Advances in Pure Mathematics, 2013, 3, 443-450 http://dx.doi.org/10.4236/apm.2013.35063 Published Online August 2013 (http://www.scirp.org/journal/apm)
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 Attribution License, which permits unrestricted use, distribution, and reproduction 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
df 1 f x f x , dx 2 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 onedimensional harmonic oscillators governed by Wigner’s commutation rules [4-6]. Put
and *
Corresponding author.
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1
π 1 2
(1)
(2)
The authors [7] have proved that the integral transform 1
where 1 2 and q is a C real-valued odd function on R. For q = 0, we regain the differential-difference operator
a
x
Xf x a Q x 1 f tx 1 t 2
df 1 f x f x f x q x f x dx 2 x
f x
Q x exp 0 q t dt , x R.
1 2
1 t dt
(3)
is the only automorphism of the space E R of C functions on R, satisfying X
d f X f and Xf 0 f 0 , dx
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 differential-difference operator Λ. More explicitly, we call 2 1 generalized wavelet each function g in L2 R , x dx satisfying almost all R : 2 da 0 Cg 0 F g a , a where F denotes the generalized Fourier transform related to Λ given by
F g R g x x x
2 1
dx , R ,
being the solution of the differential-difference equation APM
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E. A. AL ZAHRANI, M. A. MOUROU
Mf x Q x f x
f x i f x , f 0 1.
Starting from a single generalized wavelet g we construct by dilation and translation a family of generalized wavelets by putting g a ,b x tT b g a x , a 0, b R, b
F g a F g a .
2 1
.
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 t X 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 [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).
p ,
R
f x
p
x
2 1
dx
1p
, if p ,
and f , esssup xR f x . LQp R , 1 p , the class of measurable functions f on R for which f p ,Q Q f p , , where Q is given by (2). L1p Q R , 1 p , the class of measurable functions f on R for which f p ,1 Q f Q p , . Remark 1. Clearly the map
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x Q x e i x ,
(5)
where e denotes the one-dimensional Dunkl kernel defined by z j 1 iz z C , 2 1
e z j iz
j being the normalized spherical Bessel function of index given by
j z 1 n 0
1 z 2 z C . n ! n 1 2n
n
2) For all x R , C and n 0,1, , we have n n Im x x Q x x e . n
(6)
3) For each x R and C , we have the Laplace type integral representation
x a Q x 1 1 t 2 1
1 2
1 t ei xt dt ,
(7)
where a is given by (1). The generalized Fourier transform of a function f in L1Q R is defined by 2 1
R
Notation. We denote by Lp R , 1 p , the class of measurable functions f on R for which f p , , where
admits a unique C solution on R, denoted , given by
F f f x x x
2. Preliminaries
f
The following statement is proved in [7]. Lemma 1. 1) For each C , the differential-difference equation u i u, u 0 1,
Accordingly, the generalized continuous wavelet transform associated with Λ is defined for regular functions f on R by g f a, b R f x g a ,b x x
is an isometry from LQp R onto Lp R ; from Lp R onto L1p Q R .
2.1. Generalized Fourier Transform
where T 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
(4)
dx.
(8)
Remark 2. 1) By (6) and (7), it follows that the generalized Fourier transform F maps continuously and injectively L1Q R into the space C0 R of continuous functions on R vanishing at infinity. 2) Recall that the one-dimensional Dunkl transform is defined for a function f L1 R by F f R f x e i x x
2 1
dx.
(9)
Notice by (5), (8) and (9) that
F F M ,
(10)
where M is given by (4). Two standard results about the generalized Fourier APM
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E. A. AL ZAHRANI, M. A. MOUROU
transform F are as follows. Theorem 1 (inversion formula). Let f L1Q R such that F f L1 R . Then for almost all x R we have f x Q x m R F f x 2
2 1
d ,
1 2
2 2
1
2
Theorem 2 (Plancherel). 1) For every f L2Q R , we have the Plancherel formula
R
f x
2
Q x x 2
2 1
m R F f 2
dx
2 1
R T h1 y h2 y y dy 2 1 dy. R h1 y tT x h2 y y Proof. 1) By (14) and [13, Equation (8)] we have p ,1 Q
f y R f z d x , y z ,
(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 f z d x , y z . Q z
(13)
By (12) and (13) observe that T f y Q x Q y f Q y . x
(14)
The generalized dual translation operators are given by Q x x t x T f y Qf y . Q y
(15)
We claim the following statement. Proposition 1. 1) Let f be in L1p Q R , 1 p . Then for all x R , T x f is a well defined element in L1p Q R , and p ,1 Q
2Q x f
p ,1 Q
2) Let f be in LQp R , 1 p . Then for all x R , T x f is well defined as a function in LQp R , and
t
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Q
p ,
Q x x f Q p ,
2Q x f
p ,1 Q
p ,
.
