New results on the continuous Weinstein wavelet transform

We consider the continuous wavelet transform ShW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{S}_{h}^{W}$\end{document} associated with the Weinstein operator. We introduce the notion of localization operators for ShW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {S}_{h}^{W}$\end{document}. In particular, we prove the boundedness and compactness of localization operators associated with the continuous wavelet transform. Next, we analyze the concentration of ShW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{S}_{h}^{W}$\end{document} on sets of finite measure. In particular, Benedicks-type and Donoho-Stark’s uncertainty principles are given. Finally, we prove many versions of Heisenberg-type uncertainty principles for ShW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{S}_{h}^{W}$\end{document}.


Introduction
In this paper, we consider the Weinstein operator (also called the Laplace-Bessel differential operator (see [])) defined on R d- × (, ∞) by where x is the Laplace operator on R d- , and L β,x d the Bessel operator on (, ∞) given by The Weinstein operator β has several applications in pure and applied mathematics especially in fluid mechanics (see []).
The harmonic analysis associated with the Weinstein operator is studied by Ben Nahia and Ben Salem [, ]. In particular the authors have introduced and studied the generalized Fourier transform associated with the Weinstein operator. This transform is called the Weinstein transform.
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 []. The harmonic analyst Meyer and many other mathematicians became aware of this theory, and they recognized many classical results inside it (see [-]). Classical wavelets have wide applications, ranging from signal analysis in geophysics and acoustics to quantum theory and pure mathematics (see [-] and the references therein).
Next, the theory of wavelets and continuous wavelet transforms has been extended to hypergroups, in particular, to the Chébli-Trimèche hypergroups (see []).
Recently in [] the authors have introduced and studied the Weinstein wavelet transform S W h . In the same paper, for the transform S W h , the authors have proved the Plancherel and inversion formulas.
Nowadays, the continuous wavelet transform is one of the useful subjects in harmonic analysis. In this paper we present only two subjects.
The first subject is the new uncertainty principles involving time-frequency representations.
The second subject is the localization operators. These operators were initiated by Daubechies [-], and detailed in the book [] by Wong. As the harmonic analysis associated with the Weinstein operator has known remarkable development, it is a natural question whether there exist an equivalent of the theory of localization operators and new uncertainty principles for the continuous wavelet transform relating to this harmonic analysis.
The purpose of the present paper is twofold. On one hand, we want to study many versions of quantitative uncertainty principles for the continuous Weinstein wavelet transform. On the other hand, we want to study the localization operators for the continuous Weinstein wavelet transform.
The remainder of this paper is arranged as follows. In Section , we recall the main results about the harmonic analysis associated with the Weinstein operator. In Section , we introduce and study the two-localization operators associated with the Weinstein continuous wavelet transform. More precisely, the Schatten-von Neumann properties of these two-wavelet localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented. In Section , we study the quantitative analysis of the continuous Weinstein wavelet transform and time-frequency concentration. In particular, we give results on sets of finite measure and Donoho-Stark and Benedicks-type uncertainty principles. In Sections  and , we prove many versions of the Heisenberg uncertainty inequalities for the Weinstein continuous wavelet transform. Finally, some conclusions are drawn in Section .

Preliminaries
To confirm the basic and standard notations, we briefly overview the Weinstein operator and related harmonic analysis. The main references are [, ].

Harmonic analysis associated with the Weinstein operator
In this subsection, we collect some notation and results on the Weinstein kernel, the Weinstein transform, and the Weinstein convolution. We use the following notation: is the space of continuous functions on R d even with respect to the last variable. C p * (R d ) is the space of functions of class C p on R d even with respect to the last variable. E * (R d ) is the space of C ∞ -functions on R d even with respect to the last variable. S * (R d ) is the Schwartz space of rapidly decreasing functions on R d even with respect to the last variable. D * (R d ) is the space of C ∞ -functions on R d with compact support and even with respect to the last variable. The Weinstein kernel is given by where j β is the normalized Bessel function. The Weinstein kernel satisfies the following properties: and |ν| = ν  + · · · + ν d . In particular, where dλ β is the measure on R d + given by The Weinstein transform is given for Some basic properties of this transform are the following: , and for all f in S * (R d ), For a function f ∈ S * (R d ) and y ∈ R d + , the generalized translation τ y f is defined by the following relation: (.) By using the generalized translation, we define the generalized convolution product f * W g of functions f , g ∈ L  β (R d + ) as follows: This convolution is commutative and associative. Moreover, we have the following: Definition . Let E, F be two measurable subsets of R d + . Then: .
The constant C β (E, F) is called the annihilation constant of (E, F).

Proposition . [] Let E, F be two measurable subsets of
Then the pair (E, F) is a strongly annihilating pair.

