# Uncertainty Principles and Extremal Functions for the DunklL

Authors: Fethi Soltani
Journal: Journal of Operators
 (2014)

---

## Abstract

We study some class of DunklL
2-multiplier operators; and related to these
operators we establish the Heisenberg-Pauli-Weyl uncertainty principle and Donoho-Stark’s uncertainty principle. We give also an application of the theory of reproducing
kernels to the Tikhonov regularization on the Sobolev-Dunkl spaces.


---

## Body

## 1. Introduction

In this paper, we considerR
d with the Euclidean inner product ·
,
· and norm |
y
|
≔
y
,
y. For α
∈
R
d
∖
{
0
}, let σ
α be the reflection in the hyperplane H
α
⊂
R
d orthogonal to α:
(1)
σ
α
x
≔
x
-
2
α
,
x
α
2
α
.A finite setR
⊂
R
d
∖
{
0
} is called a root system, if R
∩
R.  α
=
{
-
α
,
α
} and σ
α
R
=
R for all α
∈
R. We assume that it is normalized by |
α
|
2
=
2 for all α
∈
R. For a root system R, the reflections σ
α,  α
∈
R, generate a finite group G
⊂
O
(
d
), the reflection group associated with R. All reflections in G correspond to suitable pairs of roots. For a given β
∈
R
d
∖
⋃
α
∈
R
‍
H
α, we fix the positive subsystem R
+
≔
{
α
∈
R
:
〈
α
,
β
〉
>
0
}. Then for each α
∈
R either α
∈
R
+ or -
α
∈
R
+.Letk
:
R
→
C be a multiplicity function on R (i.e., a function which is constant on the orbits under the action of G). As an abbreviation, we introduce the index γ
=
γ
k
≔
∑
α
∈
R
+
‍
k
(
α
).Throughout this paper, we will assume thatk
(
α
)
≥
0 for all α
∈
R. Moreover, let w
k denote the weight function w
k
(
x
)
≔
∏
α
∈
R
+
‍
|
〈
α
,
x
〉
|
2
k
(
α
), for all x
∈
R
d, which is G-invariant and homogeneous of degree 2
γ.Letc
k be the Mehta-type constant given by
(2)
c
k
≔
∫
R
d
e
-
x
2
/
2
w
k
x
d
x
-
1
.
We denote by μ
k the measure on R
d given by d
μ
k
(
x
)
≔
c
k
w
k
(
x
)
d
x and by L
p
(
μ
k
), 1
≤
p
≤
∞, the space of measurable functions f on R
d, such that
(3)
f
L
p
μ
k
≔
∫
R
d
f
x
p
d
μ
k
x
1
/
p
<
∞
,
55555555555555555515555555
1
≤
p
<
∞
,
f
L
∞
μ
k
≔
ess
sup
⁡
x
∈
R
d
f
x
<
∞
.Forf
∈
L
1
(
μ
k
) the Dunkl transform is defined (see [1]) by
(4)
F
k
f
y
≔
∫
R
d
E
k
-
i
x
,
y
f
x
d
μ
k
x
,
555555555555555155555555555555
y
∈
R
d
,
where E
k
(
-
i
x
,
y
) denotes the Dunkl kernel (for more details, see Section 2).Many uncertainty principles have already been proved for the Dunkl transform, namely, by Rösler [2] and Shimeno [3] who established the Heisenberg-Pauli-Weyl inequality for the Dunkl transform, by showing that, for every f
∈
L
2
(
μ
k
),
(5)
f
L
2
μ
k
2
≤
2
2
γ
+
d
x
f
L
2
μ
k
y
F
k
f
L
2
μ
k
.
Recently, the author [4, 5] proved general forms of the Heisenberg-Pauli-Weyl inequality for the Dunkl transform.Letm be a function in L
2
(
μ
k
). The Dunkl L
2-multiplier operators, T
k
,
m, are defined, for regular functions f on R
d, by
(6)
T
k
,
m
f
a
,
x
≔
F
k
-
1
m
a
x
F
k
f
x
,
5555515555555555
a
,
x
∈
0
,
∞
×
R
d
.
These operators are studied in [6, 7] where the author established some applications (Calderón’s reproducing formulas, best approximation formulas, and extremal functions…).Form
∈
L
2
(
μ
k
) satisfying the admissibility condition: ∫
0
∞
‍
|
m
(
a
x
)
|
2
(
d
a
/
a
)
=
1, a.e.  x
∈
R
d, then the operators T
k
,
m satisfy Plancherel’s formula:
(7)
T
k
,
m
f
L
2
Ω
k
2
≔
∫
R
d
∫
0
∞
T
k
,
m
f
a
,
x
2
d
Ω
k
a
,
x
=
f
L
2
μ
k
2
,
5555555555555555555555555555555555555555
f
∈
L
2
μ
k
,
where Ω
k is the measure on ]
0
,
∞
[
×
R
d given by d
Ω
k
(
a
,
x
)
≔
(
d
a
/
a
)
d
μ
k
(
x
).For the operatorsT
k
,
m we establish a Heisenberg-Pauli-Weyl uncertainty principle. More precisely, we will show, for f
∈
L
2
(
μ
k
),
(8)
f
L
2
μ
k
2
≤
2
2
γ
+
d
y
F
k
f
L
2
μ
k
·
∫
R
d
∫
0
∞
x
2
T
k
,
m
f
a
,
x
2
d
a
a
d
μ
k
x
1
/
2
,
provided m
∈
L
2
(
μ
k
) satisfying ∫
0
∞
‍
|
m
(
a
x
)
|
2
(
d
a
/
a
)
=
1, a.e. x
∈
R
d.Building on the techniques of Donoho and Stark [8], we show a continuous-time principle for the L
2 theory. Let E be measurable subset of R
d, let S be measurable subset of ]
0
,
∞
[
×
R
d, and let f
∈
L
2
(
μ
k
). If f is ɛ-concentrated on E and T
k
,
m
f is η-concentrated on S (see Section 3 for more details), then
(9)
μ
k
E
1
/
2
∫
∫
S
1
a
2
2
γ
+
d
d
Ω
k
a
,
x
1
/
2
≥
1
-
η
-
ɛ
m
L
1
μ
k
,
provided m
∈
L
1
∩
L
2
(
μ
k
) satisfying ∫
0
∞
‍
|
m
(
a
x
)
|
2
(
d
a
/
a
)
=
1, a.e. x
∈
R
d.Building on the ideas of Saitoh [9, 10], Matsuura et al. [11], and Yamada et al. [12], we give an application of the theory of reproducing kernels to the Tikhonov regularization, which gives the best approximation of the operator T
k
,
m on the Sobolev-Dunkl spaces H
s
(
μ
k
). More precisely, for all λ
>
0,  g
∈
L
2
(
Ω
k
), the infimum
(10)
inf
⁡
f
∈
H
s
μ
k
λ
f
H
s
μ
k
2
+
g
-
T
k
,
m
f
L
2
Ω
k
2
is attained at one function f
λ
,
g
*, called the extremal function.In particular forf
∈
H
s
(
μ
k
) and g
=
T
k
,
m
f, the corresponding extremal functions {
f
λ
*
}
λ
>
0 converge to f as λ
→
0
+.This paper is organized as follows. In Section2 we define the Dunkl L
2-multiplier operators T
k
,
m, and we give for them Plancherel’s formula. Some examples of Dunkl L
2-multiplier operators are given. In Section 3 we establish the Heisenberg-Pauli-Weyl uncertainty principle and Donoho-Stark’s uncertainty principle for the operators T
k
,
m. In the last section we give an application of the theory of reproducing kernels to the Tikhonov regularization related to the operators T
k
,
m on the Sobolev-Dunkl spaces H
s
(
μ
k
).


