The Continuous Wavelet Transform for A Bessel Type Operator on the Half Line

The Continuous Wavelet Transform for A Bessel

Type Operator on the Half Line

R.F. Al Subaie

,

M.A. Mourou

∗

Department of Mathematics, College of Sciences for Girls, University of Dammam, P.O.Box 1982, Dammam 31441, Saudi Arabia

∗_{Corresponding Author: mohamed [email protected]}

Copyright c⃝2013 Horizon Research Publishing All rights reserved.

Abstract

We consider a singular differential oper-ator ∆ on the half line which generalizes the Bessel operator. Using harmonic analysis tools corresponding to ∆, we construct and investigate a new continuous wavelet transform on [0,∞[ tied to ∆. We apply this wavelet transform to invert an intertwining operator between ∆ and the second derivative operatord2_/dx2_.

Keywords

Singular differential operator, general-ized wavelets, generalgeneral-ized continuous wavelet transform.

1

Introduction

Consider the second-order singular differential opera-tor on the half line

∆f(x) = d 2_f dx2 +

2α+ 1 x

df dx−

4n(α+n) x2 f(x), whereα >−1/2 andn= 0,1, .... Forn= 0, we regain the differential operator

Lαf(x) =

d2_f dx2 +

2α+ 1 x

df dx,

which is referred to as the Bessel operator of order α. A well known harmonic analysis on the half line gener-ated by the Bessel operatorLα, is amply and brilliantly

exposed by Trimeche in [14]. Selected excerpts of this harmonic analysis are presented in Section 2.

The authors have showed in [1] that the integral trans-form

X(f)(x) = √2 Γ(α+ 2n+ 1) πΓ(α+ 2n+ 1/2) x

2n

×

∫ 1

0

f(tx)(1−t2)α+2n−1/2dt

is a topological isomorphism between two suitable func-tional spaces, satisfying the intertwining relation

X ◦ d2

dx2 = ∆◦ X,

Through the intertwining operatorX, a completely new commutative harmonic analysis on the half line related

to the differential operator ∆, was initiated. A summary of this harmonic analysis is provided in Section 3. The main contribution of this work is to extend the classical theory of wavelets to the differential operator ∆. More explicitly, we call generalized wavelet each functiongin a suitable functional space, satisfying the admissibility condition

0< Cg=

∫ _∞

0

|F∆(g)(λ)|2 dλ

λ <∞,

whereF∆denotes the generalized Fourier transform re-lated to ∆ given by

F∆(g)(λ) =

∫ _∞

0

f(x)φλ(x)x2α+1dx,

with

φλ(x) =x2njα+2n(λx),

jα+2n being the normalized spherical Bessel function of

indexα+ 2n.

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

ga,b(x) =

1 a2α+2n+2T

b_(g

a)(x), a >0, b≥0,

where ga(x) =g(x/a) and Tb stand for the generalized

translation operators tied to the differential operator ∆. Thereby, the generalized continuous wavelet trans-form associated with ∆ is defined for regular functions f on [0,∞[ by

Φg(f)(a, b) =

∫ _∞

0

f(x)ga,b(x)x2α+1dx.

In Section 4, we exhibit a relationship between the generalized and Bessel continuous wavelet transforms. Such a relationship enables us to establish for the gen-eralized continuous wavelet transform a Plancherel for-mula, a pointwise reconstruction formula and a Calderon reproducing formula.

examples of use of wavelet type transforms in inverse problems the reader is referred to [6, 10, 11, 12, 13] and the references therein.

In the classical framework, the notion of wavelets was first introduced by J. Morlet a French petroleum en-gineer at ELF-Aquitaine, in connection with his study of seismic traces. The mathematical foundations were given by A. Grossmann and J. Morlet in [5]. The har-monic analyst Y. Meyer and many other mathematicians became aware of this theory and they recognized many classical results inside it (see [2, 8, 9]). Classical wavelets have wide applications, ranging from signal analysis in geophysics and acoustics to quantum theory and pure mathematics (see [3, 4, 7] and the references therein).

2

Preliminaries

In the present section we recapitulate some facts about harmonic analysis related to the Bessel opera-tor Lα. We cite here, as briefly as possible, only those

properties actually required for the discussion. For more details we refer to [14].

