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

(1)

The Continuous Wavelet Transform Associated with a

Dunkl Type Operator on the Real Line

E. A. Al Zahrani, M. A. Mourou*

Department of Mathematics, College of Sciences for Girls, University of Dammam, Dammam, KSA Email: *_{[email protected]}

Received April 16, 2013; revised May 25, 2013; accepted June 27,2013

Copyright © 2013 E. A. Al Zahrani, M. A. Mourou. This is an open access article distributed under the Creative Commons Attribu-tion License, which permits unrestricted use, distribuAttribu-tion, and reproducAttribu-tion in any medium, provided the original work is properly cited.

ABSTRACT

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.

Keywords: Differential-Difference Operator; Generalized Wavelets; Generalized Continuous Wavelet Transform

1. Introduction

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

 

d 1

 

     

d 2

f x f x

f

f x q x f

x  x

   

  _  _ 

  x

where   1 2 and q is a real-valued odd function on R. For q = 0, we regain the differential-difference operator

C

 

d 1

 

,

d 2

f x f x

f f x

x  x

   

  _  _  

which is referred to as the Dunkl operator with parameter

1 2

 associated with the reflection group Z2 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 one- dimensional harmonic oscillators governed by Wigner’s commutation rules [4-6].

Put









1

π 1 2

a_ 



  

  (1)

and

 

exp



₀x

 

d ,



.

Q x  



q t t xR (2)

The authors [7] have proved that the integral transform

 

1

 



₂



1 2





1 1 1 d

Xf x a Q x f tx t  t t

 



_

  (3)

is the only automorphism of the space E

 

R of C

functions on R, satisfying

 

d _{and 0} _{0 ,}

d

X f X f Xf f

x   

for all 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 dif-

ferential-difference operator Λ. More explicitly, we call generalized wavelet each function g in L2



R,x21dx



satisfying almost all R:

  

2 0

d

0 g ,

a

C F g a

a



 

 

_



where F_ denotes the generalized Fourier transform

related to Λ given by

  

 

2 1

d , ,

F_ g  



g x __ x x  x 

R R





 being the solution of the differential-difference equation

(2)

 

,

 

0 1

f x i f x f

    .

,



Starting from a single generalized wavelet g we con- struct by dilation and translation a family of generalized wavelets by putting

 

, , 0,

t b

a b a

g x  T g x a b R

where stand for the generalized dual translation operators tied to the differential-difference operator Λ, and ga is the dilated function of g given by the relation

t b

T

  

a

  

.

F_ g  F_ g a

Accordingly, the generalized continuous wavelet transform associated with Λ is defined for regular functions f on R by

  

 

2 1

,

g f a b f x ga b x x



 

_

R .

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 [9-11]). Classical wavelets have wide applications, ranging from signal analysis in geophysics and acoustics to quantum theory and pure mathematics (see [12-14] and the references therein).

2. Preliminaries

Notation. We denote by

 p

 

, 1 ,

L_ R   p the class of measurable functions f on R for which f _p_,_  , where

 



₂ ₁



1

, d , if

p p

p

f _ 



f x x  x p

R  ,

and f _,_ esssupxR f x

 

.

 p

 

, 1 , Q

L R   p the class of measurable functions f on R for which _, f _p_,_ ,

p Q

f  Q   where Q is given by (2).

 ₁p

 

, 1 , Q

L R   p the class of measurable functions f on R for which _p_,1_Q _, .

Remark 1. Clearly the map p

Q 

 

   

Mf x Q x f x (4) is an isometry

 from p

 

Q

L R onto L_p

 

R ;

 from p

 

L_ R onto L₁p_Q

 

R .

2.1. Generalized Fourier Transform

The following statement is proved in [7].

Lemma 1. 1) For each C, the differential-dif-

ference equation

 

, 0 1

u i u u ,

  

admits a unique C solution on R, denoted _, given

by

 

x Q x e i x

   

,

  

  (5) where e_ denotes the one-dimensional Dunkl kernel

defined by

 

_{       }

1 ,

2 1

z

e_ z j_ iz j_ iz z

 

  

 C

j_ being the normalized spherical Bessel function of

index  given by

 





   

_

2

_{ }



0

1 2

1 .

