitions and Auxiliary Notations
Throughout this paper, we denote by the letters C and Ci i is either positive integers or
variables on which C depends only positive constants depending only on the dimension n.
Their value may be different at different occurrences, even within the same formula. We will
denote the points in Sn−1 by x and y, and the elementary surface piece on Sn−1 by dω. If it is
necessary, we will write dωx referring to the variable of the integration. The notation a ≈ b
means that there exists a positive constant C such that C−1 b ≤ a ≤ Cb where C is independent
of a and b.
Next, we introduce some concepts and properties of sphere as well as caps see 7, 10.
The volume of Sn−1 is
Ωn−1 :
Sn−1
dω 2π n/2
.
Γn/2
2.1
Corresponding to dω, the inner product on Sn−1 is defined by
f, g :
Sn−1
2.2
fxgxdωx.
Denote by Lp Sn−1 the space of p-integrable functions on Sn−1 endowed with the norms
f : f ∞ n−1 : ess sup fx,
∞
L S x∈Sn−1
f : f p n−1 :
p
L S Sn−1
fxp dωx
1/p
2.3
< ∞,
1 ≤ p < ∞.
Journal of Inequalities and Applications
3
We denote by Dx0 , γ the spherical cap with center x0 and angle 0 < γ ≤ π/2, that is,
D x0 , γ : x ∈ Sn−1 : x · x0 ≥ cos γ ,
2.4
and by Dγ the volume of Dx0 , γ, that is,
D γ :
γ
n−2 n−2
S sin θdθ.
2.5
0
Then for fixed x0 and γ, Lp Dx0 , γ is a Banach space endowed with the norm ·D,p defined
by
: ess sup fx,
f D,∞
f :
D,p
x∈Dx0 ,γ
fxp dωx
2.6
1/p
,
1 ≤ p < ∞.
Dx0 ,γ
For any f ∈ Lp Dx0 , γ, we note
⎧
⎨fx, x ∈ D x0 , γ ,
f ∗ x ⎩0,
x ∈ Sn−1 \ D x0 , γ ,
2.7
and clearly, f ∗ ∈ Lp Sn−1 and f ∗ p fD,p . This allows us to introduce some operators on
spherical cap using existing operators on the sphere.
Definition 2.1. Suppose that T : Lp Sn−1 → Lp Sn−1 is an operator on Sn−1 , then
Tx0 ,γ : Lp D x0 , γ −→ Lp D x0 , γ ,
Tx0 ,γ f x T f ∗ x, x ∈ D x0 , γ
2.8
is called the operator on Dx0 , γ introduced by T . We may use the notation T instead of Tx0 ,γ
for convenience without mixing up.
We now make a brief introduction of projection operators Yj · by ultraspherical
Gegenbauer polynomials {Gλj }∞
j1 λ n − 2/2 for discussion of saturation property of
Jackson operators.
Ultraspherical polynomials {Gλj }∞
j1 are defined in terms of the generating function
see 11:
1
1 − 2tr where |r| < 1, |t| ≤ 1.
r 2 λ
∞
j0
Gλj tr j ,
2.9
4
Journal of Inequalities and Applications
For any λ > 0, we have see 11
Gλ1 t 2λt,
2.10
d λ
G t 2λGλ
1
j−1 t.
dt j
When λ n − 2/2 see 7,
Gλj t
Γ 2λ j
P n t,
Γ j 1 Γ2λ j
j 0, 1, 2, . . . ,
2.11
where Pjn t is the Legendre polynomial of degree j. Particularly,
Gλj 1
Γ 2λ j
Γ 2λ j
n
P 1 ,
Γ j 1 Γ2λ j
Γ j 1 Γ2λ
j 0, 1, 2, . . . .
2.12
Therefore,
Gλj t
Pjn t Gλj 1
2.13
.
Besides, for any j 0, 1, 2, . . . , and |t| ≤ 1, |Pjn t| ≤ 1 see 10.
The projection operators is defined by
Γλn λ
Yj f x 2π n/2
Sn−1
Gλj x · y f ∗ y dω y .
2.14
It follows from 2.10 and 2.13 that
m
1 − Pjn t
j j 2λ
lim
.
m t → 1− 1 − P n t
2λ 1
1
2.15
In the same way, we define the inner product on Dx0 , γ as follows:
f, g D :
fxgxdωx.