2) By (15) and [13, Equation (8)] we have t
Tx f
p ,Q
Q tT x f
p ,
2Q x Qf
Recall that the Dunkl translation operators , x R, are defined by
Tx f
Tx f
2Q x f Q
x
x
2 1
x
d.
p ,Q
R , p = 1 or 2, we have F tT x f x F f .
2.2. Generalized Convolution
T x f y Q x Q y R
2Q x f
3) For f L
Tx f
2) The generalized Fourier transform F extends uniquely to an isometric isomorphism from L2Q R onto L2 R .
x
p ,Q
p Q
(11)
.
Tx f
4) Let 1 p1 , p2 such that 1 p1 1 p2 1. If p h1 L1p1Q R and h2 LQ2 R , then we have the duality relation
where m
t
Q x x Q f p ,
2Q x f
p ,
. p ,Q
3) By (5), (10), (15) and [1, Theorem 11] we have
F tT x f
F Q T f Q x F Q f
t
x
x
Q x e i x F Q f x F f .
4) By (14), (15) and [1, Theorem 11] we have 2 1 x R T h1 y h2 y y dy 2 1 Q x R x h1 Q y Q y h2 y y dy 2 1 Q x R h1 Q y x Qh2 y y dy 2 1 R h1 y tT x h2 y y dy.
This concludes the proof. ■ The generalized convolution product of two functions f and g on R is defined by f # g x R tT y f x g y y
2 1
dy.
(16)
Remark 3. Recall that the Dunkl convolution product of two functions f and g on R is defined by f g x R x f y g y y
2 1
dy
(17)
By virtue of (15), (16) and (17) it is easily seen that f # g x
Qf Qg x . Q x
(18)
By use of (10), (18) and the properties of the Dunkl convolution product mentioned in [16], we obtain the APM
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E. A. AL ZAHRANI, M. A. MOUROU
next statement. Proposition 2. 1) Let p1 , p2 , p3 1, such that 1 1 1 If f LQp1 R and g LQp2 R , then 1 p1 p2 p3 f # g LQp3 R and f #g
p3 ,Q
2 f
g
p1 ,Q
p2 ,Q
F f # g F f F g .
where Fu denotes the usual Fourier transform on R given by Fu h R h x ei x dx, h L1 R . t
According to [7], the dual of the intertwining operator X given by (3), takes the form
Xf y a x y f x Q x sgn x x y 2
where * denotes the usual convolution product on R given by 6) Let f L1Q R and g L R . Then X
2 1 2
x y dx
d t f , f DR, X f tX dx f x df 1 f x f x q x f x . dx 2 x
Moreover, we have the factorizations X M V ,
(19)
X tV M ,
where V and V are respectively the Dunkl intertwining operator and its dual given by
V f x a 1 f tx 1 t 2
1 2
V f y a x y f x sgn x x y 2
2
1 t dt ,
1 2
x y dx.
Using (19) and the properties of V and V provided by [17], we easily derive the next statement. Proposition 3. 1) If f L R then Xf L1Q R and Xf ,1 Q f . t
2) If f L1Q R then t Xf L1 R and t Xf f 1,Q . 1
3) For every f L R and g L1Q R , we have the duality relation 2 1
dx R f y t Xg y dy.
4) For every f L1Q R we have the identity Copyright © 2013 SciRes.
(21)
Notation. For a function f on R put f ~ x f x , x R.
Definition 1. A Dunkl wavelet is a function g L2 R satisfying the admissibility condition
0 C g F g a 0
2
da , a
(22)
for almost all R. Notation. For a function g in L2 R and for a, b 0, R we write
g a ,b x b ga x ,
t
R Xf x g x x
t
3.1. Dunkl Wavelets
where is the dual operator of Λ defined by
t
Xg g Q2 f # 2 . Q
Xf
3. Generalized Wavelets
It was shown that t X is an automorphism of the space D R of C compactly supported functions on R, satisfying the intertwining relation
1
X f # g t Xf t Xg ,
h1 h2 x R h1 x y h2 y dy.
2.3. Intertwining Operators
t
(20)
5) Let f , g L1Q R . Then
.