Weinstein wavelets
By a simple calculations we see that, for almost all x ∈ R d + , Thus h is an example of a Weinstein wavelet on R d + .
Let b > , and let h be in L  β (R d + ). The dilation of h by b is defined by It is easy to see that h b ∈ L  β (R d + ) and We introduce the family h b,y , b >  and y ∈ R d + , of Weinstein wavelets on R d Notation We denote: where the measure dμ β is defined by It is easy to see that , and let h be a Weinstein wavelet. Then we have and

Proposition . (Covariance properties) Let h be a Weinstein wavelet. The transformation S W h is a bounded linear operator from L
into the space of continuous bounded functions on d+ . Moreover, we have the following covariance property:

Preliminaries
) are the eigenvalues of the positive self-adjoint operator |M| = √ M * M. For  ≤ p < ∞, the Schatten class S p is the space of compact operators whose singular values lie in l p . Hence S p is equipped with the norm (  .  ) In particular, S  is the space of Hilbert-Schmidt operators, and S  is the space of trace-class operators. It is well known that the trace of an operator M ∈ S  is defined by For consistency, we define

Boundedness
In this section, h and k will be two Weinstein wavelets such that Definition . The localization operator associated with the symbol a and two Weinstein wavelets is denoted by L h,k (a) and defined on L  β (R d + ) by Often, it is more convenient to interpret the definition of L h,k (a) in a weak sense, that is, Proof Thus we get The objective of this subsection is to prove that the operators are bounded for all symbols a ∈ L p μ β ( d+ ). We first consider this problem for a in L  μ β ( d+ ) and next in L ∞ μ β ( d+ ), and then by interpolation theory we deduce the result.

Proposition . Let a be in L
. Thus, Proof For all functions f and g in L  β (R d + ), by Hölder's inequality we have Applying the Plancherel formula (.) for S W h and S W k , we obtain Thus, We now can prove that L h,k (a) is in S ∞ for every symbol a in L p μ β ( d+ ),  ≤ p ≤ ∞.

Theorem . Let a be in L
Then there exists a bounded linear operator .
. We consider the operator Then, by Proposition . and Proposition ., , the result is proved.

Schatten-von Neumann properties for L h,k (a)
Let us begin with the following statement.

Proposition . Let a be a symbol in L
Hence L h,k (a n ) → L h,k (a) in S ∞ as n → ∞. On the other hand, since by Proposition . L h,k (a n ) is in S  and hence compact, it follows that L h,k (a) is compact.
Theorem . Let a be in L  μ β ( d+ ). Then where a is given by where s j , j ∈ N, are the positive singular values of L h,k (a) corresponding to φ j . Thus, we get Hence, by Fubini's theorem, Cauchy-Schwarz's inequality, Bessel inequality, and relations (.) and (.) we obtain We now prove that L h,k (a) satisfies the first member of (.). Indeed, from (.) we have Then, using Fubini's theorem, we obtain Thus, applying Plancherel's identity for S W h and S W k , we get The proof is complete.

Corollary . For a in L
Proof By Proposition ., L h,k (a) ∈ S  . Then, using (.), we get and the proof is complete.

Corollary . Let a be in L
Then, the localization operator Proof The result follows from Proposition . and Theorem . and by interpolation [, Theorems . and .].

Continuous Weinstein wavelet transform and time-frequency concentration
In this section, we suppose that the Weinstein wavelet h belongs to L  β (R d

Proposition . Let h be a Weinstein wavelet on
) is a reproducing kernel Hilbert space with kernel The kernel is pointwise bounded: Using relation (.), we obtain Applying Proposition .(iii), we find that, for all b, b >  and y, y ∈ R d + , the function . Therefore, the result is obtained.
In the following we denote: where χ U denotes the characteristic function of U ⊂ d+ with We put We further prove the concentration of S W h (f ) in small sets.

Proposition . Let h be a Weinstein wavelet, and let U ⊂ d+ with
.
Then, for all f ∈ L  β (R d + ), we have where χ U denotes the characteristic function of U.
We further prove the concentration of S W h (f ) in arbitrary sets of finite measures.

Theorem . Let h be a Weinstein wavelet, and let U
For the proof of this theorem, we need the following lemma.
Proof of Theorem . Define H  and H  by Proceeding as in [], we prove that Hence, P U P h is a Hilbert-Schmidt operator and therefore compact. Now, Lemma . implies the existence of a constant C >  such that (.) holds for P H  := P U and P H  := P h . Since this leads to (.).
Definition . Let h be a Weinstein wavelet, and let U ⊂ d+ such that  < μ β (U) < ∞. Then () We say that U is weakly annihilating if any function f ∈ L  β (R d + ) vanishes when its Weinstein wavelet transform S W h (f ) with respect to the Weinstein wavelet h is supported in U. () We say that U is strongly annihilating if there exists a constant C β (U) >  such that, for every function f ∈ L  β (R d + ), The constant C β (U) is called the annihilation constant of U.
The analogue of this definition was introduced by Ghobber and Omri [] in the cadre of windowed Hankel transform.

