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

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 onK= [0;+∞[×R_{, we study two types of generalized} wavelet packets and the corresponding generalized wavelet transforms, and we prove for these transforms, the Plancherel, Calderon and reconstruction formulas. c 1998 Elsevier Science B.V. All rights reserved.

Keywords:Laguerre functions; Generalized wavelet; Generalized wavelet packets

0. Introduction

We consider the Laguerre functions dened on K= [0;+∞[×R_{; by}

’; m(x; t) = eite−||x

2₌₂L(_m)(||x2)

L(m)(0)

;

where (; m)∈R_×N_{; ∈}R_; ¿0 and L()

m 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 dene and study in Sections 2 and 3, the generalized wavelet packets and

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

K

|f(x; t)|p_dm (x; t)

1=p

¡+∞;

dm being the measure on K dened by dm(x; t) = (1= (+ 1))x2+1dxdt. −Lp(R×N);p∈

[1;+∞], the space of functions g:R_×N_→C_{, measurable, satisfying}

kgkLp=

Z

R_×N

|g(; m)|p_d (; m)

1=p

¡+∞ if p∈[1;+∞[;

kgkL∞

= ess sup

(;m)∈R×N

|f(; m)|

d the measure onR×Ndened by d(; m) =Lm()(0)||+1d⊗m, wherem is the Dirac measure

at m.

The function ’; m;(; m)∈R×N, considered in the introduction satises 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 dened by

T₍(_{x; t})₎f(y; s) =

      

     

1 2

Z 2

0

f(

q

x2_+y2_{+ 2xycos; s+t+xysin) d if = 0;}

Z 1

0

Z 2

0

f(

q

x2_+y2_{+ 2xyrcos; s+t+xyrsin)}

×r(1−r2₎−1_{drd; if ¿0:}

Denition 1.1. Let f and g be continuous functions on K with compact support. The convolution product f∗g of f and g is dened by

f∗g(x; t) =

Z

K

T₍(_{x; t})₎f(y; s)g(y;−s) dm(y; s); (x; t)∈K:

Denition 1.2. The generalized Fourier transform F _{associated with Laguerre functions is dened}

on L1

(K) by

F_{(f)(; m) =}

Z

K

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

R×N

F_{(f)(; m)’; m(x; t) d(; m); a:e:}

(ii) For all f∈(L1

∩L2)(K); we have the Plancherel formula

kfk2

;2=kF(f)k 2

L2

:

(iii) The transform F_{can be extended to an isometric isomorphism from L}2

(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 L}2

(R×N) and we have

F_{(f∗g) =}F_(f)F_(g):

(ii) For all f and g in L2

(K); we have

Z

K

|f∗g(x; t)|2_dm

(x; t) =

Z

R_×N

|F_{(f)(; m)|}2_|F_{(g)(; m)|}2_{d(; m);}

where both members are nite or innite.

2. Generalized wavelet packets

Denition 2.1. Letg be in L2

(K), we say that g is a generalized wavelet onK if there is a constant

Cg such that for all m∈N, we have

0¡Cg=

Z

R

|F_{(g)(; m)|}2d

||¡+∞:

Example. Let r¿0; Er and g be the functions dened by

Er(x; t) =

Z

R_×N

’; m(x; t) exp −r2

m+(+ 1) 2

2!

d(; m);

g(x; t) = − d

drEr(x; t):

Then g is a generalized wavelet on K, and we have Cg= 1=4r2.

Let g be a generalized wavelet on K in L2

(K). For a∈R−{0}, we put

ga(x; t) =

1

|a|+2g

x

p |a|;

t a

!

Then we have

Proposition 2.1. Let g be a generalized wavelet on K in L2

(K); a∈R−{0} and {j}j∈Z a

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

1

(ii) is an immediate consequence of Theorem 1.1(iii).

Denition 2.2. (i) The sequence{gp_j}j∈Z is called generalized wavelet packet (also called P-wavelet

packet).

(ii) The function gp_j; j∈Z, is called generalized P-wavelet packet member of step j.

Proposition 2.2. For allj∈Z_{; and (x; t)∈K; the function g}p

Proof. The statements follow from (2.3),(2.4) and the properties of the generalized translation op-erators.

Denition 2.3. Let {gp_j}j∈Z be a generalized P-wavelet packet. The generalized P-wavelet packet

transform p

This transform can also be written in the form

_gp(f)(j;(x; t)) =f∗gpj(x; t); (2.5)

where ∗ is the convolution product given in Denition 1.1.

Theorem 2.1. (i) Plancherel formula for p

g: For all f∈L2(K); we have

(ii) Parseval formula for p

g: For all f1 and f2 in L2(K) we have

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

Z

By Fubini–Tonelli’s theorem we deduce

+∞

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

Hp; q(; m) =

Proof. From Holders inequality for the measure da, and Fubini Theorem we obtain

Z

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

Z

The other assertions are immediate.

The following result is called Calderon’s formula for p g.

