Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:

Journal of Computational and Applied Mathematics 99 (1998) 353–364

Generalized wavelet packet associated with Laguerre functions
Mohamed Si
Ecole Superieure des Sciences et Techniques de Tunis, 5 rue Taha Hussein, Tunis 1008, Tunis.
Received 2 November 1997; received in revised form 15 April 1998

Abstract
Using the harmonic analysis associated with Laguerre functions on K = [0; +∞[×R, we study two types of generalized
wavelet packets and the corresponding generalized wavelet transforms, and we prove for these transforms, the Plancherel,
c 1998 Elsevier Science B.V. All rights reserved.
Calderon and reconstruction formulas.
Keywords: Laguerre functions; Generalized wavelet; Generalized wavelet packets

0. Introduction
We consider the Laguerre functions dened on K = [0; +∞[×R; by
2

’; m (x; t) = eit e−||x =2

Lm() (||x2 )

;
Lm() (0)

where (; m) ∈ R × N;  ∈ R; ¿0 and Lm() is the Laguerre polynomial of degree m and order
. These functions satisfy a product formula which permits to build a harmonic analysis on K
(generalized convolution product, generalized Fourier transform,...) (see [5, 7]).
In [3, 5], we have studied a continuous wavelet analysis associated with Laguerre functions. This
theory has been used in [4] to invert the generalized Radon transform.
The objective of this paper is to present a general construction of generalized wavelet packets
starting from the precedent wavelet analysis.
According to [1, 2, 6] we mean by generalized wavelet packets families of functions generated
from a single one by simple transformations and whose relative bandwidth is not constant and can
be matched to a given analysed function.
It is believed that generalized wavelet expansions as discussed here will be a very useful tool in
many areas of mathematics (see [7]).
This paper is organized as follows. In Section 1 we study a harmonic analysis associated with
Laguerre functions. We dene and study in Sections 2 and 3, the generalized wavelet packets and
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 6 9 - 1

354

M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

the corresponding generalized wavelet packets transforms and we prove for these transforms the
Plancherel, Calderon and reconstruction formulas.
1. Harmonic analysis associated with Laguerre functions
In this section we recall some results on harmonic analysis associated with Laguerre functions
(for more details see [4, 5] or [7]).
Notations. We denote by −Lp (K); p ∈ [1 + ∞[, the space of measurable functions on K satisfying
kfk;p =

Z

1=p

|f(x; t)|p dm (x; t)

K

¡+∞;

dm being the measure on K dened by dm (x; t) = (1= ( + 1))x2+1 dx dt. −Lp (R × N); p ∈
[1; +∞], the space of functions g : R × N → C, measurable, satisfying
kgk =
Lp

Z

p

|g(; m)| d
 (; m)

R×N

1=p

¡+∞

if p ∈ [1; +∞[;

kgkL∞
= ess sup |f(; m)|

(;m)∈R×N

d
 the measure on R × N dened by d
 (; m) = Lm() (0)||+1 d ⊗ m , where m is the Dirac measure
at m.
The function ’; m ; (; m) ∈ R × N, considered in the introduction satises the product formula
()
’; m (x; t)’; m (y; s) = T(x;
t) ’; m (y; s);
()
where T(x;
t) is the generalized translation operators associated with Laguerre functions dened by


Z
1 2 q 2


f( x + y2 + 2xy cos ; s + t + xy sin ) d



 2Z 0Z
()
 1 2 q 2
T(x;
t) f(y; s) =

f( x + y2 + 2xyr cos ; s + t + xyr sin )





0
0



× r(1 − r 2 )−1 dr d;

if  = 0;

if ¿0:

Denition 1.1. Let f and g be continuous functions on K with compact support. The convolution
product f ∗ g of f and g is dened by
f ∗ g(x; t) =

Z

K

()

T(x;
t) f(y; s)g(y; −s) dm (y; s);

(x; t) ∈ K:

Denition 1.2. The generalized Fourier transform F associated with Laguerre functions is dened
on L1 (K) by
F(f)(; m) =

Z

K

’−; m (x; t)f(x; t) dm (x; t);

