We prove a version of Heisenberg-type uncertainty principle for the Dunkl-Wigner transform of magnitude s > 0; and we deduce a local uncertainty principle for this transform.

1. Introduction

In this paper, we consider

with the Euclidean inner product 〈·, ·〉 and norm

. For

, let σ_α be the reflection in the hyperplane

orthogonal to α:

()

A finite set is called a root system, if and for all . We assume that it is normalized by |α|² = 2 for all . For a root system , the reflections σ_α, , generate a finite group G. The Coxeter group G is a subgroup of the orthogonal group O(d). All reflections in G 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 (a function which is constant on the orbits under the action of G). As an abbreviation, we introduce the index .

Throughout this paper, we will assume that k(α) ≥ 0 for all . Moreover, let w_k denote the weight function , for all , which is G-invariant and homogeneous of degree 2γ.

Let c_k be the Mehta-type constant given by

. We denote by μ_k the measure on

given by dμ_k(y)≔c_kw_k(y)dy, by L^p(μ_k), 1 ≤ p ≤ ∞, the space of measurable functions f on

, such that

()

and by

the subspace of L^p(μ_k) consisting of radial functions.

For f ∈ L¹(μ_k) the Dunkl transform of f is defined (see [1]) by

()

where E_k(−ix, y) denotes the Dunkl kernel. (For more details see the next section.)

Many uncertainty principles have already been proved for the Dunkl transform

, namely, by Rösler [2] and Shimeno [3] who established the Heisenberg-type uncertainty inequality for this transform, by showing that for f ∈ L²(μ_k),

()

Recently, the author [4–7] proved general forms of the Heisenberg-type inequality for the Dunkl transform

The Dunkl translation operators τ_x,

, [8] are defined on L²(μ_k) by

()

Let

. The Dunkl-Wigner transform V_g is the mapping defined for f ∈ L²(μ_k) by

()

where

()

This transform is studied in [9, 10] where the author established some applications (Plancherel formula, inversion formula, Calderón’s reproducing formula, extremal function, etc.).

In this paper we use formula (4); we prove uncertainty principle intervening

and V_g of magnitudes a, b ≥ 1; that is, for every f ∈ L²(μ_k),

()

Next, we prove a Heisenberg-type uncertainty principle for the Dunkl-Wigner transform V_g of magnitude s > 0; that is, there exists a constant c(k, s) > 0 such that, for f ∈ L²(μ_k),

()

Finally, we prove a local uncertainty principle for the Dunkl-Wigner transform V_g; that is, there exists a constant b(k, s) > 0 such that, for f ∈ L²(μ_k) and for measurable subset E of

such that 0 < μ_k ⊗ μ_k(E) < ∞,

()

where χ_E is the indicator function of the set E.

In the classical case, the Fourier-Wigner transforms are studied by Weyl [11] and Wong [12]. In the Bessel-Kingman hypergroups, these operators are studied by Dachraoui [13].

This paper is organized as follows. In Section 2, we recall some properties of the Dunkl-Wigner transform V_g. In Section 3, we prove a Heisenberg-type uncertainty principle for the Dunkl-Wigner transform V_g of magnitude s > 0; and we deduce a local uncertainty principle for this transform.

2. The Dunkl-Wigner Transform

The Dunkl operators

, j = 1, …, d, on

associated with the finite reflection group G and multiplicity function k are given, for a function f of class C¹ on

, by

()

For

, the initial value problem

, j = 1, …, d, with u(0, y) = 1 admits a unique analytic solution on

, which will be denoted by E_k(x, y) and called Dunkl kernel [14, 15]. This kernel has a unique analytic extension to

(see [16]). The Dunkl kernel has the Laplace-type representation [17]

()

where

and Γ_x is a probability measure on

, such that

. In our case,

()

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 [15]. The Dunkl transform of a function f in L¹(μ_k) is defined by

()

We notice that

agrees with the Fourier transform

that is given by

()

The Dunkl transform of a function

which is radial is again radial and could be computed via the associated Fourier-Bessel transform

(see [18], Proposition 4); that is,

()

where f(x) = F(|x|) and

()

Here j_γ is the spherical Bessel function (see [19]).

Some of the properties of Dunkl transform are collected below (see [1, 15]).

Theorem 1. (i) L¹ − L^∞-Boundedness. For all f ∈ L¹(μ_k), , and

()

(ii) Inversion Theorem. Let f ∈ L¹(μ_k), such that

. Then

()

(iii) Plancherel Theorem. The Dunkl transform

extends uniquely to an isometric isomorphism of L²(μ_k) onto itself. In particular, one has

()

(iv) Parseval Theorem. For f, g ∈ L²(μ_k), one has

()

The Dunkl transform

allows us to define a generalized translation operators on L²(μ_k) by setting

()

It is the definition of Thangavelu and Xu given in [8]. It plays the role of the ordinary translation τ_xf = f(x + ·) in

, since the Euclidean Fourier transform satisfies

. Note that, from (13) and Theorem 1(iii), relation (22) makes sense, and

, for all f ∈ L²(μ_k).

Rösler [20] introduced the Dunkl translation operators for radial functions. If f are radial functions, f(x) = F(|x|), then

()

where Γ_x is the representing measure given by (12).

This formula allows us to establish the following results [8, 21].

Proposition 2. (i) For all p ∈ [1,2] and for all , the Dunkl translation is a bounded operator, and for , one has

()

(ii) Let

. Then, for all

, one has

()

The Dunkl convolution product ∗_k of two functions f and g in L²(μ_k) is defined by

()

We notice that ∗_k generalizes the convolution ∗ that is given by

()

Proposition 2 allows us to establish the following properties for the Dunkl convolution on (see [8]).

