Time-frequency analysis associated with the Laguerre wavelet transform

We define the localization operators associated with Laguerre wavelet transforms. Next, we prove the boundedness and compactness of these operators, which depend on a symbol and two admissible wavelets on Lα(K), 1 ≤ p ≤ ∞.

Then T = ∂ ∂t and The theory of harmonic analysis on L p rad (H d ) was exploited by many authors (see [23,27,32]). When one considers the problems of radial functions on the Heisenberg group H d , the underlying manifold can be regarded as the Laguerre hypergroup K := [0, ∞) × R. Stempak [33] introduced a generalized translation operator on K 32 HATEM MEJJAOLI AND KHALIFA TRIMÈCHE and established the theory of harmonic analysis on L 2 (K, dν α ), where the weighted Lebesgue measure ν α on K is given by dν α (x, t) := x 2α+1 dxdt πΓ(α + 1) , α ≥ 0.
In this paper we are interested in the Laguerre hypergroup K. We recall that (K, * α ) is a commutative hypergroup [29], on which the involution and the Haar measure are respectively given by the homeomorphism (x, t) → (x, t) − = (x, −t) and the Radon positive measure dν α (x, t). The unit element of (K, * α ) is given by e = (0, 0).
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 recognized many classical results inside it (see [6,21,26]). Classical wavelets have wide applications, ranging from signal analysis in geophysics and acoustics to quantum theory and pure mathematics (see [8,16] and the references therein).
Next, the theory of wavelets and the continuous wavelet transform has been extended to hypergroups, in particular to the Laguerre hypergroups (see [29,34]).
One of the aims of wavelet theory is the study of localization operators for the continuous wavelet transform.
Time-frequency localization operators are 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, these operators have been introduced and studied by Daubechies [9,10,11] and Ramanathan and Topiwala [30], and they are now extensively investigated as an important mathematical tool in signal analysis and other applications [17,12,13,35,7].
As the harmonic analysis on the Laguerre hypergroup has known remarkable development, it is natural to ask whether there exists the equivalent of the theory of localization operators for the continuous wavelet transform related to this harmonic analysis.
Using the properties of the generalized Fourier transform on the Laguerre hypergroup K, our main aim in this paper is to expose and study the two-wavelet localization operator on the Laguerre hypergroup.
The reason for the extension from one wavelet to two wavelets comes from the extra degree of flexibility in signal analysis and imaging when the localization operators are used as time-varying filters. It turns out that localization operators with two admissible wavelets have a richer mathematical structure than the onewavelet analogues.
The remainder of this paper is arranged as follows. Section 2 contains some basic facts about the Laguerre hypergroup, its dual, and the Schatten-von Neumann classes. In Section 3 we introduce and study the two-wavelet localization operators in the setting of the Laguerre hypergroup. More precisely, the Schatten-von Neumann properties of these two localization wavelet operators are established, and for trace class Laguerre two-wavelet localization operators, the traces and the trace class norm inequalities are presented. Section 4 is devoted to proving that under suitable conditions on the symbols and two admissible wavelets, the L p boundedness and compactness of these two-wavelet localization operators hold.

Preliminaries
In this section we set some notation and we recall some basic results in harmonic analysis related to Laguerre hypergroups and Schatten-von Neumann classes. The main references are [29,35].
• C * (K) is the space of continuous functions on R 2 , even with respect to the first variable. • C * ,c (K) is the subspace of C * (K) formed by functions with compact support.
m being the Laguerre polynomial of degree m and order α.
•K := R × N equipped with the weighted Lebesgue measure γ α onK given by It is well known (see [29]) that for all (λ, m) ∈K, the system where D 1 and D 2 are singular partial differential operators given by The harmonic analysis on the Laguerre hypergroup K is generated by the singular operator , (x, t) ∈ K, while its dualK is generated by the differential difference operator where the operators Λ 1 , Λ 2 are given, for a suitable function g onK, by and the function where the difference operators ∆ + , ∆ − are given, for a suitable function g onK, by These operators satisfy some basic properties which can be found in [29,2]; namely, one has Definition 2.1. Let f ∈ C * ,c (K). For all (x, t) and (y, s) in K, we put (2.1) where x, y r,θ := x 2 + y 2 + 2xyr cos θ. The operators τ