Note 2.1 Throughout this section assume α > −1/2. DefineLp

α,1≤p≤ ∞, as the class of measurable

func-tionsf on [0,∞[for which∥f∥p,α <∞, where

∥f∥p,α=

(∫ ∞

0

|f(x)|px2α+1dx

)1/p

, if p <∞,

and∥f∥_∞,α=∥f∥∞= ess supx_≥0|f(x)|.

The Fourier-Bessel transform of orderαis defined for a functionf ∈L1α by

Fα(f)(λ) =

∫ _∞

0

f(x)jα(λx)x2α+1dx, λ≥0, (1)

where jα is the normalized spherical Bessel function of

indexαdefined by jα(z) = Γ(α+ 1)

∞

∑

n=0

(−1)n(z/2)2n

n! Γ(n+α+ 1) (z∈C). (2) Proposition 2.1 (i) The Fourier-Bessel transformFα

maps continuously and injectively L1

α into the space

C0([0,∞[) (of continuous functions on [0,∞[ vanishing

at infinity).

(ii) If bothf andFα(f) are inL1α then

f(x) =

∫ _∞

0

Fα(f)(λ)jα(λx)dµα(λ),

for almost allx≥0, where

dµα(λ) =

1

4α_{(Γ(α+ 1))}2λ

2α+1_{dλ. (3)}

(iii) For every f ∈L1α∩L2α we have

∫ _∞

0

|f(x)|2x2α+1dx=

∫ _∞

0

|Fα(f)(λ)|2dµα(λ).

(iv) The Fourier-Bessel transform Fα extends

uniquely to an isometric isomorphism from L2

α onto

L2_{([0,∞[, µ}

α). The inverse transform is given by

F−1

α (g)(x) =

∫ _∞

0

g(λ)jα(λx)dµα(λ),

where the integral converges inL2

α.

The Bessel translation operatorsτx

α, x≥0, are defined

by

τ_αx(f)(y) =aα

∫ π

0

f(√x2_+y2_{+ 2xy cosθ)(sinθ)}2α_dθ,

(4) where

aα=

2Γ(α+ 1)

√

πΓ(α+1₂). (5) Forx, y >0, a change of variables yields

τ_αx(f)(y) =

∫ x+y

|x−y|

f(z)Wα(x, y, z)z2α+1dz, (6)

with

Wα(x, y, z) =

21−α_{[Γ(α+ 1)]}2

√

πΓ(α+1₂)

×

[

(x+y)2_−z2]α−12[_z2_−(x−y)2]α−12

(xyz)2α .

(7) The Bessel convolution product of two functions f, g on [0,∞[ is defined by the relation

f ∗αg(x) =

∫ _∞

0

ταxf(y)g(y)y2α+1dy, x≥0. (8)

Proposition 2.2 (i) Let p∈[1,∞] and f ∈Lp α. Then

for all x≥0,τx

αf ∈Lpα and

∥τ_αxf∥p,α ≤ ∥f∥p,α.

(ii) For f ∈Lp

α,p= 1 or2, we have

Fα(ταxf) (λ) =jα(λx)Fα(f)(λ).

(iii) Let p, q ∈[1,∞] such that 1_p +1_q = 1. If f ∈Lp α

andg∈Lq

α, then for every x≥0 we have

∫ _∞

0

τ_αxf(y)g(y)y2α+1dy=

∫ _∞

0

f(y)τ_αxg(y)y2α+1dy.

(iv) Let p, q, r ∈[1,∞] such that _p1 +1_q −1 = 1_r. If

f ∈Lp

α andg∈Lqα, thenf ∗αg∈Lrα and

∥f∗αg∥r,α≤ ∥f∥p,α∥g∥q,α.

(v) For f ∈L1

α andg∈Lpα,p= 1 or2, we have

Fα(f∗αg) =Fα(f)Fα(g).

