inversion of the Generalized Dunkl Intertwining Operator on R and its Dual Using Generalized Wavelets

Vol. 3, No. 6, 2010, 958-979

ISSN 1307-5543 – www.ejpam.com

SPECIAL

ISSUE ON

COMPLEX

ANALYSIS: THEORY AND

APPLICATIONS

PROFESSOR

HARI

M. SRIVASTAVA,

ON THE OCCASION OF HIS

70

Inversion of the Generalized Dunkl Intertwining Operator on

R

_{and its Dual using Generalized Wavelets}

W. Chabeh

_{, M.A. Mourou}

1 _{Preparatory Institute for Engineer Studies of Monastir, University of Monastir,5019, Monastir,}

Tunisia

2_{Department of Mathematics, College of Sciences for Girls, University of Dammam, P.O.Box838,}

Dammam31113, Saudi Arabia

Abstract. We establish an inversion formula for a continuous wavelet transform associated with a class of singular differential-difference operators onR_{. We apply this result to derive new expressions for} the inverse generalized Dunkl intertwining operator and its dual onR_.

2000 Mathematics Subject Classifications: 42B20, 42C15, 44A15, 44A35

Key Words and Phrases: Differential-difference operator, Generalized Dunkl intertwining operator, Generalized continuous wavelet transform.

1. Introduction

Consider the second-order singular differential operator on the real line

∆ = d

2

d x2 +

A′(x)

A(x)

d

d x (1)

where

A(x) =_|x|2α+1B(x), α >₋1 2,

B being a positiveC∞even function onR_{. In addition we suppose that}

∗_{Corresponding author.}

Email addresses:wafa_habbahyahoo.fr(W. Chabeh),mohamed_ali.mourouyahoo.fr(M. Mourou)

(i) Ais increasing on[0,∞[and lim_x→∞A(x) =∞;

(ii) A′/Ais decreasing on]0,∞[and lim_x→∞A′(x)/A(x) =0;

(iii) There exists a constantδ >0 such that the functioneδxB′(x)/B(x)is bounded for large x ∈]0,∞[together with its derivatives.

Lions [5] has constructed an automorphismX of the space Ee(R)ofC∞ even functions on R_{, which intertwines∆ and the second derivative operator d}2_{/d x}2_{; that is, satisfying the} intertwining relation

X d

2

d x2 f = ∆X f, f ∈ Ee(R).

It is known[14]that the Lions operatorX admits the integral representation

Xf(x) =

Z |x|

0

G(x,y)f(y)d y, x6=0,

where G(x,·) is an even positive function onR_{, continuous on]}− |_x|_,|_x|_{[and supported in}

[−|x|,|x|]. Furthermore, the dual Lions operator

t_{Xf(y) =} Z ∞

|y|

G(x,y)f(x)A(x)d x, y ∈R_,

is an automorphism of the spaceS_e(R_{)of even Schwartz functions on}R_{, satisfying the} inter-twining relation

d2 d x2

t_{Xf =} t_{X∆f, f ∈ S} e(R).

In[8]the second author has introduced on the spaceE(R)ofC∞functions onR_{, the following} operator

V f =X(f_e) + d

d xX I(fo), (2)

where

fe(x) =

f(x) +f(−x)

2 , fo(x) =

f(x)− f(−x)

2 , (3)

andI is the map defined byIh(x) =R₀xh(t)d t.

Mainly, he showed thatV is an automorphism ofE(R_{)satisfying for all f} ∈ E₍R_),

V d

d xf = ΛV f, (4)

whereΛis a first-order differential-difference operator onR_{given by}

Λf(x) = d f

d x + A′(x)

A(x)

_{f(x)−f(−x)}

2

ForA(x) =|x|2α+1,α >−1/2, the intertwining operatorV reads

V(f)(x) =_p Γ(α+1)

πΓ(α+1/2)

Z 1

−1

f(t x)(1₋t2)α−1/2(1+t)d t,

and referred to as the Dunkl intertwining operator of indexα+1/2 associated with the re-flection groupZ_2onR_{. The differential-difference operator}Λreduces to the one-dimensional Dunkl operator

Dαf =

d f

d x + (α+ 1 2)

f(x)−f(−x)

x .