Proposition 3. (i) Assume that p ∈ [1,2] and q, r ∈ [1, ∞] such that 1/p + 1/q = 1 + 1/r. Then the map (f, g) → f∗_kg extends to a continuous map from to L^r(μ_k), and

()

(ii) For all

and g ∈ L²(μ_k), one has

()

(iii) Let

and g ∈ L²(μ_k). Then f∗_kg belongs to L²(μ_k) if and only if

belongs to L²(μ_k), and

()

(iv) Let

and g ∈ L²(μ_k). Then

()

where both sides are finite or infinite.

Let

and

. The modulation of g by y is the function g_k,y defined by

()

Thus,

()

Let

. The Fourier-Wigner transform associated with the Dunkl operators is the mapping V_g defined for f ∈ L²(μ_k) by

()

In the following we recall some properties of the Dunkl-Wigner transform (Plancherel formula, inversion formula, reproducing inversion formula of Calderón’s type, etc.).

Proposition 4 (see [10].)Let . Then

(i)
.
(ii)
dμ_k(z).
(iii)
The function V_g(f) belongs to L^∞(μ_k ⊗ μ_k), and
()

Theorem 5 (see [10].)Let be a nonzero function. Then one has the following.

(i)
Plancherel formula: for every f ∈ L²(μ_k), one has
()
(ii)
Parseval formula: for every f, h ∈ L²(μ_k), one has
()
(iii)
Inversion formula: for all f ∈ L¹∩L²(μ_k) such that , one has
()

Theorem 6 (Calderón’s reproducing inversion formula; see [10]). Let , −∞ < a_j < b_j < ∞, and let be a nonzero function, such that . Then, for f ∈ L²(μ_k), the function f_Δ given by

()

belongs to L²(μ_k) and satisfies

()

3. Uncertainty Principles for the Mapping V_g

In this section we establish Heisenberg-type uncertainty principle for the Dunkl-Wigner transform V_g. We begin by the following theorem.

Theorem 7. Let be a nonzero function. Then, for f ∈ L²(μ_k), one has

()

Proof. Let f ∈ L²(μ_k). Assume that . Inequality (4) leads to

()

Integrating with respect to dμ_k(y) and using the Schwarz inequality, we get

()

But by Proposition 4(ii), Fubini-Tonelli’s theorem, (16), Proposition 2(ii), and Theorem 1(iii), we have

()

This yields the result and completes the proof of the theorem.

Theorem 8. Let be a nonzero function and s ≥ 1. Then, for f ∈ L²(μ_k), one has

()

Proof. Let s ≥ 1 and let f ∈ L²(μ_k), f ≠ 0, such that . Then, for s > 1, we have

()

where s^′ is defined as usual by 1/s + 1/s^′ = 1. By Hölder’s inequality we get

()

Thus, for all s ≥ 1, we have

()

with equality if s = 1. In the same manner and using Theorem 1(iii), we have, for s ≥ 1,

()

with equality if s = 1. By (48) and (49), for all s ≥ 1, we have

()

with equality if s = 1. Applying Theorem 7, we obtain

()

which completes the proof of the theorem.

From (48) and (49) we deduce the following remark.

Remark 9. Let be a nonzero function and a, b ≥ 1. Then, for f ∈ L²(μ_k), we have

()

For λ > 0, we define the dilation of f ∈ L²(μ_k) by

()

Then

()

Let us now turn to establishing Heisenberg-type uncertainty principle for the Dunkl-Wigner transform V_g of magnitude s > 0. Thus, we consider the following lemma.

Lemma 10. Let λ > 0 and let be a nonzero function. Then, for f ∈ L²(μ_k), one has

()

Proof. From Proposition 4(ii), we have

()

But by (55) we have

()

Thus,

()

which gives the result.

Theorem 11 (Heisenberg-type uncertainty principle for Vg). Let s > 0. Then there exists a constant c(k, s) > 0 such that, for all f ∈ L²(μ_k) and , one has

()

Proof. Let s, r₀ > 0 and , where |(x, y)| = (|x|² + |y|²) ^1/2. Fix r₀ such that . We write

()

But from Hölder’s inequality and Proposition 4(iii) we have

()

Therefore, by Theorem 5(i),

()

Using the fact that |(x, y)|^s = (|x|²+|y|²) ^s/2 ≤ 2^s/2(|x|^s+|y|^s) we deduce that

()

where

()

Replacing f and g by f_λ and g_λ, respectively, in the previous inequality, we obtain by Lemma 10 and by a suitable change of variables

()

By setting

in the right-hand side of the previous inequality we obtain the desired result.

We will now prove a local uncertainty principle for the Dunkl-Wigner transform V_g, which extends the result of Faris [22].

Theorem 12 (local uncertainty principle for Vg). Let s > 0. Then there exists a constant b(k, s) > 0 such that, for all f ∈ L²(μ_k) and and for all measurable subset E of such that 0 < μ_k ⊗ μ_k(E) < ∞, one has

()

Proof. Let s > 0 and let E be a measurable subset of such that 0 < μ_k ⊗ μ_k(E) < ∞. From Hölder’s inequality and Proposition 4(iii) we have

()

From (63) there exists b(k, s) > 0 such that

()

Therefore we obtain the desired result.

Competing Interests

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

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Uncertainty Principles for the Dunkl-Wigner Transforms

Abstract

1. Introduction

2. The Dunkl-Wigner Transform

3. Uncertainty Principles for the Mapping V_g

Competing Interests

References

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Uncertainty Principles for the Dunkl-Wigner Transforms

Abstract

1. Introduction

2. The Dunkl-Wigner Transform

3. Uncertainty Principles for the Mapping Vg

Competing Interests

References

Citing Literature

References

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3. Uncertainty Principles for the Mapping V_g