Notation:
• S * (K) is the space of functions f : R 2 → C, even with respect to the first variable, C ∞ on R 2 , and rapidly decreasing together with their derivatives, i.e., for all k, p, q ∈ N we have Equipped with the topology defined by the semi-norms N k,p,q , S * (K) is a Fréchet space. • S(K) is the space of functions g :K → C such that (i) For all m, p, q, r, s ∈ N, the function is bounded and continuous on R, C ∞ on R * such that the left and the right derivatives at zero exist.
Equipped with the topology defined by the semi-norms ν k,p,q , S(K) is a Fréchet space.
(ii) The generalized Fourier transform F α extends to an isometric isomorphism from L 2 α (K) onto L 2 α (K). Corollary 2.11. For all f and g in L 2 α (K) we have the following Parseval formula for the generalized Fourier transform F α :

Schatten-von Neumann classes. Notation:
• l p (N), 1 ≤ p ≤ ∞, is the set of all infinite sequences of real (or complex) numbers u := (u j ) j∈N such that For p = 2, we provide the space l 2 (N) with the scalar product (ii) For 1 ≤ p < ∞, the Schatten class S p is the space of all compact operators whose singular values lie in l p (N). The space S p is equipped with the norm Remark 2.14. We note that the space S 2 is the space of Hilbert-Schmidt operators, and S 1 is the space of trace class operators.

Definition 2.15. The trace of an operator
for any orthonormal basis (v n ) n of L 2 α (K). Definition 2.17. We define S ∞ := B(L 2 α (K)), equipped with the norm 2.3. Basic Laguerre wavelet theory. In this subsection we recall some results introduced in [29].
where the measure µ α is defined by Definition 2.18. A Laguerre wavelet on K is a measurable function h on K satisfying, for almost all (λ, m) ∈K\{(0, 0)}, the condition Let a ∈ R\{0} and let h be a measurable function. We consider the function h a defined by are the generalized translation operators given by (2.1).
This transform can also be written in the form wheref (x, t) = f (x, −t) and * α is the generalized convolution product given by (2.2).

Laguerre two-wavelet localization operators
In this section we will derive a host of sufficient conditions for the boundedness and Schatten class of the Laguerre two-wavelet localization operators in terms of properties of the symbol σ and the windows h and k.

Preliminaries.
Definition 3.1. Let h, k be measurable functions on K, and σ a measurable function on R × K. We define L h,k (σ), the Laguerre two-wavelet localization operator on L p α (K), 1 ≤ p ≤ ∞, by According to the different choices of the symbols σ and the different continuities required, we need to impose different conditions on h and k, and then we obtain an operator on L p α (K). It is often more convenient to interpret the definition of L h,k (σ) in a weak sense, that is, for f in L p α (K), p ∈ [1, ∞], and g in L p α (K), In what follows, such operator L h,k (σ) will be named localization operator, for the sake of simplicity. p ∈ [1, ∞). Formally, we assume that we have

Proposition 3.2. Let
Then its adjoint is the linear operator L k,h (σ) : Proof. For all f in L p α (K) and g in L p α (K) it immediately follows from (3.2) that In the rest of this section, h and k will be two Laguerre wavelets on K such that The main result of this subsection is the proof that the linear operators We first consider this problem for σ in L 1 µα (R × K) and next in L ∞ µα (R × K), and we then conclude by using interpolation theory.
Proof. For all functions f and g in L 2 α (K), we have from relations (3.2) and (2.10), For all functions f and g in L 2 α (K), we have from Hölder's inequality . Using Plancherel's formula for Φ α h and Φ α k , given by relation (2.9), we get We can now associate a localization operator L h,k (σ) : The precise result is the following theorem.
. We consider the operator

Then, by Proposition 3.3 and Proposition 3.4,
and Since (3.7) is true for arbitrary functions f in L 2 α (K), we obtain the desired result.  α (a, x, t).   Proof. Let σ be in L p µα (R × K) and let (σ n ) n∈N be a sequence of functions in

Schatten-von Neumann properties for L h,k (σ). The main result of this subsection is the proof that the localization operator
such that σ n → σ in L p µα (R × K) as n → ∞. Then by Theorem 3.5, On the other hand, as by Proposition 3.6 L h,k (σ n ) is in S 2 and hence compact, it follows that L h,k (σ) is compact.