Definition 2.1 We say that a function g ∈ L2

α is a

Bessel wavelet of orderα, if it satisfies the admissibility condition

0< C_gα=

∫ _∞

0

|Fα(g)(λ)|2

dλ

λ <∞. (9) Definition 2.2 Letg∈L2αbe a Bessel wavelet of order

α. The Bessel continuous wavelet transform is defined for suitable functionsf on[0,∞[ by

Sα_g(f)(a, b) =

∫ _∞

0

f(x)gα a,b(x)x

2α+1_{dx, (10)}

where a >0,b≥0,

g_a,bα (x) = 1 a2α+2τ

b

α(ga)(x), (11)

and

The Bessel continuous wavelet transform has been in-vestigated in depth in [14] from which we recall the fol-lowing basic properties.

Theorem 2.1 Let g ∈L2

α be a Bessel wavelet of order

α. Then

(i) For allf ∈L2_αwe have the Plancherel formula

∫ _∞

0

|f(x)|2x2α+1dx

= 1 Cα

g

∫ _∞

0

∫ _∞

0

|S_gα(f)(a, b)|2b2α+1dbda a .

(ii) Assume that ∥Fα(g)∥∞ < ∞. For f ∈ L2α and

0< ε < δ <∞, the function

fε,δ(x) = 1 Cg

∫ δ

ε

∫ _∞

0

S_gα(f)(a, b)gα_a,b(x)b2α+1dbda a ,

belongs toL2α and satisfies

lim

ε→0, δ→∞

fε,δ−f_2,α= 0.

(iii) Forf ∈L1

α such that Fα(f)∈L1α, we have

f(x) = 1 Cα

g

∫ _∞

0

(∫ ∞

0

Sαg(f)(a, b)ga,bα (x)b2α+1db

)

da a

for almost allx≥0.

3

Harmonic analysis associated

with

∆

Note 3.1 From now on assume α > −1/2 and n = 0,1,2, ... . LetMbe the map defined by

Mf(x) =x2nf(x).

LetLp

α,n,1≤p≤ ∞, be the class of measurable functions

f on[0,∞[for which||f||p,α,n=M−1f_p,α+2n<∞.

Remark 3.1 It is easily seen that M is an isometry fromLp_α+2n ontoLp

α,n.

3.1

Generalized Fourier transform

Forλ∈Candx∈R, put

φλ(x) =x2njα+2n(λx), (13)

where jα+2n is the normalized Bessel function of index

α+ 2ngiven by (2). From [1] recall the following prop-erties.

Proposition 3.1 (i) φλ possesses the Laplace type

in-tegral representation

φλ(x) =aα+2nx2n

∫ 1

0

cos(λtx)(1−t2)α+2n−1/2dt, (14)

whereaα+2n is given by (5).

(ii)φλ satisfies the differential equation

∆φλ=−λ2φλ.

(iii) For allλ∈Candx∈R,

|φλ(x)| ≤x2ne|Imλ||x|.

Definition 3.1 The generalized Fourier transform is defined for a functionf ∈L1

α,nby

F∆(f)(λ) =

∫ _∞

0

f(x)φλ(x)x2α+1dx, λ≥0. (15)

Remark 3.2 (i) By (13)and (15)observe that

F∆=Fα+2n◦ M−1, (16)

where Fα+2n is the Fourier-Bessel transform of order

α+ 2ngiven by (1). (ii) If f ∈ L1

α,n then F∆(f) ∈ C0([0,∞[) and

||F∆(f)||∞≤ ||f||1,α,n.

Theorem 3.1 Let f ∈L1

α,nsuch that F∆(f)∈L1α+2n.

Then for almost allx≥0,

f(x) =

∫ _∞

0

F∆(f)(λ)φλ(x)dµα+2n(λ),

whereµα+2n is given by (3).

Proof. By (13), (16) and Proposition 2.1(ii) we have

∫ ∞

0

F∆(f)(λ)φλ(x)dµα+2n(λ)

=x2n

∫ _∞

0

Fα+2n

(

M−1_f)_(λ)j

α+2n(λx)dµα+2n(λ)

=x2nM−1f(x) =f(x),

for almost allx≥0.

Theorem 3.2 (i) For every f ∈ L1

α,n∩L2α,n we have

the Plancherel formula

∫ _∞

0

|f(x)|2x2α+1dx=

∫ _∞

0

|F∆(f)(λ)|2dµα+2n(λ).