Dx0 ,γ
2.16
the Laplace-Beltrami operator
We denote by Δ
n
2
∂
gx
:
Δf
2 ∂x
i1
i
,
|x|1
gx f
x
,
|x|
2.17
Journal of Inequalities and Applications
5
by which we define a K-function on Dx0 , γ as
K f, δ D,p : inf f − g D,p δΔg
D,p
∈ Lp D x0 , γ
: g, Δg
.
2.18
For f ∈ L1 Dx0 , γ, the translation operator is defined by
Sθ f x :
1
|Sn−2 |sinn−2 θ
f ∗ y dω
y ,
x·ycos θ
2.19
where dω
y denotes the the elementary surface piece on the sphere {y ∈ Dx0 , γ : x · y cos θ}. Then we have
Dx0 ,γ fxdωx π
Sθ f x0 Sn−2 sinn−2 θdθ.
0
2.20
The modulus of smoothness of f is defined by
ω2 f, δ D,p : sup Sθ f − f D,p .
2.21
0<θ≤δ
Using the method of 3, we have
C1 ω2 f, δ D,p ≤ K f, δ2
D,p
≤ C2 ω2 f, δ D,p .
2.22
We introduce mth translation operator in terms of multipliers see 6, 7, 12
⎛
⎞m
∞
∞ m Gλj cos θ
n
⎝
⎠ Yj f ≡
Sm
P
Yj f ,
θ
cos
j
θ f Gλj 1
j0
j0
f ∈ Lp D x0 , γ .
2.23
It has been proved that see 7
Sθ f x 1
|Sn−2 |sinn−2 θ
x·ycos θ
∞
f ∗ y dω
y Pjn cos θYj f x
j0
S1θ f x.
2.24
, we can construct Jackson-type operator on Dx0 , γ.
With the help of Sm
θ
Definition 2.2. For f ∈ Lp Dx0 , γ, the mth Jackson-type operator of degree k on Dx0 , γ is
defined by
m
Jk,s
f x γ
0
2λ
Sm
θ f xDk,s θsin θdθ,
2.25
6
Journal of Inequalities and Applications
k,s θ A−1 sin2s kθ/2/sin2s−1 θ/2 satisfying
where λ n − 2/2, and D
γ,k,s
γ
0
k,s θ ×
D
sin2λ θdθ 1.
k,s θ is a bit different from classical Jackson kernel
Remark 2.3. We may notice that D
Dk,s θ sinkθ/2
sinθ/2
2s
.
2.26
m
This difference will help us to prove the converse inequality for Jk,s
. For sake of ensuring that
m
the converse inequality for Jk,s holds, γ has to be no more than π/2. Particularly, for m 1,
we have
γ
1
k,s θsin2λ θdθ
Jk,s f x S1θ f xD
0
γ
1
2λ
0 |Sn−2 |sin θ
1
|Sn−2 |
1
|Sn−2 |
k,s θsin2λ θdθ
f ∗ y dω
y D
x·ycos θ
k,s arccos x · y dω y
f y D
2.27
∗
Dx0 ,γ
Dx0 ,γ
k,s arccos x · y dω y .
f y D
Dx,γ
Finally, we introduce the definition of saturation for operators see 13.
Definition 2.4. Let ϕρ be a positive function with respect to ρ, 0 < ρ < ∞, tending
monotonely to zero as ρ → ∞. For ρ > 0, Iρ is a sequence of operators. If there exists
K Lp Dx0 , γ such that:
i If Iρ f − fD,p oϕρ, then Iρ f f;
ii Iρ f − fD,p Oϕρ if and only if f ∈ K;
then Iρ is said to be saturated on Lp Dx0 , γ with order Oϕρ and K is called its saturation
class.
3. Some Lemmas
m
In this section, we show some lemmas on both Sm
and Jk,s
as the preparation for the main
θ
m
results. For Sθ , we have the following.
