Research Article | Open Access

# Uncertainty Principles and Extremal Functions for the Dunkl -Multiplier Operators

**Academic Editor:**Aref Jeribi

#### Abstract

We study some class of Dunkl -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.

#### 1. Introduction

In this paper, we consider with the Euclidean inner product and norm . For , let be the reflection in the hyperplane orthogonal to :

A finite set is called a root system, if . and for all . We assume that it is normalized by for all . For a root system , the reflections , , generate a finite group , the reflection group associated with . All reflections in correspond to suitable pairs of roots. For a given , we fix the positive subsystem . Then for each either or .

Let be a multiplicity function on (i.e., a function which is constant on the orbits under the action of ). As an abbreviation, we introduce the index .

Throughout this paper, we will assume that for all . Moreover, let denote the weight function , for all , which is -invariant and homogeneous of degree .

Let be the Mehta-type constant given by We denote by the measure on given by and by , , the space of measurable functions on , such that

For the Dunkl transform is defined (see [1]) by where 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 , Recently, the author [4, 5] proved general forms of the Heisenberg-Pauli-Weyl inequality for the Dunkl transform.

Let be a function in . The Dunkl -multiplier operators, , are defined, for regular functions on , by These operators are studied in [6, 7] where the author established some applications (Calderón’s reproducing formulas, best approximation formulas, and extremal functions…).

For satisfying the admissibility condition: , a.e. , then the operators satisfy Plancherel’s formula: where is the measure on given by .

For the operators we establish a Heisenberg-Pauli-Weyl uncertainty principle. More precisely, we will show, for , provided satisfying , a.e. .

Building on the techniques of Donoho and Stark [8], we show a continuous-time principle for the theory. Let be measurable subset of , let be measurable subset of , and let . If is -concentrated on and is -concentrated on (see Section 3 for more details), then provided satisfying , a.e. .

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 on the Sobolev-Dunkl spaces . More precisely, for all , , the infimum is attained at one function , called the extremal function.

In particular for and , the corresponding extremal functions converge to as .

This paper is organized as follows. In Section 2 we define the Dunkl -multiplier operators , and we give for them Plancherel’s formula. Some examples of Dunkl -multiplier operators are given. In Section 3 we establish the Heisenberg-Pauli-Weyl uncertainty principle and Donoho-Stark’s uncertainty principle for the operators . In the last section we give an application of the theory of reproducing kernels to the Tikhonov regularization related to the operators on the Sobolev-Dunkl spaces .

#### 2. The Dunkl -Multiplier Operators on

The Dunkl operators , , on associated with the finite reflection group and multiplicity function are given, for a function of class on , by

For , the initial problem , , with admits a unique analytic solution on , which will be denoted by and called Dunkl kernel [13, 14]. This kernel has a unique analytic extension to (see [15]). In our case (see [1, 13]),

The Dunkl kernel gives rise to an integral transform, which is called Dunkl transform on , 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 , in , is defined by We notice that agrees with the Fourier transform that is given by

Some of the properties of Dunkl transform are collected bellow (see [1, 14]).

Theorem 1. *(i) *-boundedness: for all , and
*(ii)* Inversion theorem: let , such that . Then
*(iii)* Plancherel theorem: the Dunkl transform extends uniquely to an isometric isomorphism of onto itself. In particular,

Let be a function in . The Dunkl -multiplier operators, , are defined, for regular functions on , by

The operators satisfy the following integral representation.

Lemma 2. *If , then
**
where
*

*Proof. *The result follows from ((18)) and Theorem 1(ii) using Fubini-Tonnelli’s theorem.

We denote by the measure on given by and by the space of measurable functions on , such that

In the following, we give Plancherel formula for the operators .

Theorem 3 (Plancherel formula). *Let be a function in satisfying the admissibility condition:
**
Then, for , one has
*

*Proof. *From Fubini-Tonnelli’s theorem, we obtain
Then, the result follows from ((22)) and Theorem 1(iii).

As applications, we give the following examples.

*Example 4. *Let , , the function is defined by
Then(a) belongs to , and by ((2)), we have
(b) satisfies the admissibility condition ((22)), that is,

Then the associated operators satisfy Plancherel’s formula ((23)).

To express the operator , we use Lemma 2, then for , we have where is the Dunkl-type heat kernel [16, 17]. From [16] this kernel is given by

*Example 5. *Let , , be the function defined by
Then one has: (a) belongs to , and
Since
by Fubini-Tonnelli’s theorem and ((2)), we deduce that
Thus,
On the other hand,
Thus,
(b) satisfies the admissibility condition ((22)); that is,
Then the associated operators satisfy Plancherel’s formula ((23)).

