LOCALIZATION OPERATORS AND SCALOGRAM ASSOCIATED WITH THE GENERALIZED CONTINUOUS WAVELET ON THE HECKMAN–OPDAM THEORY

. We consider the generalized wavelet transform Φ Wh on R d for the Heckman–Opdam theory. We study the localization operators associated with Φ Wh ; in particular, we prove that they are in the Schatten–von Neumann class. Next we introduce some results on the scalogram for this transform.


Introduction
We consider the differential-difference operators T j , j = 1, 2, . . . , d, associated with a root system R and a multiplicity function k, introduced by Cherednik in [5], and called the Cherednik operators in the literature. These operators were helpful for the extension and simplification of the theory of Heckman-Opdam, which is a generalization of the harmonic analysis on the symmetric spaces G/K ( [33,34,37]).
The Cherednik and Heckman-Opdam theories are based on the Opdam-Cherednik hypergeometric function G λ , λ ∈ C d , which is the unique analytic solution of the system T j u(x) = −iλ j u(x), j = 1, 2, . . . , d, satisfying the normalizing condition u(0) = 1, and the Heckman-Opdam kernel F λ , λ ∈ C d , which is defined by where W is the Weyl group associated with the root system R ( [33,34]). With the kernel G λ Opdam and Cherednik have defined in [5,33] the Opdam-Cherednik transform H and have used the kernel F λ to define the hypergeometric Fourier transform H W on spaces of W -invariant functions, and have established some of their properties (see also [34]).
In the classical setting, the notion of wavelets was first introduced by Morlet, a French petroleum engineer at ELF-Aquitaine, in connection with his study of seismic traces. The mathematical foundations were given by Grossmann and Morlet in [18]. The harmonic analyst Meyer and many other mathematicians became aware of this theory and they recognized many classical results inside it (see [6,22,30]). Classical wavelets have wide applications, ranging from signal analysis in geophysics and acoustics to quantum theory and pure mathematics (see [11,16,20] and the references therein).
Recently in [19] Hassini et al., with the aid of the harmonic analysis associated to the Heckman-Opdam theory, have defined and studied the generalized wavelet transform, and they have proved Plancherel's and inversion formulas for this transform.
One of the applications of the continuous wavelet transform is time-frequency analysis, which is a mathematical tool to define a restriction of functions to a region in the time-frequency plane, that is compatible with the uncertainty principle, and to extract time-frequency features. In this sense they have been introduced and studied by Daubechies [8,9,10] and Ramanathan and Topiwala [35], and they are now extensively investigated as an important mathematical tool in signal analysis and other applications [7,12,13,17,35,41].
As the harmonic analysis associated to the Heckman-Opdam theory has known remarkable development, it is a natural question to ask whether there exists the equivalent of the theory of time-frequency analysis for the generalized wavelet transform introduced in [19].
In this paper we study only two subjects of the time-frequency analysis associated with the generalized wavelet transform. The first subject is the theory of localization operators. This theory has found many applications to time-frequency analysis, the theory of differential equations, and quantum mechanics. Many works have been written on localization operators from these points of view; we refer in particular to the papers of Balazs et al. [3,4]. The second subject is the scalogram. We note that the scalogram has many applications; for example in [2], the authors used Morlet wavelet scalograms to detect a previously unknown coordinated contractility behavior of the atrium during ventricular fibrillation, a phenomenon which is not captured in a normal electrocardiogram. Other applications can also be found in [39], where the authors applied the scalogram to biomedical signals to detect their short-lived temporal interactions. We mention that scalograms have been studied in the context of the generalized wavelet transforms by many authors; see for example [15,29].
The remainder of the paper is organized as follows. In Section 2 we recall the main results about the harmonic analysis associated with the Cherednik operators. Section 3 is devoted to the study of boundedness and compactness properties of the localization operators for the generalized continuous wavelet transform Φ W h ; we show that they are in the Schatten-von Neumann class. We also give a trace formula. In the last section we study the eigenvalues and eigenfunctions of the time-frequency localization operator. Next we study the scalogram associated with the generalized continuous wavelet transform.
2.1. Reflection groups, root systems, and multiplicity functions. The basic ingredient in the theory of Cherednik operators are root systems and finite reflection groups, acting on R d with the standard euclidean scalar product ·, · for which the basis {e i , i = 1, . . . , d} is orthogonal, and x = x, x . On C d , · denotes also the standard Hermitian norm, while z, w = d j=1 z j w j .
For α ∈ R d \{0}, let α ∨ = 2 α α be the coroot associated to α and let For a given root system R the reflections r α , α ∈ R, generate a finite group W ⊂ O(d), called the reflection group associated with R.
We fix a positive root system R + = {α ∈ R : α, β > 0} for some β ∈ R d \ α∈R H α . We denote by R 0 + the set of positive indivisible roots. Let be the positive chamber. We denote by C + its closure.
is called a multiplicity function if it is invariant under the action of the associated reflection group W . For abbreviation, we introduce the index Moreover, let A k denote the weight function .
We note that this function is W invariant and satisfies 2.2. The eigenfunctions of the Cherednik operators. The Cherednik operators T j , j = 1, . . . , d, on R d associated with the finite reflection group W and the multiplicity function k are given by The operators T j can also be written in the form In the case k(α) = 0, for all α ∈ R + , the T j , j = 1, 2, . . . , d, reduce to the corresponding partial derivatives. • For R + = {α}, we have the Cherednik operator with ρ = k(α). This operator can also be written in the form • For R + = {2α}, we have the Cherednik operator . This operator can also be written in the form with ρ = 2k(2α). • For R + = {α, 2α}, we have the Cherednik operator with ρ = k(α) + 2k(2α). It can also be written as The operators (2.1), (2.2) and (2.3) are particular cases of the differentialdifference operator with k ≥ k ≥ 0 and k = 0. This operator is called the Jacobi-Cherednik operator (cf. [14]).
The Heckman-Opdam Laplacian k on R d is defined by where and ∇ are respectively the euclidean Laplacian and the gradient operator on R d . The Heckman-Opdam Laplacian on W -invariant functions is denoted by W k and has the expression