Lemma 3.1. For f ∈ Lp Dx0 , γ, 1 ≤ p ≤ ∞, 0 < θ ≤ π,
1 m−m1
Sm
Sθ
;
i for 1 ≤ m1 < m, Sm
θ
θ
ii for m ≥ 1, Sm
fD,p ≤ fD,p ;
θ
iii for m ≥ 1, Sm
f − fD,p ≤ mSθ f − fD,p ;
θ
m fD,p ≤ Cm θ−2 fD,p , where
iv for m > 2n/2 3/n − 2, 0 < θ ≤ π/2, ΔS
θ
Cm → 0, as m → ∞;
∈ Lp Dx0 , γ, ΔS
m fD,p ≤ Δf
D,p .
v for m ≥ 1, and f which satisfies Δf
θ
Journal of Inequalities and Applications
7
Proof. i, ii and iii are clear. Using 2, Remark 3.5, we can obtain iv. For v, we have
∞
,
θ f − j j 2λ P n cos θYj f Sθ Δf
ΔS
j
3.1
j0
which implies
m ΔSθ f
D,p
m−1 f Sθ ΔS
θ
D,p
m−1 ≤ ΔS
f
θ
D,p
≤ · · · ≤ Δf
D,p
.
3.2
We need the following lemma.
Lemma 3.2. For β ≥ −2, 2s > β 2λ 1, 0 < γ ≤ π, n ≥ 3, and k ≥ 1, one has
γ
k,s θsin2λ θ dθ ≈ k−β ,
θβ D
3.3
0
where λ n − 2/2.
Proof. A simple calculation gives, for β ≥ −2,
γ
θ
0
γ
θ
0
β
sin2s kθ/2
sin2s−1 θ/2
2s
β sin kθ/2
sin2s−1 θ/2
sin θdθ ≤ C
2λ
∞
0
sin2λ θdθ ≥ C
sin2s
θ2s−2λ
β
1
kγ/4
0
dθ k2s−2λ
β
2 ,
sin2s
θ2s−2λ
β
1
3.4
dθ k2s−2λ
β
2 .
Therefore,
γ
γ
θβ sin2s kθ/2/sin2s−1 θ/2sin2λ θdθ
k,s θsin θdθ 0
θ D
≈ k−β .
γ
2s
2s−1
2λ
sin kθ/2/sin
θ/2sin θdθ
0
0
β
2λ
3.5
For Jackson-type operator, we have the following lemma.
Lemma 3.3. For f ∈ Lp Dx0 , γ, 1 ≤ p < ∞, there hold
m
fD,p ≤ fD,p ;
i Jk,s
∈ Lp Dx0 , γ, ΔJ
m fD,p ≤ Δf
D,p ;
ii for f which satisfies Δf
k,s
m fD,p ≤ Cm k2 fD,p .
iii for n ≥ 3, m > 2n/2 3/n − 2, and 0 < γ ≤ π/2, ΔJ
k,s
8
Journal of Inequalities and Applications
Proof. From the definition and ii and v of Lemma 3.1, i and ii are clear. We just have
to add the proof of iii. In fact, using Minkowski inequality, iv of Lemmas 3.1 and 3.2, we
have
m
ΔJk,s f
D,p
γ
m ≤ ΔS
f
θ
D,p
0
≤ Cm f D,p
γ
k,s θsin2 θdθ
D
k,s θsin2λ θdθ
θ−2 D
3.6
0
≈ Cm,γ k2 f D,p ,
where the constant in the approximation is independent of m and k.
The following lemma is useful in the proof of the converse inequality for Jackson-type
operator.
∞
Lemma 3.4 see 14. Suppose that for nonnegative sequences {σk }∞
k1 , {τk }k1 with σ1 0 the
inequality
p
k
σn ≤
σk τk ,
n
p > 0, 1 ≤ k ≤ n,
3.7
is satisfied for any positive integer n. Then one has
σn ≤ Cp n−p
n
kp−1 τk .
3.8
k1
m
The following lemma gives the multiplier representation of Jk,s
f, which follows from
Definition 2.2 and 2.23.
m
f has the representation
Lemma 3.5. For f ∈ Lp Dx0 , γ, Jk,s
∞
m
ξkm j Yj f x,
Jk,s
f x 3.9
j0
where
ξkm
j γ
0
m
k,s θ P n cos θ sin2λ θ dθ,
D
j
j 0, 1, 2, . . . .
3.10
The following lemma is useful for determining the saturation order. It can be deduced
by the methods of 13, 15.