Theorem 2.2. Let g be a generalized wavelet on K in L2

(K) such that F(g)∈L∞ (R×N) and

strongly in L2

Proof. It is clear that fp; q_=f

?G_{p; q}. So by Lemma 2.1 and Proposition 1.1(i), fp; q∈L2(K), and

F_(fp; q_{) =}F_(f)H

p; q:

From Theorem 1.1(ii), we obtain

kfp; q_−fk2

;2=

Z

R×N

|F_{(f)(; m)|}2_{|1−Hp; q(; m)|}2_{d(; m):}

The result follows from Lemma 2.1 and dominated convergence theorem.

Theorem 2.3. For f inLr

(K); r= 1;2 such thatF(f)belongs toL1(R×N)we have the following

reconstruction formula for p g:

f(x; t) = +∞

X

j=−∞

Z

K

_gp(j;(y; s))gp_{j;(x; t)}(y; s) dm(y; s); 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

p

g(f)(j;(y; s))g p

j;(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

_gp(f)(j;(y; s))gp_j;(x;t)(y; s) =f∗gp_j(y; s)T₍(_{y; s})₎gp_j(x; t):

If the functions (y; s)→f∗gpj(y; s) and (y; s)→T

() (y; s)g

p

j(y; s) belong to L2(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

R_×N

F_{(f)(; m)[}F_(gp

j)(; m)]2’; m(x; t) d(; m): (2.8)

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

+∞

X

j=−∞

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

R×N

|F_{(f)(; m)|}

+∞

X

j=−∞

[F_(gp

j)(; m)]2

!

d(; m)

=kF_(f)kL1

¡+∞:

Thus, the series P+∞

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

+∞

X

j=−∞

I(j;(x; t)) = +∞

X

j=−∞

Z

R×N

F_{(f)(; m)[}F_(gp

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

and the assertion follows from (2.3) and Theorem 1.1. (ii) We suppose that f∈L2

we deduce (ii) from Theorem 1.1.

3. Generalized scale discrete scaling function on K

Proposition 3.1. Let {gpj}j∈Z be the P-wavelet packet given in Denition 2.2. Then

(i) For all (; m)∈R_×N; we have

Proof. The proof is the same as for Proposition 2.1.

Denition 3.1. The sequence {G_Jp}J∈Z is called generalized scale discrete scaling function.

We consider for all J∈Z _{and (x; t)∈K, the function G}p

J;(x; t) dened on K by

G_J;p_{(x; t)}(y; s) =T₍(_{y; s})₎Gp_J(x; t): (3.5)

The following proposition give the properties of the function G_J;p_{(x; t)}.

Proposition 3.2. (i) For all j∈Z _{and(x; t)∈K; the function G}p

J;(x; t) belongs to L2(K);and we have

kGp_{J;(x; t)}k;26kGJpk;2:

(ii) For all (; m)∈R_×N_;F_(Gp

J;(x; t))(; m) =’; m(x; t)F(GpJ)(; m).

Notation. We denote by h: ; :i the scalar product on L2(K) given by

hf; gi=

Z

K

f(x; t)g(x; t) dm(x; t): (3.6)

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

kfk2

;2= lim

J→+∞

Z

K

|hf; G_J;p_{(x; t)}i|2dm(x; t):

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

Z

K

f1(x; t)f2(x; t) dm(x; t) = lim J→+∞

Z

K

hf1; GJ;p(y; s)ihf2; GJ;p(y; s)idm(y; s):

Proof. (i) We have for all J∈Z _{and (y; s)∈K,}

hf; G_J;p_{(y; s)}i=f∗G p J(x;−t):

Then using Proposition 1.1, we obtain

Z

K

|hf; Gp_{J;(x; t)}i|2dm(y; s) =

Z

R×N

|F_{(f)(; m)|}2_|F_(Gp

J)(; m)|

2_d

(; m): (3.7)

Then (i) follows from Theorem 1.1. (ii) We deduce the result from (i).

Theorem 3.2. (i) Plancherel formula associated with {G_Jp}J∈Z and_gp:For all f inL2(K)andJ∈Z;

we have

kfk2

;2=

Z

K

|hf; G_J;p_{(x; t)}i|2dm(x; t) +

+∞

X

j=J

Z

K

|_gp(f)(j;(x; t))|2_dm

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

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

Z

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

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

strongly in L2

(ii) The function

fJ; n_{(x; t) =f}∞; J_{(x; t) +} n−1

X

j=J

Z

K

p

g(f)(j(y; s))g p

j;(y; s)(x; t) dm(y; s)

belongs to L2

(K) and we have

lim

n→∞f

J; n_=f;

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}+G_{J; 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

hf; G_J;p_{(x; t)}iGpJ;(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; G_J;p_{(x; t)}iG p

J;(y; s)(x; t) dm(y; s)

+ +∞

X

j=J

Z

K

p

g(f)(j;(y; s))g p

j;(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.

Acknowledgements

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

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