Example 2.2.
For d = 1, W = Z 2 and k ≥ k ≥ 0, k = 0, the Heckman-Opdam Laplacian is the Jacobi operator defined for even functions f of class C 2 on R by We denote by G λ the eigenfunction of the operators T j , j = 1, 2, . . . , d. It is the unique analytic function on R d that satisfies the differential-difference system We consider the function F λ defined by This function is the unique analytic W -invariant function on R d that satisfies the differential equation for all W -invariant complex polynomials p on R d and p(T ) = p(T 1 , . . . , T d ).
In particular, for all λ ∈ R d we have . The function F λ is called the Heckman-Opdam hypergeometric function.
The functions G λ and F λ possess the following properties: ii) The functions G λ and F λ satisfy the estimate iv) Let p and q be polynomials of degree m and n, respectively. Then there exists a positive constant M such that for all λ ∈ C d and for all x ∈ R d , we have v) The preceding estimate holds true for F λ too. 2 , with ρ = α + β + 1 and 2 F 1 is the Gauss hypergeometric function.
In this case the Heckman-Opdam kernel F λ (x) is given for all λ ∈ C and x ∈ R by
Remark 2.4. The function C k is positive, continuous on R d , and satisfies the estimate The inverse transform is given by Using the generalized translation operator, we define the generalized convolution product of functions as follows.
Definition 2.11. The generalized convolution product of f and g in (2.5)

Basic generalized wavelet theory.
Definition 2.13. A generalized wavelet on R d is a measurable function h that is W -invariant on R d and satisfies, for almost all λ ∈ R d , the condition

This function is given by the relation
and satisfies Let a > 0 and h be in Notation. We denote by where the measure µ k is defined by .
Remark 2.17. i) The generalized continuous wavelet transform can also be written in the form where * k is the Heckman-Opdam convolution product given by (2.5). ii) Let h be a generalized wavelet. Then for all f in where for each y ∈ R d , both the inner integral and the outer integral are absolutely convergent, but possibly not the double integral.
3. Localization operators for the generalized continuous wavelet transform

Preliminaries.
Notation. We denote by: • l p (N) the set of all infinite sequences of real (or complex) numbers x := (x j ) j∈N such that For p = 2, we provide this space l 2 (N) with the scalar product where (v n ) n is any orthonormal basis of L 2 A k (R d ). Moreover, a compact operator A on the Hilbert space L 2 A k (R d ) is a Hilbert-Schmidt operator if the positive operator A * A is in the space of trace class S 1 . Then for any orthonormal basis (v n ) n of L 2 A k (R d ).