## 2. The DunklL
2-Multiplier Operators on R
d

The Dunkl operatorsD
j,  j
=
1
,
…
,
d, on R
d associated with the finite reflection group G and multiplicity function k are given, for a function f of class C
1 on R
d, by
(11)
D
j
f
x
≔
∂
∂
x
j
f
x
+
∑
α
∈
R
+
k
α
α
j
f
x
-
f
σ
α
x
α
,
x
.Fory
∈
R
d, the initial problem D
j
u
(
·
,
y
)
(
x
)
=
y
j
u
(
x
,
y
), j
=
1
,
…
,
d, with u
(
0
,
y
)
=
1 admits a unique analytic solution on R
d, which will be denoted by E
k
(
x
,
y
) and called Dunkl kernel [13, 14]. This kernel has a unique analytic extension to C
d
×
C
d (see [15]). In our case (see [1, 13]),
(12)
E
k
i
x
,
y
≤
1
,
x
,
y
∈
R
d
.The Dunkl kernel gives rise to an integral transform, which is called Dunkl transform onR
d, and was introduced by Dunkl in [1], where already many basic properties were established. Dunkl’s results were completed and extended later by de Jeu [14]. The Dunkl transform of a function f, in L
1
(
μ
k
), is defined by
(13)
F
k
f
y
≔
∫
R
d
E
k
-
i
x
,
y
f
x
d
μ
k
x
,
555555555555555555555555555555
y
∈
R
d
.
We notice that F
0 agrees with the Fourier transform F that is given by
(14)
F
f
y
≔
2
π
-
d
/
2
∫
R
d
e
-
i
x
,
y
f
x
d
x
,
x
∈
R
d
.Some of the properties of Dunkl transformF
k are collected bellow (see [1, 14]).Theorem 1.
(i)
L
1
-
L
∞-boundedness: for all f
∈
L
1
(
μ
k
), F
k
(
f
)
∈
L
∞
(
μ
k
) and
(15)
F
k
f
L
∞
μ
k
≤
f
L
1
μ
k
.
(ii) Inversion theorem: let f
∈
L
1
(
μ
k
), such that F
k
(
f
)
∈
L
1
(
μ
k
). Then
(16)
f
x
=
F
k
F
k
f
-
x
,
a
.
e
.
x
∈
R
d
.
(iii) Plancherel theorem: the Dunkl transform F
k extends uniquely to an isometric isomorphism of L
2
(
μ
k
) onto itself. In particular,
(17)
F
k
f
L
2
μ
k
=
f
L
2
μ
k
.Letm be a function in L
2
(
μ
k
). The Dunkl L
2-multiplier operators, T
k
,
m, are defined, for regular functions f on R
d, by
(18)
T
k
,
m
f
a
,
x
≔
F
k
-
1
m
a
x
F
k
f
x
,
5555555555555555
a
,
x
∈
0
,
∞
×
R
d
.The operatorsT
k
,
m satisfy the following integral representation.Lemma 2.
Ifm
,
f
∈
L
1
∩
L
2
(
μ
k
), then
(19)
T
k
,
m
f
a
,
x
=
1
a
2
γ
+
d
∫
R
d
H
k
x
a
,
y
a
,
m
f
y
d
μ
k
y
,
555555555555555555555555555
a
,
x
∈
0
,
∞
×
R
d
,
where(20)
H
k
x
,
y
,
m
=
∫
R
d
m
z
E
k
i
x
,
z
E
k
-
i
y
,
z
d
μ
k
z
.Proof.
The result follows from ((18)) and Theorem 1(ii) using Fubini-Tonnelli’s theorem.We denote byΩ
k the measure on ]
0
,
∞
[
×
R
d given by d
Ω
k
a
,
x
≔
(
d
a
/
a
)
d
μ
k
(
x
) and by L
2
(
Ω
k
) the space of measurable functions F on ]
0
,
∞
[
×
R
d, such that
(21)
F
L
2
Ω
k
≔
∫
R
d
∫
0
∞
F
a
,
x
2
d
Ω
k
a
,
x
1
/
2
<
∞
.In the following, we give Plancherel formula for the operatorsT
k
,
m.Theorem 3 (Plancherel formula).
Letm be a function in L
2
(
μ
k
) satisfying the admissibility condition:
(22)
∫
0
∞
m
a
x
2
d
a
a
=
1
,
a
.
e
.
x
∈
R
d
.
Then, for f
∈
L
2
(
μ
k
), one has
(23)
T
k
,
m
f
L
2
Ω
k
=
f
L
2
μ
k
.Proof.
From Fubini-Tonnelli’s theorem, we obtain(24)
∫
R
d
∫
0
∞
T
k
,
m
f
a
,
x
2
d
Ω
k
a
,
x
=
∫
0
∞
∫
R
d
m
a
y
2
F
k
f
y
2
d
μ
k
y
d
a
a
=
∫
R
d
F
k
f
y
2
∫
0
∞
m
a
y
2
d
a
a
d
μ
k
y
.
Then, the result follows from ((22)) and Theorem 1(iii).As applications, we give the following examples.Example 4.
Letm
t, t
>
0, the function is defined by
(25)
m
t
x
≔
-
8
t
x
2
e
-
t
x
2
,
x
∈
R
d
.
Then(a)
m
t belongs to L
1
∩
L
2
(
μ
k
), and by ((2)), we have
(26)
m
t
L
1
μ
k
=
8
t
∫
R
d
x
2
e
-
t
x
2
d
μ
k
x
=
-
8
t
∂
∂
t
∫
R
d
e
-
t
x
2
d
μ
k
x
=
2
2
γ
+
d
2
t
2
γ
+
d
,
m
t
L
2
μ
k
2
=
8
t
2
∫
R
d
x
4
e
-
2
t
x
2
d
μ
k
x
=
2
t
2
∂
2
∂
t
2
∫
R
d
e
-
2
t
x
2
d
μ
k
x
=
2
γ
+
d
γ
+
d
/
2
+
1
2
t
2
γ
+
d
.
(b)
m
t satisfies the admissibility condition ((22)), that is,
(27)
∫
0
∞
m
t
a
x
2
d
a
a
=
8
t
2
x
4
∫
0
∞
a
3
e
-
2
t
x
2
a
2
d
a
=
1
.Then the associated operatorsT
k
,
m
t satisfy Plancherel’s formula ((23)).To express the operatorT
k
,
m
t, we use Lemma 2, then for f
∈
L
1
∩
L
2
(
μ
k
), we have
(28)
T
k
,
m
t
f
a
,
x
=
8
t
a
2
γ
+
d
∫
R
d
∂
∂
t
h
k
x
a
,
y
a
,
t
f
y
d
μ
k
y
,
55555555555555555555555555555555555555555
x
∈
R
d
,
where
(29)
h
k
x
,
y
,
t
=
∫
R
d
e
-
t
z
2
E
k
i
x
,
z
E
k
-
i
y
,
z
d
μ
k
z
is the Dunkl-type heat kernel [16, 17]. From [16] this kernel is given by
(30)
h
k
x
,
y
,
t
=
1
2
t
γ
+
d
/
2
e
-
x
2
+
y
2
/
4
t
E
k
x
2
t
,
y
2
t
.Example 5.
Letm
t,  t
>
0, be the function defined by
(31)
m
t
x
≔
-
2
t
x
e
-
t
x
,
x
∈
R
d
.
Then one has:(a)m
t belongs to L
1
∩
L
2
(
μ
k
), and
(32)
m
t
L
1
μ
k
=
2
t
∫
R
d
x
e
-
t
x
d
μ
k
x
=
-
2
t
∂
∂
t
∫
R
d
e
-
t
x
d
μ
k
x
. Since
(33)
e
-
t
x
=
1
π
∫
0
∞
e
-
s
s
e
-
t
2
/
4
s
x
2
d
s
,
by Fubini-Tonnelli’s theorem and ((2)), we deduce that
(34)
∫
R
d
e
-
t
x
d
μ
k
x
=
1
π
∫
0
∞
e
-
s
s
∫
R
d
e
-
t
2
/
4
s
x
2
d
μ
k
x
d
s
=
1
π
∫
0
∞
e
-
s
s
2
s
t
2
γ
+
d
d
s
=
Γ
γ
+
d
+
1
/
2
π
2
t
2
γ
+
d
.
Thus,
(35)
m
t
L
1
μ
k
=
2
2
γ
+
d
Γ
γ
+
d
+
1
/
2
π
2
t
2
γ
+
d
.
On the other hand,
(36)
m
t
L
2
μ
k
2
=
4
t
2
∫
R
d
x
2
e
-
2
t
x
d
μ
k
x
=
∂
2
∂
t
2
∫
R
d
e
-
2
t
x
d
μ
k
x
=
∂
2
∂
t
2
Γ
γ
+
d
+
1
/
2
π
2
t
2
γ
+
d
.
Thus,
(37)
m
t
L
2
μ
k
2
=
4
2
γ
+
d
Γ
γ
+
d
+
3
/
2
π
2
t
2
γ
+
d
+
2
.
(b)m
t satisfies the admissibility condition ((22)); that is,
(38)
∫
0
∞
m
t
a
x
2
d
a
a
=
4
t
2
x
2
∫
0
∞
a
e
-
2
t
x
a
d
a
=
1
. Then the associated operators T
k
,
m
t satisfy Plancherel’s formula ((23)).
To express the operatorsT
k
,
m
t, we use Lemma 2; then for f
∈
L
1
∩
L
2
(
μ
k
), we have
(39)
T
k
,
m
t
f
a
,
x
=
2
t
a
2
γ
+
d
∫
R
d
∂
∂
t
p
k
x
a
,
y
a
,
t
f
y
d
μ
k
y
,
where
(40)
p
k
x
,
y
,
t
=
∫
R
d
e
-
t
z
E
k
i
x
,
z
E
k
-
i
y
,
z
d
μ
k
z
is the Dunkl-type Poisson kernel [18]. From ((33)) this kernel is given by
(41)
p
k
x
,
y
,
t
=
1
π
∫
0
∞
e
-
s
s
h
k
x
,
y
,
t
2
4
s
d
s
.