! 1

n n

n

z

j z z

n n

  _

 



   

  



C

2) For all xR, C and n0,1, , we have

 

_eIm _. n

n x

n x Q x x

 



 _ _

 (6)

3) For each xR_and C, we have the Laplace

type integral representation

 

1



₂



1 2





11 1 e

i xt_{d ,}

x a Q x t  t  t

 

 

 

_

  (7)

where a_ is given by (1).

The generalized Fourier transform of a function f in

 

1 Q

L R is defined by

  

 

2 1

d .

F_ f  



f x __ x x 

R x (8)

Remark 2. 1) By (6) and (7), it follows that the gener-

alized Fourier transform F_ maps continuously and

injectively 1

 

Q

L R into the space of continu-

ous functions on R vanishing at infinity.

 

0

C R

2) Recall that the one-dimensional Dunkl transform is defined for a function 1

 

by

f L_ R

  

  



2 1

d .

F_ f  



f x e_ i x x 

R x (9)

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

FF_M (10) where M is given by (4).

f  f _ 

(3)

transform F_ are as follows.

Theorem 1 (inversion formula). Let fL1Q

 

R

x

such that

 

1

 

. Then for almost all

F_ f L_ R R

we have

   





2

    

2 1

d ,

f x Q x m_



F_ f  _ x   

R where









2

2 2

1 _.

2 1

m_

 _



  (11)

Theorem 2 (Plancherel). 1) For every f L2Q

 

R , we have the Plancherel formula

 



 



  

2 2 2 1

2 2 1 d

d .

f x Q x x x

m F f

 

   

  





R

2) The generalized Fourier transform F_ extends

uniquely to an isometric isomorphism from 2

 

Q L R

onto 2

 

L R .

2.2. Generalized Convolution

Recall that the Dunkl translation operators ,x , x R



 

are defined by

 

d ,

 

x

x y ,

f y f z  z



 

_

R  (12) where x y,



 is a finite signed measure on R, of total

mass 1, with support

, , ,

x y x y x y x y

       

   

and such that x y_, 2



  . For the explicit expression of the measure x y, ,



 see [15].

Define the generalized translation operators Tx, xR,

associated with Λ by

 

   

 

_{ }

d ,

 

. (13)

x

x y

f z

T f y Q x Q y z

Q z

 



_

R

By (12) and (13) observe that

 

    



.

x x

T f y Q x Q y _ f Q y



(14)

The generalized dual translation operators are given by

 

_{ }

 

  

.

t x Q x x

T f y Qf y

Q y 



 (15)

We claim the following statement.

Proposition 1. 1) Let f be in 1

 

, p

Q

L R 1  p .

Then for all xR, T fx is a well defined element in

 

1 ,

p Q

L R and

 

_,1 ,1 2

x

p Q p Q

T f  Q x f

2) Let f be in p

 

, . Then for all Q

L R 1  p xR,

is well defined as a function in and t x

T f LQp

 

R ,

 

_,

, 2 t x

p Q p Q

T f  Q x f

3) For p

 

, Q

fL R p = 1 or 2, we have





 

   

.

F tT fx    x F f 



4) Let 1 p1, p2  such that 1 p11 p2 1. If

 

1

p

h₁L₁Q R and

 

R

2 2

p Q

h L , then we have the duality relation

    

 

  

2 1 1 2

2 1

1 2

d

d .

x

t x

T h y h y y y

h y T h y y y

 







R

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

 

  



 

,1 , ,

, ,

2 2

x x x

1 .

p Q p p

p p

T f T f Q Q x f Q

Q x f Q Q x f



Q

 

 

 

 

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

 





 

, ,

Q

2 2

t x t x x

,

.

p Q p

p p

T f Q T f Q x f

Q x Qf Q x f

 _p

Q

 





 

 

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





 





 





 

  

 

 

    

Q

i Q

.

t x t x

x

F T f F T f

Q x F f

Q x e x F f

x F f



 



 

 

 

 



 



 

 

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

    

 



     

  

  

 

 

  

2 1

1 2

2 1

1 2

2 1

1 2

2 1

1 2

d

Q d

d . x

x

t 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



 

 

 

  

 





R

This concludes the proof. ■

The generalized convolution product of two functions f and g on R is defined by

 

    

2 1

# t y d _.

f g x 



T f x g y y  y

R (16)

Remark 3. Recall that the Dunkl convolution product

of two functions f and g on R is defined by

 

    

2 1

d

x

f  g x  f y g y y  y



 

_



R (17)

By virtue of (15), (16) and (17) it is easily seen that

      

_{ }

# Qf Qg x .

f g x

Q x

 

 (18)

(4)

next statement.

