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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.

In this paper we consider the first-order singular differential-difference operator on R

where 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 associated with the reflection group Z_{2} on R. Those operators were introduced and studied by Dunkl [1-3] 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 [4-6].

Put

and

The authors [

is the only automorphism of the space of functions on R, satisfying

for all 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 satisfying almost all

where denotes the generalized Fourier transform related to Λ given by

being the solution of the differential-difference equation

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.

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 [

Notation. We denote by

• the class of measurable functions f on R for which where

and

• the class of measurable functions f on R for which where Q is given by (2).

• the class of measurable functions f on R for which

Remark 1. Clearly the map

is an isometry

• from onto;

• from onto.

The following statement is proved in [

Lemma 1. 1) For each, the differential-difference equation

admits a unique solution on R, denoted, given by

where denotes the one-dimensional Dunkl kernel defined by

being the normalized spherical Bessel function of index given by

2) For all, and we have

3) For each and, we have the Laplace type integral representation

where is given by (1).

The generalized Fourier transform of a function f in is defined by

Remark 2. 1) By (6) and (7), it follows that the generalized Fourier transform maps continuously and injectively into the space of continuous functions on R vanishing at infinity.

2) Recall that the one-dimensional Dunkl transform is defined for a function by

Notice by (5), (8) and (9) that

where M is given by (4).

Two standard results about the generalized Fourier transform are as follows.

Theorem 1 (inversion formula). Let such that. Then for almost all we have

where

Theorem 2 (Plancherel). 1) For every, we have the Plancherel formula

2) The generalized Fourier transform extends uniquely to an isometric isomorphism from onto.

Recall that the Dunkl translation operators are defined by

where is a finite signed measure on R, of total mass 1, with support

and such that. For the explicit expression of the measure see [

Define the generalized translation operators T^{x}, , associated with Λ by

By (12) and (13) observe that

The generalized dual translation operators are given by

We claim the following statement.

Proposition 1. 1) Let f be in Then for all is a well defined element in and

2) Let f be in Then for all, is well defined as a function in and

3) For p = 1 or 2, we have

4) Let, such that If and then we have the duality relation

Proof. 1) By (14) and [13, Equation (8)] we have

2) By (15) and [13, Equation (8)] we have

3) By (5), (10), (15) and [1, Theorem 11] we have

4) By (14), (15) and [1, Theorem 11] we have

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 [

Proposition 2. 1) Let such that

If and then

and

.

2) For and p = 1 or 2, we have

According to [

It was shown that is an automorphism of the space of compactly supported functions on R, satisfying the intertwining relation

where is the dual operator of Λ defined by

Moreover, we have the factorizations

where and are respectively the Dunkl intertwining operator and its dual given by

Using (19) and the properties of and provided by [

Proposition 3. 1) If then

and

2) If then and

3) For every and we have the duality relation

4) For every we have the identity

where F_{u} denotes the usual Fourier transform on R given by

5) Let. Then

where * denotes the usual convolution product on R given by

6) Let and Then

Notation. For a function f on R put

Definition 1. A Dunkl wavelet is a function satisfying the admissibility condition

for almost all

Notation. For a function g in and for we write

where are the Dunkl translation operators given by (12), and

Definition 2. Let be a Dunkl wavelet. The Dunkl continuous wavelet transform is defined for smooth functions f on R by

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 [

Theorem 3. Let be a Dunkl wavelet. Then 1) For all we have the Plancherel formula

2) For such that we have

for almost all

3) Assume that For and the function

belongs to and satisfies

Definition 3. We say that a function is a generalized wavelet if it satisfies the admissibility condition

for almost all

Remark 4. 1) The admissibility condition (27) can also be written as

2) If g is real-valued we have, so (27) reduces to

3) If is real-valued and satisfies such that, as then (27) is equivalent to

4) According to (10), (22) and (27), is a generalized wavelet if and only if, is a Dunkl wavelet, and we have

Notation. For a function g on R and, put

Remark 5. Notice by (24) and (29) that

Proposition 4. 1) Let and for some Then and

where q is such that

2) For and p = 1 or 2, we have

Proof. 1) By (30) and [13, Equation (13)], we have

2) By (10), (30) and [13, Equation (11)], we have

which achieves the proof. ■

Definition 4. Let be a generalized wavelet. We define for regular functions f on R, the generalized continuous wavelet transform by

where

and are the dual generalized translation operators given by (15).

Remark 6. A combination of (15), (23) and (32) yields

Proposition 5. Let be a generalized wavelet. Then for all p = 1 or 2, we have

where # is the generalized convolution product given by (16).

Proof. By (18), (25), (26), (30), (31) and (33), we have

which ends the proof. ■

A combination of Theorem 3 with identities (28), (33) and (34) yields the following basic results for the generalized continuous wavelet transform.

Theorem 4 (Plancherel formula). Let be a generalized wavelet. Then for all we have

Theorem 5 (inversion formula). Let be a generalized wavelet. If and then we have

for almost all

Theorem 6 (Calderon’s formula). Let be a generalized wavelet such that Then for and the function

belongs to and satisfies

In order to invert ^{t}X we need the following two technical lemmas.

Lemma 2. Let such that

and satisfying

as Let Then and

where is given by (11).

Proof. We have

As by (3) and (7),

we deduce that

with

Clearly, So it suffices, in view of (36) and Theorem 2, to prove that h belongs to We have

By (35) there is a positive constant k such that

From the Plancherel theorem for the usual Fourier transform, it follows that

which ends the proof. ■

Lemma 3. Let be real-valued such that and satisfying such that

as Let Then is a generalized wavelet and

Proof. By using (37) and Lemma 2 we see that, is bounded and

Thus, in view of Remark 4 3), the function satisfies the admissibility condition (27). ■

Recall that the classical continuous wavelet transform is defined for suitable functions f on R by

where, and is a classical wavelet on R, i.e., satisfying the admissibility condition

for almost all A more complete and detailed discussion of the properties of the classical continuous wavelet transform can be found in [

Remark 7. 1) According to [

2) In view of (20), (27) and (39), is a generalized wavelet, if and only if, is a classical wavelet and we have

In the next statement we exhibit a formula relating the generalized continuous wavelet transform to the classical one.

Proposition 6. Let g be as in Lemma 3. Let Then for all p = 1 or 2, we have

Proof. By (34) we have

But

by virtue of (3), (24) and (29). So using (21) and (38) we find that

which gives the desired result.

Combining Theorems 5, 6 with Lemma 3 and Proposition 6 we get Theorem 7. Let g be as in Lemma 3. Let. Then we have the following inversion formulas for the integral transform:

1) If and then for almost all we have

2) For and the function

satisfies

This work was funded by the Deanship of Scientific Research at the University of Dammam under the reference 2012018.