Journal of Operators

Volume 2014 (2014), Article ID 659069, 9 pages

http://dx.doi.org/10.1155/2014/659069

Research Article

Uncertainty Principles and Extremal Functions for the Dunkl -Multiplier Operators

Fethi Soltani

Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 114, Jazan, Saudi Arabia

Received 19 November 2013; Accepted 12 June 2014; Published 17 August 2014

Academic Editor: Aref Jeribi

Copyright © 2014 Fethi Soltani. 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.

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

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is partially supported by Deanship of Scientific Research, Jazan University.

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