Journal of Inequalities and Applications
9
Lemma 3.6. Suppose that {Iρ }ρ>0 is a sequence of operators on Lp Dx0 , γ, and there exists
function series {λρ j}∞
with respect to ρ, such that
j1
∞
λρ j Yj f x
Iρ f x 3.11
j0
for every f ∈ Lp Dx0 , γ. If for any j 0, 1, 2, . . . , there exists ϕρ → 0 ρ → ρ0 such that
1 − λρ j
lim
τj / 0,
ρ → ρ0
ϕ ρ
3.12
then {Iρ }ρ>0 is saturated on Lp Dx0 , γ with the order Oϕρ and the collection of all constants
is the invariant class for {Iρ }ρ>0 on Lp Dx0 , γ.
4. Main Results and Their Proofs
In this section, we will discuss the main results, that is, the lower and upper bounds as well
as the saturation order for Jackson-type operator on Lp Dx0 , γ.
m
.
The following theorem gives the Jackson-type inequality for Jk,s
m ∞
Theorem 4.1. For any integer m ≥ 1 and 0 < γ ≤ π/2, {Jk,s
}k1 is the series of Jackson-type
2
p
∈
operators on L Dx0 , γ defined previously, and g ∈ Lp Dx0 , γ : {f ∈ Lp Dx0 , γ : Δf
Lp Dx0 , γ}, 1 ≤ p ≤ ∞.
Then
m
Jk,s g − g D,p
≤ Cm,γ k−2 Δg
D,p
4.1
.
Therefore, for f ∈ Lp Dx0 , γ,
m
Jk,s f − f D,p
≤ Cm,γ ω2 f, k−1
D,p
,
4.2
where C is independent of k and f.
2
Proof. Since g ∈ Lp Dx0 , γ, we have see 13
Sθ g x − gx θ
sin
0
2−n
ν
ν
0
xdτ dν,
sinn−2 τSτ Δg
4.3
10
Journal of Inequalities and Applications
and it is true that
sup θ
−2
θ
sin
2−n
ν
sin
0
θ>0
ν
n−2
τdτ dν
< ∞.
4.4
0
Therefore explained below,
m
Jk,s g − g D,p
γ
k,s θ Sm g x − gx sinn−2 θdθ
D
θ
0
≤m
D,p
γ
k,s θsinn−2 θ
D
θ
sin
0
0
≤ m sup θ
θ>0
2−n
−2
θ
2−n
sin
0
≤ Cm,γ k−2 Δg
sin
τ Sτ Δg
D,p
n−2
sin
γ
τdτ dν Δg
D,p
0
D,p
ν
n−2
0
ν
ν
ν
dτdν dθ
k,s θsinn−2 θ dθ
θ2 D
0
,
4.5
where the Minkowski inequality, 4.3, and Lemma 3.1 are used in the first inequality, and
the second and third one are deduced from 3.3 and Lemma 3.1. From 2.22 and i of
Lemma 3.3, it is easy to deduce 4.2.
m
Next, we prove the Bernstein-type inequality for Jk,s
fx for f ∈ Lp Dx0 , γ.
m ∞
Theorem 4.2. Assume that {Jk,s
}k1 , m > 2n/2 3/n − 2 are mth Jackson-type operators on
p
Dx0 , γ. For f ∈ L Dx0 , γ, 0 < γ ≤ π/2, then there exits a constant C independent of k and f
such that
ω2 f, k−1
m
≤ C maxJv,s
f − f D,p
D,p
4.6
v≥k
holds for every f ∈ Lp Dx0 , γ and every integer k.
Proof. Li and Yang 4 have proved the Marchaud-Stec̆kin inequality for Jackson operator on
the sphere. Following the method in 4, we first prove the Marchaud-Stec̆kin inequality for
m
:
Jk,s
ω2 f, k−1
D,p
≤ C1 k−2
k m
vJv,s
f − f D,p .
4.7
v1
Let
m
σk k−2 ΔJ
f
k,s
D,p
,
m
τk Jk,s
f − f D,p
,
4.8
Journal of Inequalities and Applications
11
then for v 1, 2, . . . , k, by Lemma 3.3,
m
m
m
m Jv,s
m Jv,s
σk k−2 ΔJ
f
−
ΔJ
f
ΔJ
f
k,s
k,s
k,s
m
m
k−2 ΔJ
f
−
J
f
v,s
k,s
D,p
≤ k−2
m m
ΔJ
J
f
k,s v,s
D,p
D,p
m
m
fD,p ΔJ
Ck2 f − Jv,s
v,s f
D,p
4.9
m
m
Cf − Jv,s
fD,p k−2 ΔJ
f
v,s
D,p
2
v
Cτv σv ,
k
so we can deduce from Lemma 3.4 where p is set to be 2 that
σk ≤ Ck−2
k
vτv ,
4.10
v1
that is,
m
ΔJ f
k,s
D,p
Since there exists k/2 ≤ r ≤ k such that
m
Jr,s f − f k m
vJv,s
f − f D,p .