(; m) ∈ R × N:

M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

355

Theorem 1.1. (i) (Inversion formula) Let f be in L1 (K) such that F(f) belongs to L1 (R × N).
Then we have the following inversion formula for F:
f(x; t) =

Z

F(f)(; m)’; m (x; t) d
 (; m);

a:e:

R×N

(ii) For all f ∈ (L1 ∩ L2 )(K); we have the Plancherel formula
kfk2; 2 = kF(f)k2L2 :
(iii) The transform F can be extended to an isometric isomorphism from L2 (K) onto L2 (R × N).
Proposition 1.1. (i) Let f and g be two functions in L2 (K). The function f ∗ g belongs to L2 (K)
if and only if the function F(f)F(g) belongs to L2 (R × N) and we have
F(f ∗ g) = F(f)F(g):

(ii) For all f and g in L2 (K); we have
Z

Z

2

|f ∗ g(x; t)| dm (x; t) =

K

|F(f)(; m)|2 |F(g)(; m)|2 d
 (; m);

R×N

where both members are nite or innite.

2. Generalized wavelet packets
Denition 2.1. Let g be in L2 (K), we say that g is a generalized wavelet on K if there is a constant

Cg such that for all m ∈ N, we have
0¡Cg =

Z

|F(g)(; m)|2

R

d
¡+∞:
||

Example. Let r¿0; Er and g be the functions dened by
Er (x; t) =

Z

’; m (x; t) exp −r

R×N

g(x; t) = −

2



( + 1)
m+
2

2 !

d
 (; m);

d
Er (x; t):
dr

Then g is a generalized wavelet on K, and we have Cg = 1=4r 2 .
Let g be a generalized wavelet on K in L2 (K). For a ∈ R−{0}, we put
1
ga (x; t) = +2 g
|a|

!

x t
p ;
;
|a| a

a:e: on K:

(2.1)

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M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

Then we have
a:e: on R × N;

F(ga )(; m) = F(g)(a; m);
kga k; 2 = kF(ga )kL2 =

1
kgk; 2 :
|a|+1

(2.2)

Proposition 2.1. Let g be a generalized wavelet on K in L2 (K); a ∈ R−{0} and {j }j∈Z a decreasing sequence in ]0; +∞[ such that limj→−∞ j = + ∞; limj→+∞ j = 0; and Ij = {a ∈ R = j+1 6
|a|6j }. Then
(i) the function
1
(; m) →
Cg

Z

2 da

|F(ga )(; m)|

|a|

Ij

!1=2

belongs to L2 (R × N).
(ii) there exists a function gjp in L2 (K); such that for all (; m) ∈ R × N;
F(gjp )(;

1
m) =
Cg

Z

2 da

|F(ga )(; m)|

Ij

|a|

!1=2

:

Proof. (i) It follows from Fubini–Tonelli’s theorem and relation (2.2) that
1
Cg

Z

Z

1
=
Cg
=

|F(ga )(; m)|

Ij

R×N

!

2 da

|a|

1
da
= kgk2; 2
|F(ga )(; m)| d
 (; m)
|a| Cg
R×N

Z Z
Ij

d
 (; m)

2



Z

Ij

da
|a|2+2

#

"

2
1
1
2+1 − 2+1 kgk; 2 ¡+∞:
(2 + 1)Cg j+1
j

(ii) is an immediate consequence of Theorem 1.1(iii).
Denition 2.2. (i) The sequence {gjp }j∈Z is called generalized wavelet packet (also called P-wavelet
packet).
(ii) The function gjp ; j ∈ Z, is called generalized P-wavelet packet member of step j.
Property. For all j ∈ Z and (; m) ∈ R × N we have
06F(gjp )(; m) 6 1;

+∞
X

[F(gjp )(; m)]2 = 1:

(2.3)

j=−∞

Let {gjp }j∈Z be a generalized P-wavelet packet. We consider for all j ∈ Z and (x; t) ∈ K, the
function gj;p (x; t) given by
() p
gj;p (x; t) (y; s) = T(y;
s) gj (x; t):