(ii) The generalized Fourier transform F∆ extends

uniquely to an isometric isomorphism from L2

α,n onto

L2_{([0,∞[, µ}

α+2n). The inverse transform is given by

F−1

∆ (g)(x) =

∫ ∞

0

g(λ)φλ(x)dµα+2n(λ),

where the integral converges inL2

α,n.

Proof. Let f ∈L1α,n∩L2α,n. By (16) and Proposition

2.1(iii) we have

∫ _∞

0

|F∆(f)(λ)|2dµα+2n(λ)

=

∫ _∞

0

_Fα+2n(

M−1_f)_(λ)2 _dµ

α+2n(λ)

=

∫ _∞

0

_M−1_f(x)2_x2α+4n+1_dx

=

∫ _∞

0

|f(x)|2x2α+1dx,

3.2

Generalized convolution product

Definition 3.2 Define the generalized translation oper-atorsTx,x≥0, by the relation

Txf(y) = (xy)2nταx+2n

(

M−1_f)_{(y), y≥0, (17)}

whereτ_αx_+2n are the Bessel translation operators of order

α+ 2n given by (4).

Remark 3.3 Assume that x, y > 0. Then according to (6)and (17)we have

Tx(f)(y) =

∫ x+y

|x₋y_|

f(z)Wα,n(x, y, z)z2α+1dz,

with

Wα,n(x, y, z) = (xyz)2nWα+2n(x, y, z),

whereWα+2n(x, y, z) is given by (7).

Definition 3.3 The generalized convolution product of two functions f andg on [0,∞[is defined by

f#g(x) =

∫ ∞

0

Txf(y)g(y)y2α+1dy, x≥0. (18)

Remark 3.4 Notice by (17)that

f#g=M[(M−1f)∗α+2n

(

M−1_g)]_{, (19)}

where∗α+2n is the Bessel convolution given by (8).

Proposition 3.2 (i) Letf be inLp

α,n,1≤p≤ ∞.Then

for allx≥0, the function Tx_{f belongs to L}p α,n, and

∥Txf∥p,α,n≤x2n∥f∥p,α,n.

(ii) For f ∈Lp

α,n, p= 1or 2, we have

F∆(Txf)(λ) =φλ(x)F∆(f)(λ).

(iii) Letp, q∈[1,∞] such that 1_p+1_q = 1.If f ∈Lp α,n

andg∈Lq α,nthen

∫ _∞

0

Txf(y)g(y)y2α+1dy=

∫ _∞

0

f(y)Txg(y)y2α+1dy.

(iv) Let p, q, r ∈[1,∞] such that _p1 + 1_q −1 = 1_r. If

f ∈Lp

α,n andg∈Lqα,n thenf#g∈Lrα,n and

∥f#g∥r,α,n≤ ∥f∥p,α,n∥g∥q,α,n.

(v) Forf ∈L1α,nand g∈Lpα,n,p= 1or 2, we have

F∆(f#g) =F∆(f)F∆(g).

Proof. (i) By (17) and Proposition 2.2(i) we have

∥Txf∥p,α,n = x2nM ◦ταx+2n◦ M−1(f)_p,α,n

= x2nτ_αx_+2n◦ M−1(f)

p,α+2n

≤ x2nM−1f_p,α+2n = x2n∥f∥p,α,n.

(ii) By (13), (16), (17) and Proposition 2.2(ii) we have

F∆(Txf)(λ) = Fα+2n◦ M−1◦Tx(f)(λ)

= x2nFα+2n◦ταx+2n◦ M−

1_(f)(λ) = x2njα+2n(λx)Fα+2n◦ M−1(f)(λ)

= φλ(x)F∆(f)(λ).

(iii) By (17) and Proposition 2.2(iii) we have

∫ _∞

0

Txf(y)g(y)y2α+1dy =x2n

∫ ∞

0

τ_αx_+2n(M−1f)(y)M−1(g)(y)y2α+4n+1dy =x2n

∫ _∞

0

M−1_f(y)τx α+2n

(

M−1_g)_(y)y2α+4n+1_dy

=

∫ _∞

0

f(y)Txg(y)y2α+1dy

(iv) By (19) and Proposition 2.2(iv) we have

∥f#g∥r,α,n = (M−1f

)

∗α+2n

(

M−1_g)

r,α+2n

≤ M−1_f

p,α+2nM

−1_g

q,α+2n

= ∥f∥p,α,n∥g∥q,α,n.