## 3. Uncertainty Principle for the OperatorsT
k
,
m

### 3.1. Heisenberg-Pauli-Weyl Uncertainty Principle

This section is devoted to establish Heisenberg-Pauli-Weyl uncertainty principle for the operatorsT
k
,
m; more precisely, we will show the following theorem.Theorem 6.
Letm be a function in L
2
(
μ
k
) satisfying the admissibility condition ((22)). Then, for f
∈
L
2
(
μ
k
), one has
(42)
f
L
2
μ
k
2
≤
2
2
γ
+
d
y
F
k
f
L
2
μ
k
·
∫
R
d
∫
0
∞
x
2
T
k
,
m
f
a
,
x
2
d
a
a
d
μ
k
x
1
/
2
.Proof.
Letf
∈
L
2
(
μ
k
).
Assume thaty
F
k
f
L
2
(
μ
k
)
<
∞ and ∫
R
d
‍
∫
0
∞
‍
|
x
|
2
|
T
k
,
m
f
(
a
,
x
)
|
2
(
d
a
/
a
)
d
μ
k
(
x
)
<
∞. The inequality ((5)) leads to
(43)
∫
R
d
T
k
,
m
f
a
,
x
2
d
μ
k
x
≤
2
2
γ
+
d
∫
R
d
x
2
T
k
,
m
f
a
,
x
2
d
μ
k
x
1
/
2
·
∫
R
d
y
2
F
k
T
k
,
m
f
a
,
·
y
2
d
μ
k
y
1
/
2
.
Integrating with respect to (
d
a
/
a
) gives
(44)
T
k
,
m
f
L
2
Ω
k
2
≤
2
2
γ
+
d
∫
0
∞
∫
R
d
x
2
T
k
,
m
f
a
,
x
2
d
μ
k
x
1
/
2
·
∫
R
d
y
2
F
k
T
k
,
m
f
a
,
·
y
2
d
μ
k
y
1
/
2
d
a
a
.
From Theorem 3 and Schwarz’s inequality, we get
(45)
f
L
2
μ
k
2
≤
2
2
γ
+
d
∫
0
∞
∫
R
d
x
2
T
k
,
m
f
a
,
x
2
d
μ
k
x
d
a
a
1
/
2
·
∫
0
∞
∫
R
d
y
2
F
k
T
k
,
m
f
a
,
·
y
2
d
μ
k
y
d
a
a
1
/
2
.
But by ((18)), Fubini-Tonnelli’s theorem, and ((22)), we have
(46)
∫
0
∞
∫
R
d
y
2
F
k
T
k
,
m
f
a
,
·
y
2
d
μ
k
y
d
a
a
=
∫
0
∞
∫
R
d
y
2
m
a
y
2
F
k
f
y
2
d
μ
k
y
d
a
a
=
∫
R
d
y
2
F
k
f
y
2
d
μ
k
y
.
This yields the result and completes the proof of the theorem.


### 3.2. Uncertainty Principle of Concentration Type

LetE be a measurable subset of R
d. We say that a function f
∈
L
2
(
μ
k
) is ɛ-concentrated on E, if
(47)
f
-
χ
E
f
L
2
μ
k
≤
ɛ
f
L
2
μ
k
,
where χ
E is the indicator function of the set E.LetS be a measurable subset of ]0
,
∞
[
×
R
d and let f
∈
L
2
(
μ
k
). We say that T
k
,
m
f is η-concentrated on S, if
(48)
T
k
,
m
f
-
χ
S
T
k
,
m
f
L
2
Ω
k
≤
η
T
k
,
m
f
L
2
Ω
k
.Donoho-Stark’s uncertainty principle for the operatorsT
k
,
m is obtained.Theorem 7.
Letf
∈
L
2
(
μ
k
) and let m
∈
L
1
∩
L
2
(
μ
k
) satisfying ((22)). If f is ɛ-concentrated on E and T
k
,
m
f is η-concentrated on S, then
(49)
m
L
1
μ
k
μ
k
E
1
/
2
∫
∫
S
1
a
2
2
γ
+
d
d
Ω
k
a
,
x
1
/
2
≥
1
-
η
-
ɛ
.Proof.
Letf
∈
L
2
(
μ
k
). Assume that μ
k
(
E
)
<
∞ and ∫
‍
∫
S
‍
1
/
a
2
(
2
γ
+
d
)
d
Ω
k
(
a
,
x
)
<
∞. From ((47)), ((48)), and Theorem 3 it follows that
(50)
T
k
,
m
f
-
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
≤
T
k
,
m
f
-
χ
S
T
k
,
m
f
L
2
Ω
k
+
χ
S
T
k
,
m
f
-
χ
E
f
L
2
Ω
k
≤
η
T
k
,
m
f
L
2
Ω
k
+
T
k
,
m
f
-
χ
E
f
L
2
Ω
k
≤
η
+
ɛ
f
L
2
μ
k
.
Then the triangle inequality shows that
(51)
T
k
,
m
f
L
2
Ω
k
≤
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
+
T
k
,
m
f
-
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
≤
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
+
η
+
ɛ
f
L
2
μ
k
.
But
(52)
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
=
∫
∫
S
T
k
,
m
χ
E
f
a
,
x
2
d
Ω
k
a
,
x
1
/
2
and since m
,
χ
E
f
∈
L
1
∩
L
2
(
μ
k
), then, by Lemma 2, we have
(53)
T
k
,
m
χ
E
f
a
,
x
≤
1
a
2
γ
+
d
m
L
1
μ
k
f
L
2
μ
k
μ
k
E
1
/
2
.
Thus,
(54)
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
≤
m
L
1
μ
k
f
L
2
μ
k
μ
k
E
1
/
2
·
∫
∫
S
1
a
2
2
γ
+
d
d
Ω
k
a
,
x
1
/
2
,
T
k
,
m
f
L
2
Ω
k
≤
m
L
1
μ
k
f
L
2
μ
k
μ
k
E
1
/
2
·
∫
∫
S
1
a
2
2
γ
+
d
d
Ω
k
a
,
x
1
/
2
+
η
+
ɛ
f
L
2
μ
k
.
By applying Theorem 3, we obtain
(55)
m
L
1
μ
k
μ
k
E
1
/
2
∫
∫
S
1
a
2
2
γ
+
d
d
Ω
k
a
,
x
1
/
2
≥
1
-
η
-
ɛ
,
which gives the desired result.Remark 8.
IfS
⊂
{
(
a
,
x
)
∈
]
0
,
∞
[
×
R
d
:
a
≥
δ
} for some δ
>
0, one supposes that α
=
max
⁡
{
1
/
a
:
(
a
,
x
)
∈
S
for
some
x
∈
R
d
}. Then by Theorem 7 we deduce that
(56)
α
2
γ
+
d
m
L
1
μ
k
μ
k
E
1
/
2
Ω
k
S
1
/
2
≥
1
-
η
-
ɛ
.


