Research Article | Open Access
Uncertainty Principles and Extremal Functions for the Dunkl -Multiplier Operators
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.
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
Many uncertainty principles have already been proved for the Dunkl transform, namely, by Rösler  and Shimeno  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 , 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. , and Yamada et al. , 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 ). 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 , where already many basic properties were established. Dunkl’s results were completed and extended later by de Jeu . The Dunkl transform of a function , in , is defined by We notice that agrees with the Fourier transform that is given by
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
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
As applications, we give the following examples.
Then the associated operators satisfy Plancherel’s formula ((23)).
Example 5. Let , , be the function defined by
Then one has: (a) belongs to , and
by Fubini-Tonnelli’s theorem and ((2)), we deduce that
On the other hand,
(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 . 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 , Matsuura et al. , and Saitoh . 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:
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.