(2.4)

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M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

Proposition 2.2. For all j ∈ Z; and (x; t) ∈ K; the function gj;p (x; t) belongs to L2 (K) and we have for
all (; m) ∈ R × N;
(i) kgj;p (x; t) k; 2 6kgjp k; 2 .
(ii) F(gj;p (x; t) )(; m) = ’; m (x; t)F(gjp )(; m).
Proof. The statements follow from (2.3),(2.4) and the properties of the generalized translation operators.
Denition 2.3. Let {gjp }j∈Z be a generalized P-wavelet packet. The generalized P-wavelet packet
transform gp is dened on L2 (K) by
gp (f)( j; (x; t)) =

Z

K

f(y; −s)gj;p (x; t) (y; s) dm (y; s):

This transform can also be written in the form
gp (f)( j; (x; t)) = f ∗ gjp (x; t);

(2.5)

where ∗ is the convolution product given in Denition 1.1.
Theorem 2.1. (i) Plancherel formula for gp : For all f ∈ L2 (K); we have
Z

|f(x; t)|2 dm (x; t) =

K

+∞ Z
X

j=−∞

K

|gp (f)(j; (y; s)|2 dm (y; s):

(ii) Parseval formula for gp : For all f1 and f2 in L2 (K) we have
Z

f1 (x; t)f2 (x; t) dm (x; t) =

K

+∞ Z
X

K

j=−∞

gp (f1 )(j; (y; s))gp (f2 )( j; (y; s)) dm (y; s):

Proof. (i) Using (2.5) and Proposition 1.1, we obtain for all j ∈ Z,
Z

K

|gp (f)(j; (x; t))|2 dm (x; t) =

Z

|F(f)(; m)|2 |F(gjp )(; m)|2 d
 (; m);

R×N

By Fubini–Tonelli’s theorem we deduce
+∞ Z
X

j=−∞

K

|gp (f)( j; (x; t))|2 dm (x; t) =

Z

|F(f)((; m)|2

R×N

+∞
X

[F(gjp )(;
j=−∞

m)]2

!

d
 (; m);

the assertion follows from (2.3) and Theorem 1.1(iii).
(ii) We obtain the result from (i).
Lemma 2.1. Let g be a generalized wavelet on K in L2 (K) such that F(g) ∈ L∞
 (R × N). For
p¡q; p; q ∈ Z; (x; t) ∈ K; (; m) ∈ R × N we put
Gp; q (x; t) =

1
Cg

Z

q 6|a|6p

ga ∗ ga (x; t)

da
;
|a|

(2.6)

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M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

1
Hp; q (; m) =
Cg

Z

da
:
|a|

|F(ga )(; m)|2

q 6|a|6p

(2.7)

Then Gp; q ∈ L2 (K); Hp; q ∈ (L1 ∩ L∞
 )(R × N); and we have
(i) kHp; q kL2 61.
(ii) limq→+∞;p→−∞ Hp; q (; m) = 1; for all m ∈ N and a.e  ∈ R:
(iii) F(Gp; q ) = Hp; q .
Proof. From Holders inequality for the measure da, and Fubini Theorem we obtain
Z

2

|Gp; q (x; t)| dm (x; t)6C

Z

q 6|a|6mp

K

Z



2

|ga ∗ ga (x; t)| dm (x; t) da;

K

where
C=

1
Cg2

Z

q 6|a|6p

da
:
a2

By using Fubini–Tonelli’s theorem, (2.2) and Theorem 1.1(ii) we deduce
Z

2

|Gp; q (x; t)| dm (x; t) 6 C

Z

Z

Z

Z

q 6|a|6p

K

6C

q 6|a|6p



2

|ga ∗ ga (x; t)| dm (x; t) da

K

6 CkF(g)k2L∞


4

|F(ga )(; m)| d
 (; m) da

R×N

Z

q 6|a|6p

= CkF(g)k2L∞
kgk2; 2 |




Z

Z

2

|F(ga )(; m)| d
 (; m) da

R×N

q 6|a|6p



da
¡+∞:
|a|2+2

Gp; q ∈ L2 (K).