## 3.1. Heisenberg-Pauli-Weyl Uncertainty Principle

This section is devoted to establish Heisenberg-Pauli-Weyl uncertainty principle for the operatorsT
k
,
m; more precisely, we will show the following theorem.Theorem 6.
Letm be a function in L
2
(
μ
k
) satisfying the admissibility condition ((22)). Then, for f
∈
L
2
(
μ
k
), one has
(42)
f
L
2
μ
k
2
≤
2
2
γ
+
d
y
F
k
f
L
2
μ
k
·
∫
R
d
∫
0
∞
x
2
T
k
,
m
f
a
,
x
2
d
a
a
d
μ
k
x
1
/
2
.Proof.
Letf
∈
L
2
(
μ
k
).
Assume thaty
F
k
f
L
2
(
μ
k
)
<
∞ and ∫
R
d
‍
∫
0
∞
‍
|
x
|
2
|
T
k
,
m
f
(
a
,
x
)
|
2
(
d
a
/
a
)
d
μ
k
(
x
)
<
∞. The inequality ((5)) leads to
(43)
∫
R
d
T
k
,
m
f
a
,
x
2
d
μ
k
x
≤
2
2
γ
+
d
∫
R
d
x
2
T
k
,
m
f
a
,
x
2
d
μ
k
x
1
/
2
·
∫
R
d
y
2
F
k
T
k
,
m
f
a
,
·
y
2
d
μ
k
y
1
/
2
.
Integrating with respect to (
d
a
/
a
) gives
(44)
T
k
,
m
f
L
2
Ω
k
2
≤
2
2
γ
+
d
∫
0
∞
∫
R
d
x
2
T
k
,
m
f
a
,
x
2
d
μ
k
x
1
/
2
·
∫
R
d
y
2
F
k
T
k
,
m
f
a
,
·
y
2
d
μ
k
y
1
/
2
d
a
a
.
From Theorem 3 and Schwarz’s inequality, we get
(45)
f
L
2
μ
k
2
≤
2
2
γ
+
d
∫
0
∞
∫
R
d
x
2
T
k
,
m
f
a
,
x
2
d
μ
k
x
d
a
a
1
/
2
·
∫
0
∞
∫
R
d
y
2
F
k
T
k
,
m
f
a
,
·
y
2
d
μ
k
y
d
a
a
1
/
2
.
But by ((18)), Fubini-Tonnelli’s theorem, and ((22)), we have
(46)
∫
0
∞
∫
R
d
y
2
F
k
T
k
,
m
f
a
,
·
y
2
d
μ
k
y
d
a
a
=
∫
0
∞
∫
R
d
y
2
m
a
y
2
F
k
f
y
2
d
μ
k
y
d
a
a
=
∫
R
d
y
2
F
k
f
y
2
d
μ
k
y
.
This yields the result and completes the proof of the theorem.


## 3.2. Uncertainty Principle of Concentration Type

LetE be a measurable subset of R
d. We say that a function f
∈
L
2
(
μ
k
) is ɛ-concentrated on E, if
(47)
f
-
χ
E
f
L
2
μ
k
≤
ɛ
f
L
2
μ
k
,
where χ
E is the indicator function of the set E.LetS be a measurable subset of ]0
,
∞
[
×
R
d and let f
∈
L
2
(
μ
k
). We say that T
k
,
m
f is η-concentrated on S, if
(48)
T
k
,
m
f
-
χ
S
T
k
,
m
f
L
2
Ω
k
≤
η
T
k
,
m
f
L
2
Ω
k
.Donoho-Stark’s uncertainty principle for the operatorsT
k
,
m is obtained.Theorem 7.
Letf
∈
L
2
(
μ
k
) and let m
∈
L
1
∩
L
2
(
μ
k
) satisfying ((22)). If f is ɛ-concentrated on E and T
k
,
m
f is η-concentrated on S, then
(49)
m
L
1
μ
k
μ
k
E
1
/
2
∫
∫
S
1
a
2
2
γ
+
d
d
Ω
k
a
,
x
1
/
2
≥
1
-
η
-
ɛ
.Proof.
Letf
∈
L
2
(
μ
k
). Assume that μ
k
(
E
)
<
∞ and ∫
‍
∫
S
‍
1
/
a
2
(
2
γ
+
d
)
d
Ω
k
(
a
,
x
)
<
∞. From ((47)), ((48)), and Theorem 3 it follows that
(50)
T
k
,
m
f
-
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
≤
T
k
,
m
f
-
χ
S
T
k
,
m
f
L
2
Ω
k
+
χ
S
T
k
,
m
f
-
χ
E
f
L
2
Ω
k
≤
η
T
k
,
m
f
L
2
Ω
k
+
T
k
,
m
f
-
χ
E
f
L
2
Ω
k
≤
η
+
ɛ
f
L
2
μ
k
.
Then the triangle inequality shows that
(51)
T
k
,
m
f
L
2
Ω
k
≤
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
+
T
k
,
m
f
-
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
≤
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
+
η
+
ɛ
f
L
2
μ
k
.
But
(52)
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
=
∫
∫
S
T
k
,
m
χ
E
f
a
,
x
2
d
Ω
k
a
,
x
1
/
2
and since m
,
χ
E
f
∈
L
1
∩
L
2
(
μ
k
), then, by Lemma 2, we have
(53)
T
k
,
m
χ
E
f
a
,
x
≤
1
a
2
γ
+
d
m
L
1
μ
k
f
L
2
μ
k
μ
k
E
1
/
2
.
Thus,
(54)
χ
S
T
k
,
m
χ
E
f
L
2
Ω
k
≤
m
L
1
μ
k
f
L
2
μ
k
μ
k
E
1
/
2
·
∫
∫
S
1
a
2
2
γ
+
d
d
Ω
k
a
,
x
1
/
2
,
T
k
,
m
f
L
2
Ω
k
≤
m
L
1
μ
k
f
L
2
μ
k
μ
k
E
1
/
2
·
∫
∫
S
1
a
2
2
γ
+
d
d
Ω
k
a
,
x
1
/
2
+
η
+
ɛ
f
L
2
μ
k
.
By applying Theorem 3, we obtain
(55)
m
L
1
μ
k
μ
k
E
1
/
2
∫
∫
S
1
a
2
2
γ
+
d
d
Ω
k
a
,
x
1
/
2
≥
1
-
η
-
ɛ
,
which gives the desired result.Remark 8.
IfS
⊂
{
(
a
,
x
)
∈
]
0
,
∞
[
×
R
d
:
a
≥
δ
} for some δ
>
0, one supposes that α
=
max
⁡
{
1
/
a
:
(
a
,
x
)
∈
S
for
some
x
∈
R
d
}. Then by Theorem 7 we deduce that
(56)
α
2
γ
+
d
m
L
1
μ
k
μ
k
E
1
/
2
Ω
k
S
1
/
2
≥
1
-
η
-
ɛ
.


