Journal of Operators

Volume 2014, Article ID 659069, 9 pages

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

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

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.

#### References

- C. F. Dunkl, “Hankel transforms associated to finite reflection groups,”
*Contemporary Mathematics*, vol. 138, pp. 123–138, 1992. View at Google Scholar - M. Rösler, “An uncertainty principle for the Dunkl transform,”
*Bulletin of the Australian Mathematical Society*, vol. 59, no. 3, pp. 353–360, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - N. Shimeno, “A note on the uncertainty principle for the Dunkl transform,”
*The University of Tokyo. Journal of Mathematical Sciences*, vol. 8, no. 1, pp. 33–42, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Soltani, “Heisenberg-Pauli-Weyl uncertainty ine quality for the Dunkl transform on ℝ
^{d},”*Bulletin of the Australian Mathematical Society*, vol. 87, no. 2, pp. 316–325, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Soltani, “A general form of Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform,”
*Integral Transforms and Special Functions*, vol. 24, no. 5, pp. 401–409, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Soltani, “Best approximation formulas for the Dunkl
*L*2-multiplier operators on R^{d},”*The Rocky Mountain Journal of Mathematics*, vol. 42, no. 1, pp. 305–328, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Soltani, “Multiplier operators and extremal functions related to the dual Dunkl-Sonine operator,”
*Acta Mathematica Scientia. Series B: English Edition*, vol. 33, no. 2, pp. 430–442, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,”
*SIAM Journal on Applied Mathematics*, vol. 49, no. 3, pp. 906–931, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Saitoh, “The Weierstrass transform and an isometry in the heat equation,”
*Applicable Analysis*, vol. 16, no. 1, pp. 1–6, 1983. View at Publisher · View at Google Scholar · View at MathSciNet - S. Saitoh, “Best approximation, Tikhonov regularization and reproducing kernels,”
*Kodai Mathematical Journal*, vol. 28, no. 2, pp. 359–367, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Matsuura, S. Saitoh, and D. D. Trong, “Approximate and analytical inversion formulas in heat conduction on multidimensional spaces,”
*Journal of Inverse and Ill-Posed Problems*, vol. 13, no. 3–6, pp. 479–493, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Yamada, T. Matsuura, and S. Saitoh, “Representations of inverse functions by the integral transform with the sign kernel,”
*Fractional Calculus & Applied Analysis*, vol. 10, no. 2, pp. 161–168, 2007. View at Google Scholar · View at MathSciNet - C. F. Dunkl, “Integral kernels with reflection group invariance,”
*Canadian Journal of Mathematics*, vol. 43, no. 6, pp. 1213–1227, 1991. View at Publisher · View at Google Scholar · View at MathSciNet - M. F. E. de Jeu, “The Dunkl transform,”
*Inventiones Mathematicae*, vol. 113, no. 1, pp. 147–162, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. M. Opdam, “Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group,”
*Compositio Mathematica*, vol. 85, no. 3, pp. 333–373, 1993. View at Google Scholar · View at MathSciNet - M. Rösler and M. Voit, “Markov processes related with Dunkl operators,”
*Advances in Applied Mathematics*, vol. 21, no. 4, pp. 575–643, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Soltani, “Inversion formulas in the Dunkl -type heat conduction on R
*d*,”*Applicable Analysis*, vol. 84, no. 6, pp. 541–553, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - F. Soltani, “Littlewood-Paley
*g*-function in the Dunkl analysis on R^{d},”*Journal of Inequalities in Pure and Applied Mathematics*, vol. 6, no. 3, article 84, 13 pages, 2005. View at Google Scholar · View at MathSciNet - G. Kimeldorf and G. Wahba, “Some results on Tchebycheffian spline functions,”
*Journal of Mathematical Analysis and Applications*, vol. 33, pp. 82–95, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. Saitoh, “Approximate real inversion formulas of the Gaussian convolution,”
*Applicable Analysis*, vol. 83, no. 7, pp. 727–733, 2004. View at Publisher · View at Google Scholar · View at MathSciNet