(v) By (16), (19) and Proposition 2.2(v) we have

F∆(f#g) = Fα+2n

[(

M−1_f)_∗

α+2n

(

M−1_g)] = Fα+2n

(

M−1_f)_F

α+2n

(

M−1_g) = F∆(f)F∆(g).

This concludes the proof.

3.3

Transmutation operators

Note 3.2 We denote by E(R) the space of C∞ even functions on R, provided with the topology of compact convergence for all derivatives. For a > 0, Da(R)

des-ignates the space ofC∞ even functions onR, which are supported in [−a, a], equipped with the topology induced by E(R). Put D(R) =∪_a>0Da(R)endowed with the

in-ductive limit topology. Let En(R) (resp. Dn(R)) stand

for the subspace ofE(R)(resp. D(R)) consisting of func-tionsf such thatf(0) =· · ·=f(2n−1)_{(0) = 0.}

Definition 3.4 For a locally bounded function f on

[0,∞[, define the integral transformX by

Xf(x) =aα+2nx2n

∫ 1

0

f(tx)(1−t2)α+2n−1/2dt, (20)

where aα+2n is given by (5).

Remark 3.5 (i) Forn= 0,X reduces to the Riemann-Liouville integral transform of order αgiven by

Rα(f)(x) =aα

∫ 1

0

f(tx)(1−t2)α−1/2dt, x≥0.

(ii) It is easily checked that

X =M ◦Rα+2n. (21)

(iii) Due to (14)and (20)we have

Definition 3.5 Define the integral transform t_{X for a}

smooth functionf on[0,∞[ by

t_{Xf(y) =a} α+2n

∫ _∞

y

f(x)(x2−y2)α+2n−1/2 dx x2n₋1.

Remark 3.6 (i) Forn= 0,tX is just the Weyl integral transform of orderαgiven by

Wα(f)(y) =aα

∫ _∞

y

f(x) (x2−y2)α−1/2x dx, y≥0.

(ii) It is easily seen that

t_{X =W}

α+2n◦ M−1. (23)

Proposition 3.3 (i) If f ∈L∞([0,∞[, dx) thenXf ∈ L∞_α,n and∥Xf∥_∞,α,n ≤ ∥f∥∞.

(ii) If f ∈ L1

α,n then tXf ∈ L1([0,∞[, dx) and

∥t_Xf∥

1≤ ∥f∥1,α,n.

(iii) For anyf ∈L∞([0,∞[, dx)andg∈L1

α,nwe have

the duality relation

∫ _∞

0

Xf(x)g(x)x2α+1dx=

∫ _∞

0

f(y)tXg(y)dy.

(iv) For allf ∈L1

α,n we have

F∆(f) =Fc◦tX(f), (24)

whereFc is the cosine transform given by

Fc(f)(λ) =

∫ _∞

0

f(x) cos(λx)dx, λ≥0.

(v) Letf, g∈L1

α,n. Then

t_{X(f#g) =} t_Xf∗t_Xg,

where∗ is the symmetric convolution product on[0,∞[

defined by

h1∗h2(x) =

∫ _∞

0

σx(h1)(y)h2(y)dy,

with

σx(h1)(y) =

1

2[h1(x+y) +h1(|x−y|)]. (vi) Let f ∈L1

α,n andg∈L∞([0,∞[, dx). Then

X(t_Xf∗g)_{=f#(Xg). (25)}

Proof. (i) By (21) and [14, Equation (2.I.48)] we have

∥Xf∥_∞,α,n=∥Rα+2nf∥∞≤ ∥f∥∞.

(ii) By (23) and [14, Equation (2.II.3)] we have

t_Xf

1≤M

−1_f

1,α+2n =∥f∥1,α,n.