## 4. Extremal Functions for the OperatorsT
k
,
m

### 4.1. Sobolev-Dunkl Spaces

Lets
≥
0. We define the Sobolev-Dunkl space of order s, which will be denoted by H
s
(
μ
k
), as the set of all f
∈
L
2
(
μ
k
) such that (
1
+
|
z
|
2
)
s
/
2
F
k
(
f
)
∈
L
2
(
μ
k
). The space H
s
(
μ
k
) is provided with the inner product
(57)
f
,
g
H
s
μ
k
=
∫
R
d
1
+
z
2
s
F
k
f
z
F
k
g
z
¯
d
μ
k
z
and the norm
(58)
f
H
s
μ
k
=
∫
R
d
1
+
z
2
s
F
k
f
z
2
d
μ
k
z
1
/
2
.
The space H
s
(
μ
k
) satisfies the following properties.(a)
ConsiderH
0
(
μ
k
)
=
L
2
(
μ
k
).
(b)
For alls
>
0, the space H
s
(
μ
k
) is continuously contained in L
2
(
μ
k
) and f
L
2
μ
k
≤
f
H
s
(
μ
k
).
(c)
For alls
,
t
>
0, such that t
>
s, the space H
t
(
μ
k
) is continuously contained in H
s
(
μ
k
) and f
H
s
μ
k
≤
f
H
t
(
μ
k
).
(d)
The spaceH
s
(
μ
k
), s
≥
0, provided with the inner product 〈
·
,
·
〉
H
s
(
μ
k
) is a Hilbert space.Remark 9.
Fors
>
γ
+
d
/
2, the function y
→
(
1
+
|
z
|
2
)
-
s
/
2 belongs to L
2
(
μ
k
).
Hence for allf
∈
H
s
(
μ
k
), one has F
k
f
L
2
μ
k
≤
f
H
s
(
μ
k
), and by Hölder’s inequality
(59)
F
k
f
L
1
μ
k
≤
∫
R
d
d
μ
k
z
1
+
z
2
s
1
/
2
f
H
s
μ
k
.
Then the function F
k
(
f
) belongs to L
1
∩
L
2
μ
k, and therefore
(60)
f
x
=
∫
R
d
E
k
i
x
,
z
F
k
f
z
d
μ
k
z
,
5555555555555555555555555
a
.
e
.
x
∈
R
d
.
Letλ
>
0. We denote by 〈
·
,
·
〉
λ
,
H
s
(
μ
k
) the inner product defined on the space H
s
(
μ
k
) by
(61)
f
,
g
λ
,
H
s
μ
k
≔
λ
f
,
g
H
s
μ
k
+
T
k
,
m
f
,
T
k
,
m
g
L
2
Ω
k
,
and the norm f
λ
,
H
s
(
μ
k
)
≔
〈
f
,
f
〉
λ
,
H
s
(
μ
k
).
Next we suppose thatm
∈
L
2
(
μ
k
) satisfying ((22)). By Theorem 3, the inner product 〈
·
,
·
〉
λ
,
H
s
(
μ
k
) can be written as
(62)
f
,
g
λ
,
H
s
μ
k
=
λ
f
,
g
H
s
μ
k
+
f
,
g
L
2
μ
k
.Theorem 10.
Letλ
>
0 and s
>
γ
+
d
/
2 and let m
∈
L
2
(
μ
k
) satisfying ((22)). The space (
H
s
(
μ
k
)
,
〈
·
,
·
〉
λ
,
H
s
(
μ
k
)
) has the reproducing kernel
(63)
K
s
x
,
y
=
∫
R
d
E
k
i
x
,
z
E
k
-
i
y
,
z
1
+
λ
1
+
z
2
s
d
μ
k
z
;
that is,(i)
for ally
∈
R
d, the function x
→
K
s
(
x
,
y
) belongs to H
s
(
μ
k
);
(ii)
the reproducing property, for allf
∈
H
s
(
μ
k
) and y
∈
R
d, is
(64)
f
,
K
s
·
,
y
λ
,
H
s
μ
k
=
f
y
.Proof.
(i) Lety
∈
R
d. From ((12)), the function Φ
y
:
z
→
E
k
(
-
i
y
,
z
)
/
1
+
λ
1
+
z
2
s belongs to L
1
∩
L
2
(
μ
k
). Then, the function K
s is well defined and by Theorem 1(ii), we have
(65)
K
s
x
,
y
=
F
k
-
1
Φ
y
x
,
x
∈
R
d
.
From Theorem 1(iii), it follows that K
s
(
·
,
y
) belongs to L
2
(
μ
k
), and we have
(66)
F
k
K
s
·
,
y
z
=
E
k
-
i
y
,
z
1
+
λ
1
+
z
2
s
,
z
∈
R
d
.
Then by ((12)), we obtain
(67)
F
k
K
s
·
,
y
z
≤
1
λ
1
+
z
2
s
,
K
s
·
,
y
H
s
μ
k
2
≤
1
λ
2
∫
R
d
d
μ
k
z
1
+
z
2
s
<
∞
.
This proves that for all y
∈
R
d the function K
s
(
·
,
y
) belongs to H
s
(
μ
k
).
(ii) Letf
∈
H
s
(
μ
k
) and y
∈
R
d. From ((62)) and ((66)), we have
(68)
f
,
K
s
·
,
y
λ
,
H
s
μ
k
=
∫
R
d
E
k
i
y
,
z
F
k
f
z
d
μ
k
z
,
and from Remark 9, we obtain the following reproducing property:
(69)
f
,
K
s
·
,
y
λ
,
H
s
μ
k
=
f
y
.
This completes the proof of the theorem.