So
The other assertions are immediate.
The following result is called Calderon’s formula for gp .
Theorem 2.2. Let g be a generalized wavelet on K in L2 (K) such that F(g) ∈ L∞
 (R × N) and
p
2
{gj }j∈Z be a generalized P-wavelet packet. Then for all f in L (K); p¡q; p; q ∈ Z and (x; t) ∈ K
the function
fp; q (x; t) =

q−1 Z
1 X
 P (f)( j; (x; t))gj;P(x;t) (y; s) dm (x; t)
Cg j=p K g

belongs to L2 (K) and satises
lim

q→+∞;p→−∞

fp; q = f

strongly in L2 (K).

M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

359

Proof. It is clear that fp; q = f? Gp; q . So by Lemma 2.1 and Proposition 1.1(i), fp; q ∈ L2 (K), and
F(fp; q ) = F(f)Hp; q :
From Theorem 1.1(ii), we obtain
kf

p; q

−

fk2; 2

Z

=

|F(f)(; m)|2 |1 − Hp; q (; m)|2 d
 (; m):

R×N

The result follows from Lemma 2.1 and dominated convergence theorem.
Theorem 2.3. For f in Lr (K); r = 1; 2 such that F(f) belongs to L1 (R×N) we have the following
reconstruction formula for gp :
f(x; t) =

+∞ Z
X

j=−∞

gp ( j; (y; s))gj;p (x; t) (y; s) dm (y; s);

K

a:e:;

where for each (x; t) in K; both the integral and the series are absolutely convergent; but possibly
not the series of integrals.
Proof. We put
I ( j; (x; t)) =

Z

K

gp (f)(j; (y; s))gj;p (x; t) (y; s) dm (y; s):

(i) We suppose that f ∈ L1 (K) such that F(f) ∈ L1 (R × N). From the relations (2.4) and (2.5), we
have
() p
gp (f)( j; (y; s))gj;p (x;t) (y; s) = f ∗ gjp (y; s)T(y;
s) gj (x; t):
() p
2
If the functions (y; s) → f ∗ gjp (y; s) and (y; s) → T(y;
s) gj (y; s) belong to L (K), then from Cauchy–
Schwarz’s inequality the integral I ( j; (x; t)) is absolutely convergent. Using Theorem 1.1 we obtain

I ( j; (x; t)) =

Z

F(f)(; m)[F(gjp )(; m)]2 ’; m (x; t) d
 (; m):

(2.8)

R×N

So from Fubini–Tonelli’s theorem and relation (2.3) we have
+∞
X

|I (j; (x; t))| 6

= kF(f)kL1 ¡+∞:

Thus, the series

j=−∞

|F(f)(; m)|

R×N

j=−∞

+∞
X

Z

P+∞

j=−∞

I (j; (x; t)) =

+∞
X

[F(gjp )(;
j=−∞

m)]

2

!

d
 (; m)

I ( j; (x; t)) is absolutely convergent. Therefore Eq. (2.8) leads to

+∞ Z
X

j=−∞

R×N

F(f)(; m)[F(gjp )(; m)]2 ’; m (x; t) d
 (; m)):

(2.9)

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M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

Applying Fubini theorem to the second member of (2.9), we obtain
+∞
X

I ( j; (x; t)) =

Z

+∞
X

[F(gjp )(;
j=−∞

F(f)(; m)

R×N

j=−∞

m)]

2

!

’; m (x; t)d
 (; m);

and the assertion follows from (2.3) and Theorem 1.1.
(ii) We suppose that f ∈ L2 (K) such that F(f) ∈ L1 (R × N).
From relation (2.3) we have for all j ∈ Z,
kF(f)F(gjP )kL2 6 kF(f)kL2 ¡+∞:
Then Proposition 1.1 implies that the function f ∗ gjp belongs to L2 (K). In the same way again (i),
we deduce (ii) from Theorem 1.1.

