The Continuous Wavelet Transform Associated with a Dunkl Type Operator on the Real Line

We consider a singular differential-difference operator Λ on R which includes as a particular case the one-dimensional Dunkl operator. By using harmonic analysis tools corresponding to Λ, we introduce and study a new continuous wavelet transform on R tied to Λ. Such a wavelet transform is exploited to invert an intertwining operator between Λ and the first derivative operator d/dx.


Introduction
In this paper we consider the first-order singular differential-difference operator on R where 1 2

  
and q is a real-valued odd function on R. For q = 0, we regain the differential-difference operator which is referred to as the Dunkl operator with parameter 1 2   associated with the reflection group Z 2 on R.Those operators were introduced and studied by Dunkl [1][2][3] in connection with a generalization of the classical theory of spherical harmonics.Besides its mathematical interest, the Dunkl operator has quantum-mechanical applications; it is naturally involved in the study of onedimensional harmonic oscillators governed by Wigner's commutation rules [4][5][6]. Put and The authors [7] have proved that the integral transform is the only automorphism of the space of f E  R The intertwining operator X has been exploited to initiate a quite new commutative harmonic analysis on the real line related to the differential-difference operator Λ in which several analytic structures on R were generalized.A summary of this harmonic analysis is provided in Section 2. Through this paper, the classical theory of wavelets on R is extended to the differential-difference operator Λ.More explicitly, we call generalized wavelet each function g in where F  denotes the generalized Fourier transform related to Λ given by Starting from a single generalized wavelet g we construct by dilation and translation a family of generalized wavelets by putting where stand for the generalized dual translation operators tied to the differential-difference operator Λ, and g a is the dilated function of g given by the relation Accordingly, the generalized continuous wavelet transform associated with Λ is defined for regular functions f on R by In Section 3, we exhibit a relationship between the generalized and Dunkl continuous wavelet transforms.Such a relationship allows us to establish for the generalized continuous wavelet transform a Plancherel formula, a point wise reconstruction formula and a Calderon reproducing formula.Finally, we exploit the intertwining operator X to express the generalized continuous wavelet transform in terms of the classical one.As a consequence, we derive new inversion formulas for dual operator of X.

t X
In the classical setting, the notion of wavelets was first introduced by J. Morlet, a French petroleum engineer at ELF-Aquitaine, in connection with his study of seismic traces.The mathematical foundations were given by A. Grossmann and J. Morlet in [8].The harmonic analyst Y. Meyer and many other mathematicians became aware of this theory and they recognized many classical results inside it (see [9][10][11]).Classical wavelets have wide applications, ranging from signal analysis in geophysics and acoustics to quantum theory and pure mathematics (see [12][13][14] and the references therein).

Preliminaries
Notation.We denote by

Generalized Fourier Transform
The following statement is proved in [7].Lemma 1. 1) For each   C , the differential-dif- ference equation where e  denotes the one-dimensional Dunkl kernel defined by 3) For each x  R and   C , we have the Laplace type integral representation where a  is given by (1).The generalized Fourier transform of a function f in Remark 2. 1) By ( 6) and (7), it follows that the generalized Fourier transform F  maps continuously and injectively Notice by ( 5), ( 8) and ( 9) that where M is given by ( 4).
Two standard results about the generalized Fourier transform F  are as follows.

Theorem 1 (inversion formula)
Theorem 2 (Plancherel).1) For every 2) The generalized Fourier transform F  extends uniquely to an isometric isomorphism from

Generalized Convolution
Recall that the Dunkl translation operators , , and such that , 2 x y    .For the explicit expression of the measure , , x y   see [15].Define the generalized translation operators T x , x  R , associated with Λ by 12) and ( 13) observe that The generalized dual translation operators are given by We claim the following statement.
x  R , is well defined as a function in and x

T h y h y y y Q x h y Q y h y y y Q x h y Qh y y y h y T h y y y
This concludes the proof.■ The generalized convolution product of two functions f and g on R is defined by Remark 3. Recall that the Dunkl convolution product of two functions f and g on R is defined by By virtue of ( 15), ( 16) and ( 17) it is easily seen that By use of ( 10), (18) and the properties of the Dunkl convolution product mentioned in [16], we obtain the next statement.

Intertwining Operators
According to [7], the dual of the intertwining operator X given by (3), takes the form It was shown that t is an automorphism of the space of C compactly supported functions on R, satisfying the intertwining relation Moreover, we have the factorizations where V  and t V  are respectively the Dunkl inter- twining operator and its dual given by Using (19) and the properties of V  and t V  pro- vided by [17], we easily derive the next statement.
4) For every we have the identity where F u denotes the usual Fourier transform on R given by where * denotes the usual convolution product on R given by

Generalized Wavelets
Notation.For a function f on R put , .
which can also be written in the form where   is the Dunkl convolution product given by (17).
The Dunkl continuous wavelet transform has been investigated in depth in [17] from which we recall the following fundamental properties.
2) For such that

Generalized Wavelets
Definition 3. We say that a function for almost all .
  be a generalized wavelet such that  .

Inversion of the Intertwining Operator t X Using Generalized Wavelets
In order to invert t X we need the following two technical lemmas.
where m  is given by (11).
Proof.We have As by ( 3) and (7), R So it suffices, in view of (36) and Theorem 2, to prove that h belongs to From the Plancherel theorem for the usual Fourier transform, it follows that Proof.By using (37) and Lemma 2 we see that Thus, in view of Remark 4 3), the function where , and R is a classical wavelet on R, i.e., satisfying the admissibility condition for almost all .
  R A more complete and detailed discussion of the properties of the classical continuous wavelet transform can be found in [10].
Remark 7. 1) According to [10], each function satisfying the conditions of Lemma 3 is a classical wavelet.

 
g D  R is a gen- eralized wavelet, if and only if, t Xg is a classical wavelet and we have   .d  d .

Proposition 6 .
In the next statement we exhibit a formula relating the generalized continuous wavelet transform to the classical one.Let g be as inLemma 3 virtue of (3), (24) and (29).So using (21) and (38) we find that

 7 .
which gives the desired result.Combining Theorems 5, 6 with Lemma 3 and Proposition 6 we get Theorem Let g be as in Lemma 3. Let 2 24) 