Remark .
() It is clear that every strongly annihilating set is also a weakly.
is strongly annihilating. () As the operator P U P h is Hilbert-Schmidt and hence compact, from [] we have that if U is weakly annihilating, then it is also strongly annihilating.
() Following [, p.], we have that if U is strongly annihilating, then P U P h < .
In the following, we prove the Benedicks uncertainty principle for the Weinstein wavelet transform under some condition on the Weinstein wavelet. We note that the Benedickstype uncertainty principle for the windowed Hankel transform was studied by Ghobber and Omri [].

Theorem . Let h be a Weinstein wavelet such that
For any subset U ⊂ d+ such that Then by the definition of the operators Then we have On the other hand, using (.) and hypothesis (.), we get Using Proposition ., we obtain that F b =  for every b > , and hence F = .
Consequently, we obtain the following improvement.

Corollary . Let h be a Weinstein wavelet on
Then, for any subset U ⊂ d+ such that  < μ β (U) < ∞, there exists a constant C := C(h, U) >  such that, for all f ∈ L  β (R d + ), we have Now we will derive a sufficient condition by means of which we can recover a signal F belonging to L  μ β ( d+ ) from the knowledge of its truncated version following the Donoho-Stark criterion [].
Let h be a Weinstein wavelet function. A signal F ∈ L  μ β ( d+ ) is transmitted to a receiver who knows that F ∈ S W h (L  β (R d + )). Suppose that the observation of F is corrupted by a noise n ∈ L  μ β ( d+ ) (which is nonetheless assumed to be small) and unregistered values on U ∈ d+ . Thus, the observable function r satisfies Here we have assumed without loss of generality that n =  on U. Equivalently, We say that F can be stably reconstructed from r if there exist a linear operator and a constant C(U, h) such that Proceeding as in [], it is easy to prove the following:

Proposition . Let h be a Weinstein wavelet function such that
Let U ⊂ d+ with  < μ β (U) < ∞. Then F can be stably reconstructed from r. The constant C(U, h) in (.) is not larger than ( -P U P h ) - .
The identity suggests an algorithm for computing Q(r). Finally, using a method similar to that in [], we give an algorithm for computing L U,h . Indeed, put Then F k → Q(r) as k → ∞. Now and so on. The iteration converges at a geometric rate to the fixed point Algorithms of type (.), have been applied to a host of problems in signal recovery; see [] and others.

Heisenberg-type uncertainty inequalities for the Weinstein wavelet transform
In this section, we establish many Heisenberg-type uncertainty inequalities for the Weinstein wavelet transform. .
Thus we have obtained the result with C  (β, s) := r s C hμ β (U r ). Now we will prove (). Indeed, using the fact that (  b , y) s ≤  s (|b| -s + y s ) in (.), we get .
Replacing f by δ t f : in the previous inequality, by (.) and by a suitable change of variables we obtain: .
Then (.) follows by minimizing the left-hand side of that inequality over t > . Now we will prove (). Indeed, using the estimates we get Finally, the result immediately follows from (.).
Proof Let us assume the nontrivial case that both integrals on the left-hand side of (.) are finite. We get from the admissibility condition (.) for h that Using relation (.), we obtain Moreover, using (.), we get ∀b > , Integrating with respect to db b β+d+ , we obtain The left-hand side of this inequality may be estimated from above using Hölder's inequality. The right-hand side can be rewritten by Plancherel's formula for S W h . Therefore, from (.) we get .
This proves the result.
6 Heisenberg-type uncertainty inequalities for the modified Weinstein wavelet transform There is no uncertainty principle of Heisenberg type for S W h (f )(b, y) with respect to b. For this, we consider the Weinstein wavelet h defined by relation (.) and introduce the modified Weinstein continuous wavelet transform S W h given by Using this transform, we obtain the following theorem. Proof In the following, we assume that

Theorem . For s, t >  and every f in L
Otherwise, (.) is trivially satisfied. Using Fubini's theorem and (.), we have Proof The result follows from Corollary . and the fact that (b, y) s ≥ y s and (b, y) t ≥ b t .
In the following, we give the local-type uncertainty principle.
On the other hand, from Corollary . we have The result immediately follows.

Conclusions
This paper is devoted to developing the localization operator theory for the continuous Weinstein wavelet transform. Given a symbol a and two Weinstein wavelets h, k, we investigate the multilinear mapping from (a, h, k) ∈ L p μ β ( d+ ) × L  β (R d + ) × L  β (R d + ),  ≤ p ≤ ∞, to the localization operator L h,k (a), and we give sufficient conditions for L h,k (a) to be bounded or to belong to a Schatten class. Our results are formulated in terms of timefrequency analysis.
The uncertainty principles in the context of the continuous Weinstein wavelet transform are also established.