3. Generalized scale discrete scaling function on K
Proposition 3.1. Let {gjp }j∈Z be the P-wavelet packet given in Denition 2.2. Then
(i) For all (; m) ∈ R × N; we have
J −1
X

[F(gjp )(;
j=−∞

1
m)] =
Cg
2

Z

|F(ga )(; m)|2

|a|¿J

da
:
|a|

(3.1)

J −1
(ii) The function (; m) → ( j=−∞
[F(gjp )(; m)]2 ))1=2 belongs to L2 (R × N).
(iii) There exists a function Gjp in L2 (K) such that for all (; m) ∈ R × N;

P

F(Gjp )(;

m) =

J −1
X

[F(gjp )(;
j=−∞

m)]

2

!1=2

:

(3.2)

Proof. The proof is the same as for Proposition 2.1.
Denition 3.1. The sequence {GJp }J ∈Z is called generalized scale discrete scaling function.
Property. For all j; J ∈ Z and (; m) ∈ R × N, we have
(i) 06F(GJp )(; m)61;
lim F(GJp )(; m) = 1:
J →+∞

(ii) [F(GJP )(; m)]2 +

+∞
X

[F(gjp )(; m)]2 = 1:

j= J

p
)(; m)]2 − [F(Gjp )(; m)]2 :
(iii) [F(gjp )(; m)]2 = [F(Gj+1

(iv)

+∞
X

p
([F(Gj+1
)(; m])2 − [F(Gjp )(; m]2 ) = 1:

j=−∞

(3:3)
(3:4)

M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

361

We consider for all J ∈ Z and (x; t) ∈ K, the function GJ;p (x; t) dened on K by
()
p
GJ;p (x; t) (y; s) = T(y;
s) GJ (x; t):

(3.5)

The following proposition give the properties of the function GJ;p (x; t) .
Proposition 3.2. (i) For all j ∈ Z and (x; t) ∈ K; the function GJ;p (x; t) belongs to L2 (K); and we have
kGJ;p (x; t) k; 2 6kGJp k; 2 :
(ii) For all (; m) ∈ R × N; F(GJ;p (x; t) )(; m) = ’; m (x; t)F(GJp )(; m).
Notation. We denote by h: ; :i the scalar product on L2 (K) given by
hf; gi =

Z

f(x; t)g(x; t) dm (x; t):

(3.6)

K

Theorem 3.1. (i) Plancherel formula associated with {GJp }j∈Z : For all f in L2 (K); we have
kfk2; 2 = lim

J →+∞

Z

K

|hf; GJ;p (x; t) i |2 dm (x; t):

(ii) Parseval formula associated with {GJp }J ∈Z : For all f1 and f2 in L2 (K) we have
Z

f1 (x; t)f2 (x; t) dm (x; t) = lim

J →+∞

K

Z

K

hf1 ; GJ;p (y; s) i hf2 ; GJ;p (y; s) i dm (y; s):

Proof. (i) We have for all J ∈ Z and (y; s) ∈ K,
hf; GJ;p (y; s) i = f ∗ GJp (x; −t):
Then using Proposition 1.1, we obtain
Z

K

|hf; GJ;p (x; t) i |2

dm (y; s) =

Z

|F(f)(; m)|2 |F(GJp )(; m)|2 d
 (; m):

(3.7)

R×N

Then (i) follows from Theorem 1.1.
(ii) We deduce the result from (i).
Theorem 3.2. (i) Plancherel formula associated with {GJp }J ∈Z and gp : For all f in L2 (K) and J ∈ Z;
we have
kfk2; 2 =

Z

K

|hf; GJ;p (x; t) i |2 dm (x; t) +

+∞ Z
X
j= J

K

|gp (f)(j; (x; t))|2 dm (x; t):

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M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

(ii) Parseval formula associated with {GJp }J ∈Z and gp : For all f1 ; f2 in L2 (K); we have
Z

f1 (x; t)f2 (x; t) dm (x; t) =

K

Z

K

+

hf1 ; GJ;p (x; t) i hf2 ; GJ;p (x; t) i dm (x; t)
+∞ Z
X

K

j=J

gp (f1 )( j; (x; t))gp (f2 )(x; t) dm (x; t):