4.11
m
min Jv,s
f − f D,p
4.12
≤C
D,p
v1
k/2≤v≤k
then,
K f, k−2
D,p
m
m
≤ Jr,s
f − f D,p k−2 ΔJ
f
r,s
D,p
≤ 4k−2
k/2≤v≤k
≤ Ck−2
k m
m
vJv,s
f − f D,p Ck−2 vJv,s
f − f D,p
v1
4.13
k m
vJv,s
f − f D,p .
v1
By 2.22, we obtain that
ω2 f, k−1
D,p
≤ CK f, k−2
D,p
k m
≤ Ck
vJv,s
f − f D,p .
−2
v1
4.14
12
Journal of Inequalities and Applications
So 4.7 holds, and it implies that
ω2 f, k−1
≤ C1 k−1−1/2
D,p
k
m
v1/2 Jv,s
f − f D,p .
4.15
v1
In order to prove 4.6, we have to show that
ω2 f, k−1
m
1
max v2 Jv,s
f − f D,p
2
k 1≤v≤k
≈
D,p
1
m
max v2
1/4 Jv,s
f
k2
1/4 1≤v≤k
≈
4.16
− f
D,p
.
We first prove
ω2 f, k−1
D,p
≈
1
2 m
.
J
max
v
f
−
f
v,s
D,p
k2 1≤v≤k
4.17
It follows from 4.2 and 4.15 that
ω2 f, k−1
D,p
≤ C1 k−2 k1/2
k
v1
≤ C1 k
−2
m
v−3/2 v2 Jv,s
f − f D,p
m
v2 Jv,s
max
1≤v≤k
f − f D,p
k
1/2
k
v−3/2
v1
m
≤ C3 k−2 max v2 Jv,s
f − f D,p
1≤v≤k
−2
≤ C4 k max v ω
2
2
f, v
1≤v≤k
−1
4.18
D,p
−2
≤ C5 k max v
2
1≤v≤k
≤ 2C5 ω2 f, k−1
D,p
2 k
1
ω2 f, k−1
D,p
v
.
Then we prove
ω2 f, k−1
D,p
≈
1
m
max v2
1/4 Jv,s
f − f D,p .
k2
1/4 1≤v≤k
4.19
Journal of Inequalities and Applications
13
In fact, the proof is similar to that of 4.17
ω2 f, k−1
D,p
≤ C1 k−2−1/4 k3/4
k
m v−7/4 v2
1/4 Jv,s
f − f D,p
k1
≤ C1 k
−2−1/4
max
1≤v≤k
m
v2
1/4 Jv,s
f − f D,p
k
3/4
k
v−7/4
v1
m
f − f D,p
≤ C6 k−2−1/4 max v2
1/4 Jv,s
1≤v≤k
≤ C7 k−2−1/4 max v2
1/4 ω2 f, v−1
1≤v≤k
≤ C8 k
−2−1/4
D,p
max v
2
1/4
1≤v≤k
2 k
1
ω2 f, k−1
≤ 2C8 ω2 f, k−1
.
D,p
D,p
v
4.20
Hence,
ω2 f, k−1
D,p
≈
m
m
1
1
max v2 Jv,s
f − f D,p ≈ 2
1/4 max v2
1/4 Jv,s
f − f D,p .
2
1≤v≤k
k 1≤v≤k
k
4.21
Now we can complete the proof of 4.6. Let
m
max v2
1/4 Jv,s
f − f D,p k12
1/4 Jkm1 ,s f − f D,p
1≤v≤k
1 ≤ k1 ≤ k.
,
4.22
It follows from 4.16 that
k−2 k12 Jkm1 ,s f − f D,p
m
≤ k−2 max v2 Jv,s
f − f D,p
1≤v≤k
≤
m
C9
max v2
1/4 Jv,s
f
2
1/4
1≤v≤k
k
− f D,p
2
1/4 m
k
f
−
f
J
k
,s
1
1
2
1/4
C9
k
D,p
4.23
.