### 4.2. Tikhonov Regularization

The main result of this subsection can then be stated as follows.Theorem 11.
Lets
>
γ
+
d
/
2 and let m
∈
L
2
(
μ
k
) satisfying ((22)). For any g
∈
L
2
(
Ω
k
) and for any λ
>
0, there exists a unique function f
λ
,
g
*, where the infimum
(70)
inf
⁡
f
∈
H
s
μ
k
λ
f
H
s
μ
k
2
+
g
-
T
k
,
m
f
L
2
Ω
k
2
is attained. Moreover, the extremal function f
λ
,
g
* is given by
(71)
f
λ
,
g
*
y
=
∫
R
d
∫
0
∞
∫
R
d
g
a
,
x
m
a
z
¯
E
k
-
i
x
,
z
E
k
i
y
,
z
1
+
λ
1
+
z
2
s
·
d
μ
k
z
d
Ω
k
a
,
x
.Proof.
The existence and unicity of the extremal functionf
λ
,
g
* satisfying ((70)) are given by Kimeldorf and Wahba [19], Matsuura et al. [11], and Saitoh [20]. Moreover, by Theorem 10 we deduce that
(72)
f
λ
,
g
*
y
=
g
,
T
k
,
m
K
s
·
,
y
L
2
Ω
k
,
where K
s is the kernel given by ((63)).
But by Theorems1(ii) and ((66)), we have
(73)
T
k
,
m
K
s
·
,
y
a
,
x
=
∫
R
d
m
a
z
F
k
K
s
·
,
y
z
E
k
i
x
,
z
d
μ
k
z
=
∫
R
d
m
a
z
E
k
i
x
,
z
E
k
-
i
y
,
z
1
+
λ
1
+
z
2
s
d
μ
k
z
.
This clearly yields the result.Remark 12.
The extremal functionf
λ
,
g
* satisfies the following inequality:
(74)
f
λ
,
g
*
y
≤
g
L
2
Ω
k
2
λ
∫
R
d
d
μ
k
z
1
+
z
2
s
1
/
2
.Proof.
From ((72)) and Theorem 3, we have
(75)
f
λ
,
g
*
y
≤
g
L
2
Ω
k
T
k
,
m
K
s
·
,
y
L
2
Ω
k
≤
g
L
2
Ω
k
K
s
·
,
y
L
2
μ
k
.
Then, by Theorems 1(iii) and ((66)),
(76)
f
λ
,
g
*
y
≤
g
L
2
Ω
k
F
k
K
s
·
,
y
L
2
μ
k
≤
g
L
2
Ω
k
∫
R
d
d
μ
k
z
1
+
λ
1
+
z
2
s
2
1
/
2
.
Using the fact that [
1
+
λ
(
1
+
|
z
|
2
)
s
]
2
≥
4
λ
(
1
+
|
z
|
2
)
s, we obtain the result.If we take in ((72)), g
=
T
k
,
m
f, where f
∈
H
s
(
μ
k
), we denote by f
λ
*
=
f
λ
,
T
k
,
m
f
*.Then by Theorem3, we have
(77)
f
λ
*
y
=
T
k
,
m
f
,
T
k
,
m
K
s
·
,
y
L
2
Ω
k
=
f
,
K
s
·
,
y
L
2
μ
k
.Corollary 13.
Lets
>
γ
+
d
/
2, f
∈
H
s
(
μ
k
), and λ
>
0. The extremal function f
λ
* given by ((77)) satisfies the following properties:(i)
f
λ
*
(
y
)
=
∫
R
d
‍
E
k
i
y
,
z
(
F
k
f
z
/
(
1
+
λ
(
1
+
z
2
)
s
)
)
d
μ
k
(
z
)
;
(ii)
F
k
(
f
λ
*
)
(
z
)
=
F
k
f
z
/
1
+
λ
1
+
z
2
s
;
(ii)
f
λ
*
H
s
μ
k
≤
1
/
2
λ
f
L
2
(
μ
k
)
.Proof.
(i) It follows from ((77)) by using Theorems 1(iii) and ((66)).
(ii) The functionF
:
z
→
F
k
(
f
)
(
z
)
/
(
1
+
λ
(
1
+
|
z
|
2
)
s
) belongs to L
1
∩
L
2
(
μ
k
). Then by Theorem 1(ii), we have
(78)
f
λ
*
y
=
F
k
-
1
F
y
.
From Theorem 1(iii), it follows that f
λ
* belongs to L
2
(
μ
k
), and
(79)
F
k
f
λ
*
z
=
F
k
f
z
1
+
λ
1
+
z
2
s
.
(iii) By relation (ii) we have(80)
f
λ
*
H
s
μ
k
2
=
∫
R
d
1
+
z
2
s
F
k
f
λ
*
z
2
d
μ
k
z
=
∫
R
d
1
+
z
2
s
1
+
λ
1
+
z
2
s
2
F
k
f
z
2
d
μ
k
z
.
Using inequality [
1
+
λ
(
1
+
|
z
|
2
)
s
]
2
≥
4
λ
(
1
+
|
z
|
2
)
s, we obtain
(81)
f
λ
*
H
s
μ
k
2
≤
1
4
λ
∫
R
d
F
k
f
z
2
d
μ
k
z
=
1
4
λ
f
L
2
μ
k
2
.
This completes the proof of the corollary.Corollary 14.
Lets
>
γ
+
d
/
2, f
∈
H
s
(
μ
k
), and λ
>
0. The extremal function f
λ
* given by ((77)) satisfies
(82)
lim
⁡
λ
→
0
+
f
λ
*
-
f
H
s
μ
k
=
0
.
Moreover, {
f
λ
*
}
λ
>
0 converges uniformly to f as λ
→
0
+.Proof.
From Corollary13(ii),
(83)
F
k
f
λ
*
-
f
z
=
-
λ
1
+
z
2
s
1
+
λ
1
+
z
2
s
F
k
f
z
.
Consequently,
(84)
f
λ
*
-
f
H
s
μ
k
2
=
∫
R
d
λ
2
1
+
z
2
3
s
F
k
f
z
2
1
+
λ
1
+
z
2
s
2
d
μ
k
z
.
Using the dominated convergence theorem and the fact that
(85)
λ
2
1
+
z
2
3
s
F
k
f
z
2
1
+
λ
1
+
z
2
s
2
≤
1
+
z
2
s
F
k
f
z
2
,
we deduce that
(86)
lim
⁡
λ
→
0
+
f
λ
*
-
f
H
s
μ
k
2
=
0
.
On the other hand, from Remark 9, the function F
k
(
f
)
∈
L
1
∩
L
2
(
μ
k
). Then by ((83)) and Theorem 1(ii),
(87)
f
λ
*
y
-
f
y
=
∫
R
d
-
λ
1
+
z
2
s
F
k
f
z
1
+
λ
1
+
z
2
s
E
k
i
y
,
z
d
μ
k
z
.
So
(88)
f
λ
*
-
f
L
∞
μ
k
≤
∫
R
d
λ
1
+
z
2
s
F
k
f
z
1
+
λ
1
+
z
2
s
d
μ
k
z
.
Again, by dominated convergence theorem and the fact that
(89)
λ
1
+
z
2
s
F
k
f
z
1
+
λ
1
+
z
2
s
≤
F
k
f
z
,
we deduce that
(90)
lim
⁡
λ
→
0
+
f
λ
*
-
f
L
∞
μ
k
=
0
,
which ends the proof.As application, we give the following examples.Example 15.
Ifm
t
(
x
)
≔
-
8
t
|
x
|
2
e
-
t
|
x
|
2 and s
>
γ
+
d
/
2, then
(91)
f
λ
,
g
*
y
=
-
8
t
∫
R
d
∫
0
∞
∫
R
d
a
2
e
-
t
a
2
z
2
z
2
g
a
,
x
1
+
λ
1
+
z
2
s
·
E
k
-
i
x
,
z
E
k
i
y
,
z
d
μ
k
·
z
d
Ω
k
a
,
x
,
and by ((28)), ((72)), and the fact that K
s
(
y
,
z
)
=
K
s
(
z
,
y
)
¯ we obtain
(92)
f
λ
,
g
*
y
=
8
t
∫
R
d
∫
0
∞
∫
R
d
g
a
,
x
a
2
γ
+
d
∂
∂
t
h
k
x
a
,
z
a
,
t
·
K
s
y
,
z
d
μ
k
z
d
Ω
k
a
,
x
.Example 16.
Ifm
t
(
x
)
≔
-
2
t
|
x
|
e
-
t
|
x
| and s
>
γ
+
d
/
2, then
(93)
f
λ
,
g
*
y
=
-
2
t
∫
R
d
∫
0
∞
∫
R
d
a
e
-
t
a
z
z
g
a
,
x
1
+
λ
1
+
z
2
s
·
E
k
-
i
x
,
z
E
k
i
y
,
z
d
μ
k
·
z
d
Ω
k
a
,
x
,
and by ((39)) and ((72)) we deduce that
(94)
f
λ
,
g
*
y
=
2
t
∫
R
d
∫
0
∞
∫
R
d
g
a
,
x
a
2
γ
+
d
∂
∂
t
p
k
x
a
,
z
a
,
t
·
K
s
y
,
z
d
μ
k
z
d
Ω
k
a
,
x
.