Proof. (i) From (3.2) and (3.7), (2.5), Proposition 1.1(ii) and Fubini–Tonelli’s theorem we have
Z

K

hf; GJ;p (x; t) i |2

dm (x; t) +

+∞ Z
X
j= J

=

Z

2

|F(f)(; m)|

R×N

K

|gp (f)( j (x; t))|2 dm (x; t)

+∞
X

[F(gjp )(;
j=−∞

m)]

2

!

d
 (; m):

Then (3.4) and Theorem 1.1 give (i).
(ii) follows from (i).
Lemma 3.1. Let g be a generalized wavelet on K in L2 (K); such that F(g) ∈ L∞
 (R × N). For
n ∈ Z; (x; t) ∈= K and (; m) ∈ R × N; we put
1
G∞; n (x; t) =
Cg

Z

ga ∗ ga (x; t)

|a|¿n

1
H∞; n (; m) =
Cg

Z

da
;
|a|

|F(ga )(; m)|2

|a|¿n

da
:
|a|

Then G∞; n ∈ L2 (K); H∞; n ∈ (L1 ∩ L∞
 )(R × N); and we have
(i) limn→+∞ H∞; n (; m) = 1; for all m ∈ N and a.e  ∈ R.
(ii) kH∞; n kL∞
61; F(G∞; n ) = H∞; n :

Proof. The proof is the same as for Lemma 2.1.
Theorem 3.3. Let g be a generalized wavelet on K in L2 (K); such that F(g) ∈ L∞
 (R × N). For
all f in L2 (K); J ¡n; J; n ∈ Z and (x; t) ∈ K;
(i) The function
f

∞; J

(x; t) =

Z

K

hf; GJ;p (x; t) i GJ;p (x; t) (:) dm (x; t)

belongs to L2 (K) and we have
lim f∞; J = f

J →∞

strongly in L2 (K).

M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

363

(ii) The function
J; n

f (x; t) = f

∞; J

(x; t) +

n−1 Z
X
j= J

K

gp (f)( j (y; s))gj;p (y; s) (x; t) dm (y; s)

belongs to L2 (K) and we have
lim fJ; n = f;

n→∞

strongly in L2 (K).
Proof. (i) It is clear that
f∞; J = f? G∞; J :
Since the functions f; G∞; J and F(f)F(G∞; J ) ∈ L2 (K) then f∞; J ∈ L2 (K) and we have
F(f∞; J ) = F(f)F(G∞; J ) = F(f)H∞; J :
We complete the proof by using Theorem 1.1(ii), Lemma 3.1 and dominated convergence theorem.
(ii) We remark that
fJ; n = f? (G∞; n + GJ; n );

F(fJ; n ) = F(f)H∞; n :

Then (ii) follows from Lemma 3.1.
Theorem 3.4. For f in Lp (K); p = 1; 2; such that F(f) belongs to L1 (R × N); we have the
following reconstruction formulas:
(i)
f(x; t) = lim

J →+∞

Z

K

p
hf; GJ;p (x; t) i GJ;(y;
s) (x; t) dm (y; s); a:e:

where for each (x; t) ∈ K; the integral is absolutely convergent.
(ii)
f(x; t) =

Z

K

+

hf; GJ;p (x; t) i GJ;p (y; s) (x; t) dm (y; s)
+∞ Z
X
j= J

K

gp (f)( j; (y; s))gj;p (y; s) (x; t) dm (y; s);

a:e:

where for each (x; t) ∈ K; in the second term of the second member both the integral and the series
are absolutely convergent; but possibly not the series of integrals.
Proof. The proof is the same as for Theorem 2.3.
Comments. As suggested by the referee, I think that it is possible to develop anologues for the
QMFs as in [1], of the wavelets packets associated with Laguerre functions. I plan to do this in a
forthcoming paper.

364

M. Si / Journal of Computational and Applied Mathematics 99 (1998) 353–364

Acknowledgements
The author thanks Professor K. Trimeche for his help and comments and the referees for their
suggestions and remarks.
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