Such operators have been introduced by Dunkl in connection with a generalization of the classical theory of spherical harmonics (see[1, 11]and the references therein). During the last years, the theory of Dunkl operators has found a wide area of applications in mathematics and mathematical physics. In fact, Dunkl operators have been used in the study of multivariable orthogonality structures with certain reflection symmetries [12, 16]. Moreover, they have been successfully involved in the description and solution of Calogero-Moser-Sutherland type quantum many body systems[4].

Define the dual operator tV ofV on the space_S(R_{)of Schwartz functions on}R_{, by the} relation

t_{V f =} t_X(f e) +

d d x

t_XJ(f

o), (6)

whereJ is the map defined by

J h(x) =

Z x

−∞

h(y)d y, x _∈R_{. (7)}

In this paper, it is shown that the dual operatortV is an automorphism ofS(R_{)which satisfies} the intertwining relation

d d x

t_{V f =} t_{VΛf, f ∈ S(R).}

Moreover, the following inversion formulas forV andtVon certain specific subspaces ofS(R₎ are provided

f =VK tV f; f =MVtV f; f = tVMV f; f =_K tV V f;

analysis results related to the differential-difference operatorΛ. Next we list some basic prop-erties of the generalized Dunkl intertwining operator V and its dual tV. In section 3 we introduce the generalized continuous wavelet transform associated withΛ, and we prove for this transform Plancherel and reconstruction formulas. Using generalized wavelets, we obtain in Section 4 formulas which give the inverse operatorsV−1andtV−1on Schwartz type spaces.

2. Preliminaries

In this section we provide some facts about harmonic analysis related to the differential-difference operator Λ. We cite here, as briefly as possible, only those properties actually required for the discussion. For more details we refer to[8].

Notation. We denote by

- S(R_{) the space of C}∞ _{functions f on} R_{, which are rapidly decreasing together with} their derivatives, i.e., such that for allm,n=0, 1, . . .,

Pm,n(f) =sup x∈R

(1+x2)m

dn d xnf(x)

<∞.

The topology ofS(R_{)is defined by the semi-normsPm,n,m,n=0, 1, . . . .}

- Se(R)(resp. So(R)) the subspace ofS(R)consisting of even (rep. odd) functions. - B(R_{)the subspace of}S₍R_{)consisting of functions f such that for alln=0, 1, . . .,}

Z

R

f(x)bn(x)A(x)d x=0,

with bn(x) =V _yn

n!

(x), V being the generalized Dunkl intertwining operator given by (2).

- W(R_{)the subspace of}S₍R_{)consisting of functions f such that for alln=0, 1 . . .,} Z

R

f(x)xnd x=0.

- H(R_{)the subspace of}S₍R_{)consisting of functions f such that for alln=0, 1 . . .,} dn

d xnf(0) =0. Put

Be(R) =Se(R)∩ B(R), Bo(R) =So(R)∩ B(R),

We(R) =Se(R)∩ W(R), Wo(R) =So(R)∩ W(R),

Remark 1.

(i) Due to our assumptions on the function A there is a positive constant k such that

A(x)_∼k|x|2α+1, as|x| → ∞. (ii) It follows from (4) that

Λbn+1= bn (8)

for all n∈N_{. Further, by[9]we have for any n}∈N_{and x} ∈R_,

|b_n(x)_{| ≤}k|x|n, k being a positive constant depending only on n.

(iii) It is easily checked that the spaceS(R)is invariant under the differential-difference oper-atorΛ.

For eachλ_∈C_{the differential-difference equation}

Λu=iλu, u(0) =1, (9)

admits a uniqueC∞solution onR_{, denotedΨλgiven by}

Ψλ(x) =

¨

ϕλ(x) + _i1_{λd x}d ϕλ(x) ifλ6=0,

1 ifλ=0, (10)

whereϕλ designates the solution of the differential equation

∆u=−λ2u, u(0) =1, u′(0) =0, (11)

∆being the differential operator defined by (1). Remark 2.

(i) If A(x) =|x|2α+1,α >−1/2, then

Ψ_λ(x) = j_α(λx) + iλx

2(α+1)jα+1(λx),

where jγ (γ >−1/2)stands for the normalized spherical Bessel function of indexγgiven

by

jγ(z) = Γ(γ+1)

∞ X

n=0

(−1)n(z/2)2n

(ii) It follows by (4) and (9) that

Ψ_λ(x) =V€ei뷊(x) (12) for all x∈R_andλ∈C_.

The next statement provides a new estimate for the eigenfunctionΨλ(x).