## 4.1. Sobolev-Dunkl Spaces

Lets
≥
0. We define the Sobolev-Dunkl space of order s, which will be denoted by H
s
(
μ
k
), as the set of all f
∈
L
2
(
μ
k
) such that (
1
+
|
z
|
2
)
s
/
2
F
k
(
f
)
∈
L
2
(
μ
k
). The space H
s
(
μ
k
) is provided with the inner product
(57)
f
,
g
H
s
μ
k
=
∫
R
d
1
+
z
2
s
F
k
f
z
F
k
g
z
¯
d
μ
k
z
and the norm
(58)
f
H
s
μ
k
=
∫
R
d
1
+
z
2
s
F
k
f
z
2
d
μ
k
z
1
/
2
.
The space H
s
(
μ
k
) satisfies the following properties.(a)
ConsiderH
0
(
μ
k
)
=
L
2
(
μ
k
).
(b)
For alls
>
0, the space H
s
(
μ
k
) is continuously contained in L
2
(
μ
k
) and f
L
2
μ
k
≤
f
H
s
(
μ
k
).
(c)
For alls
,
t
>
0, such that t
>
s, the space H
t
(
μ
k
) is continuously contained in H
s
(
μ
k
) and f
H
s
μ
k
≤
f
H
t
(
μ
k
).
(d)
The spaceH
s
(
μ
k
), s
≥
0, provided with the inner product 〈
·
,
·
〉
H
s
(
μ
k
) is a Hilbert space.Remark 9.
Fors
>
γ
+
d
/
2, the function y
→
(
1
+
|
z
|
2
)
-
s
/
2 belongs to L
2
(
μ
k
).
Hence for allf
∈
H
s
(
μ
k
), one has F
k
f
L
2
μ
k
≤
f
H
s
(
μ
k
), and by Hölder’s inequality
(59)
F
k
f
L
1
μ
k
≤
∫
R
d
d
μ
k
z
1
+
z
2
s
1
/
2
f
H
s
μ
k
.
Then the function F
k
(
f
) belongs to L
1
∩
L
2
μ
k, and therefore
(60)
f
x
=
∫
R
d
E
k
i
x
,
z
F
k
f
z
d
μ
k
z
,
5555555555555555555555555
a
.
e
.
x
∈
R
d
.
Letλ
>
0. We denote by 〈
·
,
·
〉
λ
,
H
s
(
μ
k
) the inner product defined on the space H
s
(
μ
k
) by
(61)
f
,
g
λ
,
H
s
μ
k
≔
λ
f
,
g
H
s
μ
k
+
T
k
,
m
f
,
T
k
,
m
g
L
2
Ω
k
,
and the norm f
λ
,
H
s
(
μ
k
)
≔
〈
f
,
f
〉
λ
,
H
s
(
μ
k
).
Next we suppose thatm
∈
L
2
(
μ
k
) satisfying ((22)). By Theorem 3, the inner product 〈
·
,
·
〉
λ
,
H
s
(
μ
k
) can be written as
(62)
f
,
g
λ
,
H
s
μ
k
=
λ
f
,
g
H
s
μ
k
+
f
,
g
L
2
μ
k
.Theorem 10.
Letλ
>
0 and s
>
γ
+
d
/
2 and let m
∈
L
2
(
μ
k
) satisfying ((22)). The space (
H
s
(
μ
k
)
,
〈
·
,
·
〉
λ
,
H
s
(
μ
k
)
) has the reproducing kernel
(63)
K
s
x
,
y
=
∫
R
d
E
k
i
x
,
z
E
k
-
i
y
,
z
1
+
λ
1
+
z
2
s
d
μ
k
z
;
that is,(i)
for ally
∈
R
d, the function x
→
K
s
(
x
,
y
) belongs to H
s
(
μ
k
);
(ii)
the reproducing property, for allf
∈
H
s
(
μ
k
) and y
∈
R
d, is
(64)
f
,
K
s
·
,
y
λ
,
H
s
μ
k
=
f
y
.Proof.
(i) Lety
∈
R
d. From ((12)), the function Φ
y
:
z
→
E
k
(
-
i
y
,
z
)
/
1
+
λ
1
+
z
2
s belongs to L
1
∩
L
2
(
μ
k
). Then, the function K
s is well defined and by Theorem 1(ii), we have
(65)
K
s
x
,
y
=
F
k
-
1
Φ
y
x
,
x
∈
R
d
.
From Theorem 1(iii), it follows that K
s
(
·
,
y
) belongs to L
2
(
μ
k
), and we have
(66)
F
k
K
s
·
,
y
z
=
E
k
-
i
y
,
z
1
+
λ
1
+
z
2
s
,
z
∈
R
d
.
Then by ((12)), we obtain
(67)
F
k
K
s
·
,
y
z
≤
1
λ
1
+
z
2
s
,
K
s
·
,
y
H
s
μ
k
2
≤
1
λ
2
∫
R
d
d
μ
k
z
1
+
z
2
s
<
∞
.
This proves that for all y
∈
R
d the function K
s
(
·
,
y
) belongs to H
s
(
μ
k
).
(ii) Letf
∈
H
s
(
μ
k
) and y
∈
R
d. From ((62)) and ((66)), we have
(68)
f
,
K
s
·
,
y
λ
,
H
s
μ
k
=
∫
R
d
E
k
i
y
,
z
F
k
f
z
d
μ
k
z
,
and from Remark 9, we obtain the following reproducing property:
(69)
f
,
K
s
·
,
y
λ
,
H
s
μ
k
=
f
y
.
This completes the proof of the theorem.