2) For f L1Q R and g LQp R , p = 1 or 2, we have
t
F f Fu t X f ,
where (12), and
b
(23)
are the Dunkl translation operators given by g a x
1 a
2 2
x g , x R. a
(24)
Definition 2. Let g L2 R be a Dunkl wavelet. The Dunkl continuous wavelet transform is defined for smooth functions f on R by S g f a, b f x g a ,b x x
2 1
R
dx,
(25)
which can also be written in the form
b ,
S g f a, b f g a
~
(26)
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. APM
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E. A. AL ZAHRANI, M. A. MOUROU
Theorem 3. Let g L2 R be a Dunkl wavelet. Then 1) For all f L2 R we have the Plancherel formula
R f x
1 Cg
2
x
2 1
dx
0 R S g f a, b
2
2 1
b
db
da . a 1
0
R S g f a, b g a,b x b
2 1
db
da a
for almost all x R. 3) Assume that F g L R . For f L2 R and 0 , the function f , x
1 Cg
R S g f a, b g a,b x b
2 1
db
da a
belongs to L2 R and satisfies lim
0,
f
,
f
2,
0.
Definition 3. We say that a function g L R is a generalized wavelet if it satisfies the admissibility condition
da , a
2
0
(27)
for almost all R. Remark 4. 1) The admissibility condition (27) can also be written as
0 Cg 0 F g 0 F g
Proposition 4. 1) Let a 0 and g LQp R for some 1 p . Then g a LQp R and ga
p ,Q
a
2 1 q
g
2
d
d
2
F g a F g a .
Proof. 1) By (30) and [13, Equation (13)], we have ga
p ,Q
Qg a
Qg a
p ,
2 1 q
Qg
p ,
p ,
a
2 1 q
0 Cg 0 F g
2
.
d
.
3) If 0 g L R is real-valued and satisfies 0 such that F g F g 0 O , as 0 , then (27) is equivalent to F g 0 0. 4) According to (10), (22) and (27), g L2Q R is a generalized wavelet if and only if, Qg L2 R is a Dunkl wavelet, and we have 2 Q
g
p ,Q
.
Qg
a
F Qg a F g a ,
which achieves the proof. ■ Definition 4. Let g L2Q R be a generalized wavelet. We define for regular functions f on R, the generalized continuous wavelet transform by g f a, b R f x g a ,b x Q x x 2
2 1
Cg CQg .
Notation. For a function g on R and a 0 , put
(28)
dx, (31)
where a 0, b R , g a ,b x tT b g a x ,
2) If g is real-valued we have F g F g , so (27) reduces to
p ,Q
where q is such that 1 p 1 q 1. 2) For a 0 and g LQp R , p = 1 or 2, we have
F g a F Qg a F
0 C g F g a
(30)
2) By (10), (30) and [13, Equation (11)], we have
2 Q
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Remark 5. Notice by (24) and (29) that
a
3.2. Generalized Wavelets
(29)
Qg a x ga x . Q x
2) For f L R such that F f L R , we have 1 Cg
Q x a g x a , x R. a 2 2 Q x
1
f x
ga x
(32)
and tT b are the dual generalized translation operators given by (15). Remark 6. A combination of (15), (23) and (32) yields g a ,b x
Q b
Q x
Qg a,b x .
(33)
Proposition 5. Let g L2Q R be a generalized wavelet. Then for all f LQp R , p = 1 or 2, we have g f a, b Q b SQg Qf a, b
Q b f # ga b , 2
~
(34)
where # is the generalized convolution product given by (16). Proof. By (18), (25), (26), (30), (31) and (33), we have APM
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E. A. AL ZAHRANI, M. A. MOUROU
g f a, b Q b R Q x f x Qg a ,b x x
2 1
F G
dx
Q b ) Qf Qg a
~
~
b
b
As by (3) and (7),
Q b Qf Q g a b
x,
x X e i
Q b f # g a b , ~
we deduce that
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 f L2Q R we have
R f x Q x 2
1 Cg
0 R
2
x
2 1
g f a, b
dx 2
b
2 1
Q b
2
db
da . a
Theorem 5 (inversion formula). Let g L R be a generalized wavelet. If f L1Q R and F f L1 R then we have
0
2 1 da b g f a, b g a ,b x b d 2 R Q b a
for almost all x R. Theorem 6 (Calderon’s formula). Let g L2Q R be a generalized wavelet such that F g L R . Then for f L2Q R and 0 , the function f
,
1 x Cg
2 1
da R g f a, b g a,b x Q b 2 db a b
belongs to L2Q R and satisfies lim
0,
f
,
f
2,Q
3.5. Inversion of the Intertwining Operator tX Using Generalized Wavelets
(36)
with h
Fu g 2 π m
2 1
.
Clearly, h L1 R . So it suffices, in view of (36) and Theorem 2, to prove that h belongs to h L2 R . We have 2
2πm
2
2πm
2
2πm
2
2 1
d
R
1
2 1
Fu g d 2
1
2 1
Fu g d 2
I1 I 2 .