Thus, C9−4 k ≤ k1 . Since k1 ≤ k, then k ≈ k1 , we obtain from 4.16 that
ω2 f, k−1
D,p
m
≤ C10 k−2−1/4 max v2
1/4 Jv,s
f − f D,p
1≤v≤k
C10 k−2−1/4 k12
1/4 Jkm1 ,s f − f D,p
m
≤ C10 k−2−1/4 max v2
1/4 Jv,s
f − f D,p
k1 ≤v≤k
m
≤ C10 max Jv,s
f − f D,p .
k1 ≤v≤k
4.24
14
Journal of Inequalities and Applications
Noticing that k ≈ k1 , we may rewrite the previous inequality as
ω2 f, k−1
D,p
m
≤ C11 maxJv,s
f − f D,p .
v≥k
4.25
This completes the proof.
We thus obtain the corollary of Theorems 4.1 and 4.2.
m ∞
Corollary 4.3. Suppose that {Jk,s
}k1 , m > 2n/2 3/n − 2, are Jackson-type operators on the
spherical cap Dx0 , γ, 0 < γ ≤ π/2, then the following are equivalent for any f ∈ Lp Dx0 , γ,
α > 0,
m
f − fD,p Ok−α , k → ∞;
i Jk,s
ii ω2 f, δD,p Oδ α , δ → 0
.
m ∞
Theorem 4.4. Suppose that {Jk,s
}k1 , m ≥ 1 are Jackson-type operators on the spherical cap Dx0 , γ,
m ∞
0 < γ ≤ π/2. Then {Jk,s }k1 are saturated on Lp Dx0 , γ with order k−2 and the collection of
constants is their invariant class.
Proof. We obtain from Lemma 3.2 that, for v 1, 2, . . . ,
1−
ξkm 1
γ
k,s θ 1 −
D
0
Gλ1 cos θ
m sin2λ θ dθ ≈
Gλ1 1
γ
k,s θsin2 θ sin2λ θ dθ ≈ k−2 .
D
2
0
4.26
By Lemma 3.6, if it is true that for j 0, 1, 2, . . . ,
1 − ξkm j
j j 2λ
lim
,
m
k → ∞ 1 − ξ 1
2λ 1
k
4.27
then the proof is completed.
In fact, for any 0 < δ < γ, it follows from 3.3 that
γ
δ
γ 3
θ
k,s θsin2λ θ dθ
D
δ δ
γ
k,s θsin2λ θ dθ ≤ Cδ,s k−3 .
≤ δ−3 θ 3 D
k,s θsin2λ θ dθ ≤
D
4.28
0
We deduce from 2.15 that, for any ε > 0, there exists δ > 0, for 0 < θ < δ, it holds that
m j j 2λ m m n
n
−
.
1 − P1 cos θ
1 − Pj cos θ
≤ ε 1 − P1n cos θ
2λ 1
4.29
Journal of Inequalities and Applications
15
Then it follows that
j j 2λ m
m
1 − ξk 1 1 − ξk j −
2λ 1
γ
m k,s θ 1 − P n cos θ
D
sin2λ θ dθ
j
0
j
j
2λ
m
k,s θ 1 − P n cos θ
− D
sin2λ θ dθ
1
2λ
1
0
4.30
γ
m m j j 2λ
D
− 1 − P1n cos θ
sin2λ θ dθ
θ 1 − Pjn cos θ
0 k,s
2λ 1
δ
γ
j j 2λ
2λ
2λ
k,s θ ε sin θ dθ 2 D
k,s θsin θ 1 ≤ D
dθ
2λ 1
0
δ
γ
≤ εk−2 Cδ,s k−3 .
So,
j j 2λ
1 − ξk j
lim
0.
/
k → ∞ 1 − ξk 1
2λ
−2
4.31
m
is O1 − ξkm 1 ≈
Therefore, we obtain by Lemma 3.6 that the saturation order for Jk,s
k .
Acknowledgments
The research was supported by the National Natural Science Foundation of China no.
60873206,the Natural Science Foundation of Zhejiang Province of China no. Y7080235, the
Key Foundation of Department of Education of Zhejiang Province of China no. 20060543,
and the Innovation Foundation of Post-Graduates of Zhejiang Province of China no.
YK2008066.
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