## 4.2. Tikhonov Regularization

The main result of this subsection can then be stated as follows.Theorem 11.
Lets
>
γ
+
d
/
2 and let m
∈
L
2
(
μ
k
) satisfying ((22)). For any g
∈
L
2
(
Ω
k
) and for any λ
>
0, there exists a unique function f
λ
,
g
*, where the infimum
(70)
inf
⁡
f
∈
H
s
μ
k
λ
f
H
s
μ
k
2
+
g
-
T
k
,
m
f
L
2
Ω
k
2
is attained. Moreover, the extremal function f
λ
,
g
* is given by
(71)
f
λ
,
g
*
y
=
∫
R
d
∫
0
∞
∫
R
d
g
a
,
x
m
a
z
¯
E
k
-
i
x
,
z
E
k
i
y
,
z
1
+
λ
1
+
z
2
s
·
d
μ
k
z
d
Ω
k
a
,
x
.Proof.
The existence and unicity of the extremal functionf
λ
,
g
* satisfying ((70)) are given by Kimeldorf and Wahba [19], Matsuura et al. [11], and Saitoh [20]. Moreover, by Theorem 10 we deduce that
(72)
f
λ
,
g
*
y
=
g
,
T
k
,
m
K
s
·
,
y
L
2
Ω
k
,
where K
s is the kernel given by ((63)).
But by Theorems1(ii) and ((66)), we have
(73)
T
k
,
m
K
s
·
,
y
a
,
x
=
∫
R
d
m
a
z
F
k
K
s
·
,
y
z
E
k
i
x
,
z
d
μ
k
z
=
∫
R
d
m
a
z
E
k
i
x
,
z
E
k
-
i
y
,
z
1
+
λ
1
+
z
2
s
d
μ
k
z
.
This clearly yields the result.Remark 12.
The extremal functionf
λ
,
g
* satisfies the following inequality:
(74)
f
λ
,
g
*
y
≤
g
L
2
Ω
k
2
λ
∫
R
d
d
μ
k
z
1
+
z
2
s
1
/
2
.Proof.
From ((72)) and Theorem 3, we have
(75)
f
λ
,
g
*
y
≤
g
L
2
Ω
k
T
k
,
m
K
s
·
,
y
L
2
Ω
k
≤
g
L
2
Ω
k
K
s
·
,
y
L
2
μ
k
.
Then, by Theorems 1(iii) and ((66)),
(76)
f
λ
,
g
*
y
≤
g
L
2
Ω
k
F
k
K
s
·
,
y
L
2
μ
k
≤
g
L
2
Ω
k
∫
R
d
d
μ
k
z
1
+
λ
1
+
z
2
s
2
1
/
2
.
Using the fact that [
1
+
λ
(
1
+
|
z
|
2
)
s
]
2
≥
4
λ
(
1
+
|
z
|
2
)
s, we obtain the result.If we take in ((72)), g
=
T
k
,
m
f, where f
∈
H
s
(
μ
k
), we denote by f
λ
*
=
f
λ
,
T
k
,
m
f
*.Then by Theorem3, we have
(77)
f
λ
*
y
=
T
k
,
m
f
,
T
k
,
m
K
s
·
,
y
L
2
Ω
k
=
f
,
K
s
·
,
y
L
2
μ
k
.Corollary 13.
Lets
>
γ
+
d
/
2, f
∈
H
s
(
μ
k
), and λ
>
0. The extremal function f
λ
* given by ((77)) satisfies the following properties:(i)
f
λ
*
(
y
)
=
∫
R
d
‍
E
k
i
y
,
z
(
F
k
f
z
/
(
1
+
λ
(
1
+
z
2
)
s
)
)
d
μ
k
(
z
)
;
(ii)
F
k
(
f
λ
*
)
(
z
)
=
F
k
f
z
/
1
+
λ
1
+
z
2
s
;
(ii)
f
λ
*
H
s
μ
k
≤
1
/
2
λ
f
L
2
(
μ
k
)
.Proof.
(i) It follows from ((77)) by using Theorems 1(iii) and ((66)).
(ii) The functionF
:
z
→
F
k
(
f
)
(
z
)
/
(
1
+
λ
(
1
+
|
z
|
2
)
s
) belongs to L
1
∩
L
2
(
μ
k
). Then by Theorem 1(ii), we have
(78)
f
λ
*
y
=
F
k
-
1
F
y
.
From Theorem 1(iii), it follows that f
λ
* belongs to L
2
(
μ
k
), and
(79)
F
k
f
λ
*
z
=
F
k
f
z
1
+
λ
1
+
z
2
s
.
(iii) By relation (ii) we have(80)
f
λ
*
H
s
μ
k
2
=
∫
R
d
1
+
z
2
s
F
k
f
λ
*
z
2
d
μ
k
z
=
∫
R
d
1
+
z
2
s
1
+
λ
1
+
z
2
s
2
F
k
f
z
2
d
μ
k
z
.
Using inequality [
1
+
λ
(
1
+
|
z
|
2
)
s
]
2
≥
4
λ
(
1
+
|
z
|
2
)
s, we obtain
(81)
f
λ
*
H
s
μ
k
2
≤
1
4
λ
∫
R
d
F
k
f
z
2
d
μ
k
z
=
1
4
λ
f
L
2
μ
k
2
.
This completes the proof of the corollary.Corollary 14.
Lets
>
γ
+
d
/
2, f
∈
H
s
(
μ
k
), and λ
>
0. The extremal function f
λ
* given by ((77)) satisfies
(82)
lim
⁡
λ
→
0
+
f
λ
*
-
f
H
s
μ
k
=
0
.
Moreover, {
f
λ
*
}
λ
>
0 converges uniformly to f as λ
→
0
+.Proof.
From Corollary13(ii),
(83)
F
k
f
λ
*
-
f
z
=
-
λ
1
+
z
2
s
1
+
λ
1
+
z
2
s
F
k
f
z
.
Consequently,
(84)
f
λ
*
-
f
H
s
μ
k
2
=
∫
R
d
λ
2
1
+
z
2
3
s
F
k
f
z
2
1
+
λ
1
+
z
2
s
2
d
μ
k
z
.
Using the dominated convergence theorem and the fact that
(85)
λ
2
1
+
z
2
3
s
F
k
f
z
2
1
+
λ
1
+
z
2
s
2
≤
1
+
z
2
s
F
k
f
z
2
,
we deduce that
(86)
lim
⁡
λ
→
0
+
f
λ
*
-
f
H
s
μ
k
2
=
0
.
On the other hand, from Remark 9, the function F
k
(
f
)
∈
L
1
∩
L
2
(
μ
k
). Then by ((83)) and Theorem 1(ii),
(87)
f
λ
*
y
-
f
y
=
∫
R
d
-
λ
1
+
z
2
s
F
k
f
z
1
+
λ
1
+
z
2
s
E
k
i
y
,
z
d
μ
k
z
.
So
(88)
f
λ
*
-
f
L
∞
μ
k
≤
∫
R
d
λ
1
+
z
2
s
F
k
f
z
1
+
λ
1
+
z
2
s
d
μ
k
z
.
Again, by dominated convergence theorem and the fact that
(89)
λ
1
+
z
2
s
F
k
f
z
1
+
λ
1
+
z
2
s
≤
F
k
f
z
,
we deduce that
(90)
lim
⁡
λ
→
0
+
f
λ
*
-
f
L
∞
μ
k
=
0
,
which ends the proof.As application, we give the following examples.Example 15.
Ifm
t
(
x
)
≔
-
8
t
|
x
|
2
e
-
t
|
x
|
2 and s
>
γ
+
d
/
2, then
(91)
f
λ
,
g
*
y
=
-
8
t
∫
R
d
∫
0
∞
∫
R
d
a
2
e
-
t
a
2
z
2
z
2
g
a
,
x
1
+
λ
1
+
z
2
s
·
E
k
-
i
x
,
z
E
k
i
y
,
z
d
μ
k
·
z
d
Ω
k
a
,
x
,
and by ((28)), ((72)), and the fact that K
s
(
y
,
z
)
=
K
s
(
z
,
y
)
¯ we obtain
(92)
f
λ
,
g
*
y
=
8
t
∫
R
d
∫
0
∞
∫
R
d
g
a
,
x
a
2
γ
+
d
∂
∂
t
h
k
x
a
,
z
a
,
t
·
K
s
y
,
z
d
μ
k
z
d
Ω
k
a
,
x
.Example 16.
Ifm
t
(
x
)
≔
-
2
t
|
x
|
e
-
t
|
x
| and s
>
γ
+
d
/
2, then
(93)
f
λ
,
g
*
y
=
-
2
t
∫
R
d
∫
0
∞
∫
R
d
a
e
-
t
a
z
z
g
a
,
x
1
+
λ
1
+
z
2
s
·
E
k
-
i
x
,
z
E
k
i
y
,
z
d
μ
k
·
z
d
Ω
k
a
,
x
,
and by ((39)) and ((72)) we deduce that
(94)
f
λ
,
g
*
y
=
2
t
∫
R
d
∫
0
∞
∫
R
d
g
a
,
x
a
2
γ
+
d
∂
∂
t
p
k
x
a
,
z
a
,
t
·
K
s
y
,
z
d
μ
k
z
d
Ω
k
a
,
x
.



---

*Source: 659069-2014-08-17.xml*