Proposition 2. 1) Let p p p1, 2, 3 

 

1, such that

1 2

1 1 1

1

3

p  p   p If

 

1 p Q

f L R and p2

 

, Q

gL R then

 

3

# p

Q

f gL R and

3, 1,

# 2

2, p Q p Q

f g  f g _{p Q}. 2) For 1

 

Q

f L R and gLpQ

 

R , p = 1 or 2, we have



#



   

.

F_ f g F_ f F_ g

2.3. Intertwining Operators

According to [7], the dual of the intertwining operator X given by (3), takes the form

 

   

 









1 2 2 2 sgn

d

t

x y

Xf y a f x Q x x x y

x y x

 

 

 

 



It was shown that t _{is an automorphism of the}

space of C compactly supported functions on R, satisfying the intertwining relation

X

 

R

D 

 

d

, ,

d

t t

X f X f f D

x    R



where  is the dual operator of Λ defined by

 

d 1

 

     

.

d 2

f x f x f

f x q

x  x

 

 

  _  _ 

 

 _{x f x}

Moreover, we have the factorizations

, ,

t t

X M V

X V M

 

 



 (19)

where V_ and tV are respectively the Dunkl inter-

twining operator and its dual given by

 

1

 



₂



1 2





1 1 1

V f x_ a_



_ f tx t  t d , t

 



₂ ₂



1 2





sgn d .

t

x y

V f y_ a_



_ f x x x y  xy x

Using (19) and the properties of V_ and tV_ provided by [17], we easily derive the next statement.

Proposition 3. 1) If f L

 

R then Xf L1Q

 

  R

and _,1 .

Q

Xf _  f _

2) If 1

 

Q

f L R then tXf L1

 

R and

1, 1 f .

t

Q

Xf

3) For every fL

 

R and gL1Q

 

R , we have the duality relation

   

2 1

 

d t _{d .}

Xf x g x x  x f y Xg y y



R



R

4) For every 1

 

we have the identity Q

f L R

 

t



,

u



F_ f F  X f (20)

where Fu denotes the usual Fourier transform on R given

by

  

 

_e i x_{d ,} 1



u

F h  



h x  x hL

R R



.

5) Let _, 1

 

Q

f gL R . Then



#



,

t t t

X f g  Xf  Xg

where * denotes the usual convolution product on R

given by

 



  

1 2 1 2 d .

h h x 



h xy h y y

R 6) Let 1

 

Q

fL R and gL

 

R . Then





2

2 #

t Xg _.

X Xf g Q f Q  

  _ _

  (21)

3. Generalized Wavelets

Notation. For a function f on R put

 

~ _, _.

f x  f x xR

3.1. Dunkl Wavelets

Definition 1. A Dunkl wavelet is a function gL2_

 

R

satisfying the admissibility condition

  

2 0

d

0 C_g F g a a ,

a



 



 

_

 (22)

for almost all R.

Notation. For a function g in L2_

 

R and for

  

a b,  0, 



R we write

 

, ,

b

a b a

g x  g x (23) where b



 _{are the Dunkl translation operators given by}

(12), and

 

2 2

1

, .

a

x

g x g x

a a





  _  

  R (24) 

Definition 2. Let be a Dunkl wavelet.

The Dunkl continuous wavelet transform is defined for smooth functions f on R by

 

2 gL R

  

 

2 1

,

, d

g a b

S f a b 



f x g x x  x

R ,

a 

(25) which can also be written in the form

  

 

~

 

, ,

g

S f a b  f _ g b (26)

where _ is the Dunkl convolution product given by (17).

(5)

Theorem 3. Let be a Dunkl wavelet.

Then

 

2 gL R

1) For all 2

 

we have the Plancherel formula f L R

 

  

2 2 1

2 2 1 0

d

1 _, _d d

g g

f x x x

a

S f a b b b

a C



 





 





 

R

R .