(iii) By (21), (23) and [14, Equation (2.II.2)] we have

∫ _∞

0

Xf(x)g(x)x2α+1dx

=

∫ _∞

0

Rα+2n(f)(x)M−1g(x)x2α+4n+1dx

=

∫ _∞

0

f(y)Wα+2n

(

M−1_g)_(y)dy

=

∫ _∞

0

f(y)tXg(y)dy.

(iv) By (16), (23) and [14, Equation (5.II.14)] we have

Fc◦tX(f) = Fc◦Wα+2n◦ M−1(f)

= Fα+2n◦ M−1(f) =F∆(f).

(v) By (19), (23) and [14, Equation (5.III.15)] we have

t_{X(f#g) = W} α+2n

[(

M−1_f)_∗

α+2n

(

M−1_g)] = (Wα+2nM−1f

)

∗(Wα+2nM−1g

)

= tXf∗tXg.

(vi) By (19), (21), (23) and [14, Equation (7.IV.9)] we have

f#(Xg) = M[(M−1f)∗α+2n

(

M−1_Xg)] = M[(M−1f)∗α+2n(Rα+2ng)

]

= MRα+2n

[(

Wα+2nM−1f

)

∗g] = X(tXf∗g).

This achieves the proof.

X and t_{X are intertwining operators between ∆ and}

the second derivative operator d2_/dx2 _{by virtue of the} following theorem proved in [1].

Theorem 3.3 (i) The integral transform X is an iso-morphism from E(R) onto En(R) satisfying the

inter-twining relation

X ◦ d2

dx2(f) = ∆◦ X(f), f ∈ E(R).

(ii) The integral transformtX is an isomorphism from

Dn(R) ontoD(R)satisfying the intertwining relation

d2 dx2 ◦

t_{X(f) =}t_{X ◦∆(f), f ∈ D} n(R).

4

Generalized wavelets

Definition 4.1 A generalized wavelet is a functiong∈ L2α,n satisfying the admissibility condition

0< Cg=

∫ _∞

0

|F∆(g)(λ)|2dλ

λ <∞. (26) Remark 4.1 (i) Let0̸=g∈L2α,n satisfying

∃η >0 such that F∆(g)(λ)− F∆(g)(0) =O(λη),

asλ→0.Then (26)is equivalent to F∆(g)(0) = 0. (ii) By (9),(16) and (26),g ∈ L2

α,n is a generalized

wavelet if and only if,M−1_{gis a Bessel wavelet of order} α+ 2n, and we have

Cg=C_Mα+2−1n_g. (27)

Note 4.1 Forg∈L2

α,n and(a, b)∈]0,∞[×[0,∞[ put

ga,b(x) =

1 a2α+2n+2 T

b

(ga)(x), (28)

where ga is given by (12) and Tb are the generalized

translation operators defined by (17).

Proposition 4.1 For alla >0andb≥0 we have

ga,b(x) = (bx)2n

(

M−1_g)α+2n

Proof. Using (11), (17) and (28) we have ga,b(x) =

1 a2α+2n+2T

b_(g a)(x)

= (bx) 2n

a2α+2n+2τ

b α+2n

(

M−1_g

a

)

(x) = (bx)

2n

a2α+4n+2τ

b α+2n

(

M−1_g)

a(x)

= (bx)2n(M−1g)α_a,b+2n(x), which ends the proof.

Definition 4.2 Let g ∈ L2

α,n be a generalized wavelet.

We define for regular functions f on [0,∞[, the gener-alized continuous wavelet transform by

Φg(f)(a, b) =

∫ _∞

0

f(x)ga,b(x)x2α+1dx, (30)

which can also be written in the form

Φg(f)(a, b) =

1

a2α+2n+2f#ga(b), (31)

where # is the generalized convolution product given by (18).

Proposition 4.2 We have

Φg(f)(a, b) =b2nS_Mα+2−1n_g

(

M−1_f)_{(a, b). (32)} Proof. From (10), (29) and (30) we deduce that

Φg(f)(a, b) =

∫ _∞

0

f(x)ga,b(x)x2α+1dx

= b2n

∫ _∞

0

(

M−1_f)_{(x) (M−}1_g)α+2n

a,b (x)

×x2α+4n+1dx = b2nS_Mα+2₋1n_g

(

M−1_f)_{(a, b),} which concludes the proof.