By (35) there is a positive constant k such that
I1 k
1
2 2 1
d
k .
From the Plancherel theorem for the usual Fourier transform, it follows that I 2 1
Fu g d R Fu g d
2 1
2
2
2π R g x dx , 2
(35)
as 0. Let G Xg Q 2 . Then G L2Q R and Copyright © 2013 SciRes.
d , a.e.
Fu g O ,
In order to invert tX we need the following two technical lemmas. Lemma 2. Let 0 g L1 L2 R such that Fu g L1 R and satisfying
,
2 1
which ends the proof. ■ Lemma 3. Let 0 g L1 L2 R be real-valued such that Fu g L1 R and satisfying 2 1 such that
0.
such that Fu g O
Xg x m R h x
R h
2 Q
1 f x Cg
,
1 Fu g ei x d , a.e. 2π R
g x
~
2
2 1
2 πm
where m is given by (11). Proof. We have
Q b SQg Qf a, b Q b Qf Qg a
Fu g
(37)
as 0. Let G Xg Q 2 . Then G L2Q R is a generalized wavelet and F G L R . Proof. By using (37) and Lemma 2 we see that G L2Q R , F G is bounded and
F G O 2 1 as 0.
Thus, in view of Remark 4 3), the function Xg Q2 APM
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E. A. AL ZAHRANI, M. A. MOUROU
satisfies the admissibility condition (27). ■ Recall that the classical continuous wavelet transform is defined for suitable functions f on R by 1 xb Wg f a, b R f x g dx, a a
Then we have the following inversion formulas for the integral transform t X : 1) If f L1Q R and F f L1 R then for almost all x R we have
(38) f x
where a 0 , b R , and g L2 R is a classical wavelet on R, i.e., satisfying the admissibility condition
0 c g Fu g a
2
0
da , 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 satisfying the conditions of Lemma 3 is a classical wavelet. 2) In view of (20), (27) and (39), g D R is a generalized wavelet, if and only if, t Xg is a classical wavelet and we have c
t
1 X Wg a 2 1
t
a, b .
Xf
G f a, b Q b f # Ga b . 2
2) For f L1Q L2Q R and 0 , the function f , x
~
X g a 2 Q
~
by virtue of (3), (24) and (29). So using (21) and (38) we find that
X g ~ a G f a, b Q b f # 2 (b) Q ~ X t Xf g a (b) 1 2 1 X Wg t Xf a, (b), a 2
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 G Xg Q 2 . Copyright © 2013 SciRes.
1 CG
R X Wg
b
2 1
Q b
2
lim
0,
db
t
Xf
a, b G x a ,b
da a 2 2
f , f
2,Q
0.
4. Acknowledgements This work was funded by the Deanship of Scientific Research at the University of Dammam under the reference 2012018.
[1]
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[2]
C. F. Dunkl, “Integral Kernels with Reflection Group Invariance,” Canadian Journal of Mathematics, Vol. 43, No. 6, 1991, pp. 1213-1227. doi:10.4153/CJM-1991-069-8
[3]
C. F. Dunkl, “Hankel Transforms Associated to Finite Reflection Groups,” Contemporary Mathematics, Vol. 138, No. 1, 1992, pp. 128-138.
[4]
S. Kamefuchi and Y. Ohnuki, “Quantum Field Theory and Parastatistics,” University of Tokyo Press, SpringerVerlag, Berlin, 1982.
[5]
M. Rosenblum, “Generalized Hermite Polynomials and the Bose like Oscillator Calculus,” In: Operator Theory: Advances and Applications, Birkhauser Verlag, 1994, pp. 369-396.
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L. M. Yang, “A Note on the Quantum Rule of the Harmonic Oscillator,” Physical Review, Vol. 84, No. 4, 1951, pp. 788-790. doi:10.1103/PhysRev.84.788
[7]
E. A. Al Zahrani, H. El Mir and M. A. Mourou, “Intertwining Operators Associated with a Dunkl Type Operator on the Real Line and Applications,” Far East Journal
But
Ga
REFERENCES
Proof. By (34) we have
~
satisfies
Xg Cg .
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 G Xg Q 2 . Then for all f LQp R , p = 1 or 2, we have G f a, b
t 0 R X Wg Xf a, b Ga,b x 2 1 da b b d . 2 2 2 Q b a 1 CG
APM
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E. A. AL ZAHRANI, M. A. MOUROU of Applied Mathematics and Applications, Vol. 64, No. 2, 2012, pp. 129-144.
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