2) For 1

 

such that

fL_ R F_

 

f L1_

 

R , we

have

 



  

 

2 1



, 0

1 _, _d d

g a b

g

a

f x S f a b g x b b

a C



 



 



_{ }

R

for almost all xR.

3) Assume that F_

 

g L

 

R . For f L2_

 

R and 0     , the function

 

  

 

2 1

,

, 1

, d

g a b

g

a

f x S f a b g x b b

a C

 

   

 

 

_{ }

R

d

belongs to 2

 

L_ R and satisfies

, 2, 0,

lim f  f 0.



   

3.2. Generalized Wavelets

Definition 3. We say that a function gL2Q

 

R is a generalized wavelet if it satisfies the admissibility condition

  

2 0

d

0 C_g F g a a

a



 

 

_

 , (27) for almost all R.

Remark 4. 1) The admissibility condition (27) can

also be written as

  

2 0

2

0

d 0

d .

g

C F g

F g

 

  



 

 

 



 

2) If g is real-valued we have F_

  

g   F_

  

g  ,

so (27) reduces to

  

2 0

d

0 Cg F g .

 



 

 

_

 

3) If ₀ 2

 

Q g L

  R is real-valued and satisfies 0



  such that F_

  

g  F_

  

g 0 O

 



  

0 0. F g

, as then (27) is equivalent to

0

_ _, _

4) According to (10), (22) and (27), 2

 

gL_Q R is a

generalized wavelet if and only if, 2

 

QgL_ R is a

Dunkl wavelet, and we have . g Qg

C C (28)

Notation. For a function g on R and a0, put

 

   

2 2

_{ }

,

a

Q x a g x a

g x x

a  Q x .

 R (29)

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

     

_{ }

a .

a

Qg x

g x

Q x



 (30)

Proposition 4. 1) Let a0 and gLQp

 

R for some 1  p . Then gaLQp

 

R and

 

2 1 , ,

q

a p Q p Q

g a  g

where q is such that 1 p1q1.

2) For a0 and gLpQ

 

  

a

 



.

F_ g  F_ g a

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

 

   

, , _,

2 1 2 1

, , .

a p Q a p a

p

q q

p p Q

g Qg Qg

a Qg a g



 _

 



   

 

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

  



 



 



 

  

,

a a a

F g F Qg F Qg

F Qg a F g a



 



  

 



 

which achieves the proof. ■

Definition 4. Let be a generalized wave-

let. We define for regular functions f on R, the general-

ized continuous wavelet transform by

 

2 Q gL R

  

 

   





2 2 1 ,

, d

g f a b f x ga b x Q x x x,

 

 

_

R (31)

where a0, bR,

 



, ,

t b

a b a



g x  T g x (32)

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

t b

T

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

yields

 

_{     }

, , .

a b a b

Q b

g x Qg

Q x



 x (33)

Proposition 5. Let gL2Q

 

R



be a generalized wavelet. Then for all p



,

Q

f L R p = 1 or 2, we have

  

 

  

 





2

   

~

, ,

# ,

g Qg

a

f a b Q b S Qf a b

Q b f g b



 

 (34)

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

(6)

  

 

       

 

  

   





 

  

   

 





   

2 1 ,

~ ~

~

2 ~

,

d ,

)

# ,

g

a b

Qg

a

a f a b

Q b Q x f x Qg x x x

Q b S Qf a b

Q b Qf Qg b

Q b Qf Qg b

Q b Qf Q g b

Q b f g b

 



 

 

 

 

 _{ } _

 

 

 _{ } _





R

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 gL2Q

 

R be a generalized wavelet. Then for all 2

 

Q

f L R we

have

 



 



  

 





2 2 2 1

2 1 2

2 0

d

1 d

, d

g g

f x Q x x x

b a

f a b b

C _{Q b} a





 

 



 

R

R .

Theorem 5 (inversion formula). Let gL2Q

 

R be a generalized wavelet. If 1

 

Q

f L R and F

 

f L1_

 

R then we have

 

  

 





2 1

, 2

0

1 _, d

g a b

g

b a

f x f a b g x b

C _{Q b}

 

_ _

 



 

 

R d  _a

for almost all xR.