Theorem 4.1 (Plancherel formula) Let g∈L2_α,n be a generalized wavelet. For every f ∈ L2α,n we have the

Plancheral formula

∫ _∞

0

|f(x)|2x2α+1dx

= 1 Cg

∫ _∞

0

∫ _∞

0

|Φg(f)(a, b)|2b2α+1db

da a.

Proof. By (27), (32) and Theorem 2.1(i) we have

∫ _∞

0

∫ _∞

0

|Φg(f)(a, b)|2b2α+1db

da a

=

∫ _∞

0

∫ _∞

0

Sα_M+2−1n_g

(

M−1_f)_{(a, b)}2_b2α+4n+1_dbda a

=C_Mα+2−1n_g

∫ ∞

0

_M−1_f(x)2_x2α+4n+1_dx

=Cg

∫ _∞

0

|f(x)|2x2α+1dx,

which ends the proof.

Theorem 4.2 (Calder´on’s formula) Letg∈L2

α,nbe

a generalized wavelet such that ∥F∆(g)∥_∞ <∞. Then forf ∈L2

α,nand0< ε < δ <∞, the function

fε,δ(x) = 1 Cg

∫ δ

ε

∫ _∞

0

Φg(f)(a, b)ga,b(x)b2α+1db

da a

belongs to L2

α,n and satisfies

lim

ε_→0, δ_→∞

fε,δ−f_2,α,n = 0.

Proof. By (27), (29) and (32) we have

fε,δ(x) = x 2n

C_Mα+2₋1n_g

∫ δ

ε

∫ _∞

0

S_Mα+2−1n_g

(

M−1_f)_{(a, b)}

×(M−1_g)α+2n a,b (x)b

2α+4n+1_dbda a.

The result is then a direct consequence of and Theorem 2.1(ii) .

Theorem 4.3 (inversion formula) Letg∈L2

α,nbe a

generalized wavelet. If f ∈ L1

α,n and F∆(f) ∈ L1α+2n

then we have

f(x) = 1 Cg

∫ _∞

0

(∫ ∞

0

Φg(f)(a, b)ga,b(x)b2α+1db

)

da a

for almost allx≥0.

Proof. By (27), (29) and (32) we have 1

Cg

∫ _∞

0

( ∫ ∞

0

Φg(f)(a, b)ga,b(x)b2α+1db

)_da

a

= x

2n

C_Mα+2−1n_g

∫ _∞

0

( ∫ ∞

0

S_Mα+2₋1n_g

(

M−1_f)_{(a, b)}

×(M−1

g)α_a,b+2n(x)b2α+4n+1db

)_da

a . The result follows now from Theorem 2.1(iii).

5

Inversion of the intertwining

operator

t

X

through the

gener-alized wavelet transform

To obtain inversion formulas for t_{X involving}

gener-alized wavelets, we have to establish some preliminary lemmas.

Lemma 5.1 Let 0 ̸=g∈L1_∩L2_{([0,∞[, dx)such that}

Fc(g)∈L1([0,∞[, dx)and satisfying

∃η > α+ 2n such that Fc(g)(λ) =O(λη) (33)

asλ→0. ThenXg∈L2

α,nand

F∆(Xg)(λ) = 2

2α+4n+1_{(Γ(α+ 2n+ 1))}2

π λ2α+4n+1 Fc(g)(λ). Proof. We have

g(x) = 2 π

∫ _∞

0

So by (22),

Xg(x) =

∫ _∞

0

h(λ)φλ(x)dµα+2n(λ), a.e. (34)

where

h(λ) = 2

2α+4n+1_{(Γ(α+ 2n+ 1))}2

π λ2α+4n+1 Fc(g)(λ) and µα+2n is given by (3). Clearly, h is an element of

L1_{([0,∞[, µ}

α+2n). So in view of (34) and Theorem 3.2,

it suffices to prove thath∈L2_{([0,∞[, µ}

α+2n). We have

∫ _∞

0

|h(λ)|2dµα+2n(λ)

=m(α, n)

∫ _∞

0

λ−2α−4n−1|Fc(g)(λ)|2dλ

=m(α, n)