9)
where σ is given by where s j , j = 1, 2, . . . , are the positive singular values of L h,k (σ) corresponding to φ j . Then, we get

HATEM MEJJAOLI AND KHALIFA TRIMÈCHE
Thus, by Fubini's theorem, Cauchy-Schwarz's inequality, Bessel's inequality, and relations (2.8) and (2.6), we get . We now prove that L h,k (σ) satisfies the first inequality of (3.9). It is easy to see that σ belongs to L 1 α (K), and using formula (3.10) we get Then from Fubini's theorem, we obtain Thus using Plancherel's formula for Φ α h , Φ α k we get The proof is complete.  α (a, x, t).
In the following we give the main result of this subsection.   α (a, x, t). Now we state a result concerning the trace of products of localization operators. Corollary 3.12. Let σ 1 and σ 2 be any real-valued and non-negative functions in L 1 µα (R × K). We assume that h = k and that h is a function in L 2 α (K) such that h L 2 α (K) = 1. Then, the localization operators L h,k (σ 1 ), L h,k (σ 2 ) are positive trace class operators and, for any natural number n, n S1 .

HATEM MEJJAOLI AND KHALIFA TRIMÈCHE
Proof. By Theorem 1 in Liu's paper [22] we know that if A and B are in the trace class S 1 and are positive operators, then ∀ n ∈ N, tr(AB) n ≤ tr(A) n tr(B) n .
So, if we take A = L h,k (σ 1 ), B = L h,k (σ 2 ), and we invoke the previous remark, the desired result is obtained and the proof is complete.
4. L p α boundedness and compactness of L h,k (σ) In this section we will derive a host of sufficient conditions for the boundedness and compactness of the localization operators L h,k (σ) on L p α (K), 1 ≤ p ≤ ∞, in terms of properties of the symbol σ and the windows h and k.

Boundedness of
, and h ∈ L p α (K). We are going to show that L h,k (σ) is a bounded operator on L p α (K). Let us start with the following propositions.
For every function f in L 1 α (K), from Fubini's theorem and the relations (3.1), (2.11), and (2.7), we have Proof. Let f be in L ∞ α (K). As above, from Fubini's theorem and the relations (3.1), (2.11), and (2.7), we have . With a Schur technique, we can obtain an L p α -boundedness result as in the previous theorem, but the estimate for the norm L h,k (σ) B(L p α (K)) is cruder.
Then there exists a unique bounded linear operator (4.1) We have By simple calculations, it is easy to see that Thus by Schur's lemma (see [15]), we can conclude that L h,k (σ) : L p α (K) → L p α (K) is a bounded linear operator for 1 ≤ p ≤ ∞, and we have Proof. For any f ∈ L p α (K), consider the linear functional Using Fubini's theorem and the relation (2.11), we get Thus, the operator I f is a continuous linear functional on L p α (K), and the operator norm satisfies , which establishes the proposition.
Combining Proposition 4.1 and Proposition 4.7, we have the following theorem.
. We can now state and prove the main results in this subsection. Theorem 4.9. Let σ be in L r µα (R × K), r ∈ [1,2], and h, k ∈ L 1 Proof. Consider the linear functional By Proposition 4.1 and Theorem 3.5, we have and Therefore, by (4.2), (4.3), and the multi-linear interpolation theory (see [5,Section 10.1] for reference), we get a unique bounded linear operator By the definition of I, we have As the adjoint of L h,k (σ) is L k,h (σ), L h,k (σ) is a bounded linear map on L r α (K) with its operator norm satisfying where Using an interpolation of (4.4) and (4.5), we have that, for any p ∈ [r, r ], r+1 , 2r r−1 , and we have where In order to prove this theorem we need the following lemmas.
Then there exists a unique bounded linear operator Proof. Consider the linear functional Then by Proposition 4.1 and Theorem 3.5, and where · B(L p µα (R×K),B(L q α (K))) denotes the norm in the Banach space of the bounded linear operators from L p µα (R × K) into B(L q α (K)), 1 ≤ p, q ≤ ∞. Using an interpolation of (4.7) and (4.8) we get the result.
Proof. As the adjoint of is the bounded linear operator the result follows from duality and the previous lemma.
Proof. The proof follows from Theorem 4.8 and Theorem 3.5 with p = 1, q instead of p, and interpolation theory.
In the following we give two results for compactness of localization operators.  Proof. The result is an immediate consequence of an interpolation of Corollary 3.10 and Proposition 4.14. See again [4, pp. 202-203] for the interpolation used.