To express the operators , we use Lemma 2; then for , we have
where
is the Dunkl-type Poisson kernel [18]. From ((33)) this kernel is given by

#### 3. Uncertainty Principle for the Operators

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

This section is devoted to establish Heisenberg-Pauli-Weyl uncertainty principle for the operators ; more precisely, we will show the following theorem.

Theorem 6. *Let be a function in satisfying the admissibility condition ((22)). Then, for , one has
*

*Proof. *Let .

Assume that and . The inequality ((5)) leads to
Integrating with respect to gives
From Theorem 3 and Schwarz’s inequality, we get
But by ((18)), Fubini-Tonnelli’s theorem, and ((22)), we have
This yields the result and completes the proof of the theorem.

##### 3.2. Uncertainty Principle of Concentration Type

Let be a measurable subset of . We say that a function is -concentrated on , if where is the indicator function of the set .

Let be a measurable subset of ] and let . We say that is -concentrated on , if

Donoho-Stark’s uncertainty principle for the operators is obtained.

Theorem 7. *Let and let satisfying ((22)). If is -concentrated on and is -concentrated on , then
*

*Proof. *Let . Assume that and . From ((47)), ((48)), and Theorem 3 it follows that
Then the triangle inequality shows that
But
and since , then, by Lemma 2, we have
Thus,
By applying Theorem 3, we obtain
which gives the desired result.

*Remark 8. *If for some , one supposes that . Then by Theorem 7 we deduce that

#### 4. Extremal Functions for the Operators

##### 4.1. Sobolev-Dunkl Spaces

Let . We define the Sobolev-Dunkl space of order , which will be denoted by , as the set of all such that . The space is provided with the inner product and the norm The space satisfies the following properties.(a)Consider .(b)For all , the space is continuously contained in and .(c)For all , such that , the space is continuously contained in and .(d)The space , , provided with the inner product is a Hilbert space.

*Remark 9. *For , the function belongs to .

Hence for all , one has , and by Hölder’s inequality
Then the function belongs to , and therefore

Let . We denote by the inner product defined on the space by
and the norm .

Next we suppose that satisfying ((22)). By Theorem 3, the inner product can be written as

Theorem 10. *Let and and let satisfying ((22)). The space has the reproducing kernel
**
that is,*(i)*for all , the function belongs to ;*(ii)*the reproducing property, for all and , is
*

*Proof. *(i) Let . From ((12)), the function belongs to . Then, the function is well defined and by Theorem 1(ii), we have
From Theorem 1(iii), it follows that belongs to , and we have
Then by ((12)), we obtain
This proves that for all the function belongs to .

(ii) Let and . From ((62)) and ((66)), we have
and from Remark 9, we obtain the following reproducing property:
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. *Let and let satisfying ((22)). For any and for any , there exists a unique function , where the infimum
**
is attained. Moreover, the extremal function is given by
*

*Proof. *The existence and unicity of the extremal function satisfying ((70)) are given by Kimeldorf and Wahba [19], Matsuura et al. [11], and Saitoh [20]. Moreover, by Theorem 10 we deduce that
where is the kernel given by ((63)).

But by Theorems 1(ii) and ((66)), we have

This clearly yields the result.

*Remark 12. *The extremal function satisfies the following inequality:

*Proof. *From ((72)) and Theorem 3, we have
Then, by Theorems 1(iii) and ((66)),
Using the fact that , we obtain the result.

If we take in ((72)), , where , we denote by .

Then by Theorem 3, we have

Corollary 13. *Let , , and . The extremal function given by ((77)) satisfies the following properties:**(i)**;**(ii) **;**(ii) **. *

*Proof. *(i) It follows from ((77)) by using Theorems 1(iii) and ((66)).

(ii) The function belongs to . Then by Theorem 1(ii), we have
From Theorem 1(iii), it follows that belongs to , and

(iii) By relation (ii) we have
Using inequality , we obtain
This completes the proof of the corollary.

Corollary 14. *Let , , and . The extremal function given by ((77)) satisfies
**
Moreover, converges uniformly to as .*

*Proof. *From Corollary 13(ii),
Consequently,
Using the dominated convergence theorem and the fact that
we deduce that
On the other hand, from Remark 9, the function . Then by ((83)) and Theorem 1(ii),
So
Again, by dominated convergence theorem and the fact that
we deduce that
which ends the proof.

As application, we give the following examples.

*Example 15. *If and , then
and by ((28)), ((72)), and the fact that we obtain

*Example 16. *If and , then
and by ((39)) and ((72)) we deduce that