(∫ 1

0 +

∫ _∞

1

)

λ−2α−4n−1|Fc(g)(λ)|2dλ

=m(α, n) (I1+I2),

wherem(α, n) = 4α+2n+1π−2(Γ(α+ 2n+ 1))2. By (33) there is a positive constantksuch that

I1≤k

∫ 1

0

λ2η−2α−4n−1dλ= k

2(η−α−2n) <∞. From the Plancherel theorem for the cosine transform, it follows that

I2 =

∫ ∞

1

λ−2α−4n−1|Fc(g)(λ)|2dλ

≤

∫ _∞

0

|Fc(g)(λ)|2dλ=

π 2

∫ _∞

0

|g(x)|2dx <∞,

which achieves the proof.

Lemma 5.2 Let 0 ̸=g ∈L1∩L2([0,∞[, dx) such that

Fc(g)∈L1([0,∞[, dx)and satisfying

∃η >2α+ 4n+ 1 such that Fc(g)(λ) =O(λη) (35)

asλ→0. ThenXg∈L2α,n is a generalized wavelet and

F∆(Xg)∈L∞([0,∞[, dx).

Proof. By (35) and Lemma 5.1 we see thatXg∈L2

α,n,

F∆(Xg) is bounded and

F∆(Xg)(λ) =O(λη−2α−4n−1) as λ→0. Thus, in view of Remark 4.1(i), the functionXgsatisfies the admissibility condition (26).

Recall that the classical continuous wavelet transform on [0,∞[ is defined for suitable functions by

Wg(f)(a, b) =

1 a

∫ _∞

0

f(x)σb(ga)(x)dx, (36)

where a > 0, b ≥ 0 and g ∈ L2_{([0,∞[, dx) is a} clas-sical wavelet on [0,∞[, i.e., satisfying the admissibility condition

0< C(g) =

∫ _∞

0

|Fc(g)(λ)|2

dλ

λ <∞. (37) A more complete and detailed discussion of the prop-erties of the classical continuous wavelet transform on [0,∞[ can be found in [14].

Remark 5.1 (i) According to [14], each function sat-isfying the conditions of Lemma 5.2 is a classical wavelet on [0,∞[.

(ii) In view of (24),(26)and (37),g∈ D(R)is a gen-eralized wavelet, if and only if, tXg is a classical wavelet and we have

C(tXg)=Cg.

The following statement provides a formula relating the generalized continuous wavelet transform to the clas-sical one.

Lemma 5.3 Let g be as in Lemma 5.2. Then for all

f ∈Lpα,n,p= 1or2, we have

Φ_Xg(f)(a, b) =

1 a2α+4n+1X

[

Wg

(_t

Xf)(a,·)](b).

Proof. By (31) we have Φ_Xg(f)(a, b) =

1

a2α+2n+2f# (Xg)a(b). But

(Xg)a=

1

a2n X(ga)

by virtue of (12) and (20). So using (25) and (36) we find that

Φ_Xg(f)(a, b) =

1

a2α+4n+2f#[X(ga)](b)

= 1

a2α+4n+2X

[t_X

f∗ga

]

(b)

= 1

a2α+4n+1X

[

Wg

(t_Xf)_(a,·)]_(b),

which completes the proof.

A combination of Theorems 4.2−4.3 with Lemmas 5.2−5.3 yields

Theorem 5.1 Letgbe as in Lemma 5.2. Then we have the following inversion formulas fort_X:

(i) If f ∈ L1

α,n and F∆(f)∈ L1α+2n then for almost

all x≥0 we have

f(x) = 1 C_Xg

∫ _∞

0

( ∫ ∞

0

X[Wg

(t_X

f)(a,·)](b)

×(Xg)a,b(x)b2α+1db

) _da

a2α+4n+2.

(ii) For f ∈ L1

α,n∩L2α,n and 0 < ε < δ < ∞, the

function

fε,δ(x) = 1 C_Xg

∫ δ

ε

∫ _∞

0

X[Wg

(_t

Xf)(a,·)](b)

×(Xg)a,b(x)b2α+1db

da a2α+4n+2

satisfies

lim

ε→0, δ→∞

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