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HATEM MEJJAOLI AND KHALIFA TRIMÈCHE Definition 3.5. We define S ∞ := B(L 2 A k (R d )), equipped with the norm In this section, h will be a generalized wavelet on R d such that 3.2. Boundedness. In this subsection we define the localization operators for the generalized continuous wavelet transform and we show that they are bounded.
Definition 3.6. The localization operator with symbol σ associated with the generalized continuous wavelet transform, denoted by L h (σ), is defined on Often it is more convenient to interpret the definition of L h (σ) in a weak sense, that is, In this section we prove that the linear operators We consider first this problem for σ in L 1 µ k (R d+1 + ) W and next in L ∞ µ k (R d+1 + ) W and we conclude by using interpolation theory.
Proof. For all functions f and g in L 2 A k (R d ) W , we have from the relations (3.2) and (2.8): Thus, Proof. For all functions f and g in L 2 A k (R d ) W , we have from Hölder's inequality: Using Plancherel's formula for Φ W h , given by the relation (2.9), we get We can now associate a localization operator The precise result is the following theorem.
Then there exists a unique bounded linear operator L h (σ) : Proof. Let f be in L 2 A k (R d ) W . We consider the operator T : . Then, by Proposition 3.7 and Proposition 3.8, and Since (3.6) is true for arbitrary functions f in L 2 A k (R d ) W , we obtain the desired result.

Schatten-von Neumann properties for L h (σ). In this subsection we will prove that the localization operator
The first result on the Schatten property of localization operators is given in the following theorem.

Then the bounded localization operator
Proof. First let us assume that σ is a nonnegative real-valued symbol, thus the localization operator L h (σ) is positive. Let {u j , j = 1, 2, . . . } be any orthonormal basis for L 2 A k (R d ) W . Then from Fubini's theorem, the Parseval identity, and relations (2.6) and (2.7), we get

Thus we get
Using now the relation (2.6), we deduce that

Then, by [41, Proposition 2.4], the operator
, so if we consider {u j , j = 1, 2, . . . } an orthonormal basis for L 2 A k (R d ) W consisting of eigenvectors of the positive compact operator L h (σ) * L h (σ) and let s j , j = 1, 2, . . . , be the eigenvalues of |L h (σ)| corresponding to u j , then For σ a real-valued function, we write σ = σ + − σ − , with Finally, when σ = σ 1 + iσ 2 is a complex-valued function with σ 1 and σ 2 the real and imaginary parts of σ, we have that L h (σ) is in S 1 and Proof. From the previous theorem, the localization operator L h (σ) belongs to S 1 ; then by the definition of trace given by the relation (3.1), we have The result is obtained by the relation (3.7).

Proposition 3.12. Let σ be a symbol in L
Then the localization operator L h (σ) is compact.
Proof. Let σ be in L p µ k (R d+1 + ) W and let (σ n ) n∈N be a sequence of functions in On the other hand, as by Theorem 3.10 L h (σ n ) is in S 1 , hence compact, it follows that L h (σ) is compact.
In the following theorem we improve the constant given in Theorem 3.10. First, we begin by investigating the case σ in L 1 µ k (R d+1 + ) W and we give, in addition, a lower bound of the norm L h (σ) S1 .
where σ is given by where s j , j = 1, 2, . . . , are the positive singular values of L h (σ) corresponding to u j . Then we get Thus, by Fubini's theorem, Schwarz's inequality, Bessel's inequality, and the relations (2.6) and (2.7), we get It is easy to see that σ belongs to L 1 A k (R d ), and using formula (3.8) we obtain The proof is complete.
In the following theorem we give the main result of this section.