Theorem 6 (Calderon’s formula). Let gL2Q

 

R

 

be a generalized wavelet such that L

 

R . ,

F_ g

Then for 2

 

and 0

Q

f L R      the function

 

  

 





2 1 ,

, 2

1 d

, d

g a b

g

b a

f x f a b g x

C _{Q b} a

 

 



 

_{ }



R b

belongs to 2

 

and satisfies Q

L R

, 2, 0,

lim 0.

Q

f  f

   

3.5. Inversion of the Intertwining Operator tX

Using Generalized Wavelets

In order to invert t

X we need the following two technical lemmas.

Lemma 2. Let 0 g L1L2

 

R such that

 

1

 

u

F g L R and satisfying

  

 

such that F gu O ,



   

   (35)

as 0. Let 2_.

G Xg Q Then GL2Q

 

R and

  

2 1,

2π

u

F g

F G

m_ 

 



  

where m_ is given by (11). Proof. We have

 

1

  

e d , a.e. 2π

i x u

g x 



F g   

R

As by (3) and (7),

 

i

 

, x X e x

 

 

we deduce that

 

   

2 1

d , a.e.

Xg x m_



h  _ x   

R (36)

with

 

  

2 1. 2π

u F g h

m   

  

Clearly, 1

 

_.

hL_ R So it suffices, in view of (36)

and Theorem 2, to prove that h belongs to hL2_

 

R .

We have

 





  









  









2 2 1

2 2 1 2

1 1

2 1 2

d

2π d

2π .

u

h

m F g

m F

m I I

  



 _ _



  

g

  



  

 





 



R

By (35) there is a positive constant k such that 2 2 1

1 ₁ d .

k I k _    

   



  







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

  

 

2 2

2 1

2 ₁

2

d d

2π d ,

u u

I F g F g

g x x



    

  

 

  



R



which ends the proof. ■ Lemma 3. Let 0 g L1L2

 

R be real-valued

such that

 

1

 

L R

u

F g  and satisfying   21 such that

  

 

,

u

F g  O  (37)

as 0 . Let 2_.

GXg Q Then GL2Q

 

R is a generalized wavelet and F

 

G L

 

R .

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

 

2 Q

GL R , F

 

G is bounded and

  



2 1



_as _{0 .}

F_ G  O     

(7)

satisfies the admissibility condition (27). ■

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

  

,

 

1 d

g

x b

W f a b f x g x

a a



 

 _ _

 



R , (38)

where a0, bR, and gL2

 

R is a classical

wavelet on R, i.e., satisfying the admissibility condition

 

  

2

0

d

0 c g F_u g a a ,

a





 

_

 (39)

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 satis-

fying the conditions of Lemma 3 is a classical wavelet. 2) In view of (20), (27) and (39), gD

 

R is a gen-

eralized wavelet, if and only if, t

Xg is a classical wave-

let and we have

 

t .

g

c Xg C

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 2_.

G Xg Q Then for all fLQp

 

  

2 1

 

   

1

, t ,

G f a b X Wg Xf a b .

a 

 

  _  _

Proof. By (34) we have

  



 



2

   

~

, #

G f a b Q b f Ga b .

 

But

 

~ ~

2 a

a

X g G

Q



 

 

 



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

  



 



 

~ 2

2 ~

2 1

, #

( )

1 _, _{( ),}

a

G

t

a

t g

X g

( )

f a b Q b f b

Q

X Xf g b

X W Xf a b

a



 

     

  _ _

 

 

 

 _ _

 

 

 _  _



which gives the desired result.

Combining Theorems 5, 6 with Lemma 3 and Proposi- tion 6 we get

Theorem 7. Let g be as in Lemma 3. Let GXg Q2.

Then we have the following inversion formulas for the integral transform t :

X 1) If 1

 

Q

fL R and F

 

f L1

 

R then for al-

most all xR we have

 

   

 





, 0

2 2 2

1 _,

d

d .

t

g a b

G

2 1

f x X



   

 

R W Xf a b G x

C

b a

Q b 





 

 _ _

  

 

2) For 1 2

 

Q Q

f L L R and 0     , the func-

tion

 

   

 





,

, 2 1

2 2 2

1 _,

d d

t

g a

G

b

f x X W Xf a b G

C

b a

b a Q b

  

 

 



 

 _ _



 

R x

satisfies

, 2, 0,

lim 0.

Q f  f

   

4. Acknowledgements

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

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