Generalized wavelet scalograms
4.1. The range of the wavelet transform. We denote by In other words, we can write where χ U denotes the characteristic function of the subset U of R d+1 In this section we shall keep our focus on localization operators L h (σ) with symbol σ = χ U , where U is a subset of R d+1 + with finite measure µ k (U ) given by In what follows, such operator will be denoted L h (U ) for the sake of simplicity.
Using the relation (2.9), we obtain On the other hand, by using Proposition 2.12, one can see that for every a, a > 0, x, x ∈ R d the function Therefore, we obtain the result. Remark 4.2. i) We note that Hence, P U P h is a Hilbert-Schmidt operator and therefore it is a compact operator. ii) We note that with integral kernel K h . iii) As K h is the integral kernel of an orthogonal projection, it satisfies

Definition 4.3. Let h be a generalized wavelet on
We define the generalized wavelet scalogram of f as It justifies the interpretation of a scalogram as a time-frequency energy density. Note that also by (3.2) we have

Definition 4.5.
We define the Calderón-Toeplitz operator Thus we deduce (4.2), and T h,U is bounded and positive. Now, we want to prove (4.3). Indeed, using Φ W h and (Φ W h ) * , the time-frequency localization operator . Consequently, the time-frequency operator L h (U ) and the Calderón-Toeplitz operator T h,U are related by From the above proposition we deduce that T h,U and L h (U ) enjoy the same spectral properties; in particular, we have the following proposition.

Proposition 4.8. The Calderón-Toeplitz operator T h,U is compact and even trace class with
Proof. We know that the operator T h, Therefore, by Definition 3.3 and Remark 3.4, the operator T h,U is trace class with µ k (R d+1 + ) W and consequently its spectral properties can be easily related to its integral kernel.
Since T h,U is positive and trace class, using the decomposition we deduce that V h,U is also positive and trace class with In addition, we have the following result.
Proposition 4.9. The trace of T 2 h,U is given by Proof. As V h,U is positive, we have On the other hand, using the fact that the space That is, V h,U has integral kernel Therefore, where by using the properties of the kernel of the reproducing kernel Hilbert space Using (4.1), we get and we conclude the proof.

Eigenvalues and eigenfunctions.
Since the localization operator L h (U ) = (Φ W h ) * χ U Φ W h that we consider is a compact and self-adjoint operator, the spectral theorem gives the following spectral representation: where {s n (U )} ∞ n=1 are the positive eigenvalues arranged in a nonincreasing manner and {ϕ U n } ∞ n=1 is the corresponding orthonormal set of eigenfunctions. Note that s n (U ) 0 and by (3.3) we have, for all n ≥ 1, By using this, together with (4.3), we can deduce that the Toeplitz operator can be diagonalized as Proof. From Proposition 4.1, we have that for all z = (a, x) ∈ R d+1 . Therefore using the properties of the kernel of the reproducing kernel Hilbert space, we get Using this, we compute again and the conclusion follows.
Then by an easy adaptation of the proof of Lemma 3.3 in [1] we obtain the following estimate for the eigenvalue distribution.

Scalogram of a subspace. Given an
, Recall that if {v n } N n=1 is an orthonormal basis of V , then The kernel G V is independent of the choice of orthonormal basis for V .
Then we have the following result.
Lemma 4.13. The scalogram SCAL k h V is given by Proof. We have Definition 4.14. We define the time-frequency concentration of a subspace V in U as Then, from Lemma 4.13, x) dµ k (a, x). Proof. We have Moreover, the min-max lemma for self-adjoint operators states that (cf. [36]) So the eigenvalues of L h (U ) determine the number of orthogonal functions that have a well-concentrated scalogram in U . Thus, s n (U ).
The min-max characterization of the eigenvalues of compact operators implies that the first N eigenfunctions of the time-frequency operator L h (U ) have optimal cumulative time-frequency concentration inside U , in the sense that Therefore no N -dimensional subset V of L 2 A k (R d ) can be better concentrated in U than V N , i.e., ξ U,h (V ) ≤ ξ U,h (V N ).
As N k (h, U ) ≥ l k (h, U ), we get As N k (ε, U ) ≥ M k (h, U ), we obtain the desired result.
Consequently, when the eigenvalues {s n (U )} n(ε,U ) n=0 are close to 1, E(h, U ) → 0. Moreover, we have the following result bounding the error between ρ (h,U ) and Θ.