Lemma 1. For allλ, x∈R_{, we have}

|Ψ_λ(x)_{| ≤}1. Proof. Forλ=0, the result is obvious. Forλ6=0, set

uλ(x) =|Ψλ(x)|2=

ϕλ(x) +

1 iλ

d

d xϕλ(x)

2

= (ϕλ(x))2+

1 λ2

_d

d xϕλ(x)

2 .

Notice thatuλ(x)is even in x. By (11),

d

d xuλ(x) =2ϕλ(x) d

d xϕλ(x) + 2 λ2

d d xϕλ(x)

d2

d x2ϕλ(x) =− 2 λ2

A′(x)

A(x)

_d

d xϕλ(x)

2 .

As the functionAis increasing on[0,∞[, it follows thatu_λ is decreasing on]0,∞[. As uλ(0) =1, we deduce thatuλ(x)≤1 for allx ≥0. This ends the proof.

Notation. For a positive Borel measureµon R_{, andp} =1 or 2, we write Lp(R_,dµ)for the class of measurable functions f onR_{for which}

kfkp,µ=

‚Z

R

|f(x)_|pdµ(x)

Œ1/p <_∞.

Definition 1. The generalized Fourier transform of a function f in L1(R_{,A(x)d x)is defined by}

FΛ(f)(λ) =

Z

R

f(x)Ψ_−λ(x)A(x)d x. (13)

Remark 3. Let f ∈L1(R_{,A(x)d x). By Lemma 1, it follows that}F_{Λ(f)is continuous on}R_and

||FΛ(f)||∞≤ kfk1,A.

An outstanding result about the generalized Fourier transform_F is as follows. Theorem 1. [8]

(i) For every f _∈L1_∩L2(R_{,A(x)d x)we have the Plancherel formula} Z

R

|f(x)|2A(x)d x=

Z

R

where

dσ(λ) = dλ

|c(_|λ_|)_|2,

c(z)being a continuous functions on]0,∞[such that c(z)−1 _∼ k1zα+

1

2, as z→ ∞,

c(z)−1 ∼ k₂zα+12, as z→0,

for some k1,k2∈C.

(ii) The generalized Fourier transform_FΛ extends uniquely to a unitary isomorphism from L2(R_,A(x)d x)onto L2(R_,dσ). The inverse transform is given by

F−1

Λ g(x) =

Z

R

g(λ)Ψ_λ(x)dσ(λ)

where the integral converges in L2(R_,A(x)d x). Remark 4.

(i) The tempered measure σ is called the spectral measure associated with the differential-difference operatorΛ.

(ii) For A(x) =_|x|2α+1,α >₋1/2, we have

c(s) = 2

α+1_Γ(α+1)

sα+1/2 .

The following lemma will play a key role in the remainder of this section. Lemma 2. The map J , given by (7), is a topological isomorphism

- fromS_o(R_)ontoS_e(R_);

- fromB_o(R_)ontoB_e(R_).

Proof.

(i) It is sufficient to show that J maps continuously S_o(R_{) into} S_e(R_{). Let f} ∈ S_o(R_). ClearlyJ f is a C∞even function onR_{. Forn=1, 2, . . ., Pm,n(J f) = Pm,n−1(f).} More-over,

(1+x2)m|J f(x)| ≤ (1+x2)m

Z ∞

|x| |

≤

Z ∞

|x|

(1+t2)m_|f(t)_|d t

≤ P_m+1,0(f)

Z ∞

|x| d t

(1+t2₎ HenceP_m,0(J f)_≤ π

2 Pm+1,0(f).

(ii) Let f ∈ B_o(R_{). By using (8) and by integrating by parts we have for anyn=0, 1, . . .,} Z

R

J f(x)bn(x)A(x)d x = Z

R

J f(x) Λbn+1(x)A(x)d x

= −

Z

R

ΛJ f(x)b_n+1(x)A(x)d x

= −

Z

R

f(x)b_n+1(x)A(x)d x=0,

which shows thatJ f _{∈ Be}(R_{). Conversely, let f} ∈ B_e(R_{). Identity (8) together with an} integration by parts yields for anyn=1, 2, . . .,

Z

R

f′(x)b_n(x)A(x)d x =

Z

R

Λf(x)b_n(x)A(x)d x

= −

Z

R

f(x)Λb_n(x)A(x)d x

= ₋

Z

R

f(x)bn−1(x)A(x)d x=0, which shows that f′∈ Bo(R).

Proposition 1.

(i) For all f inS(R_{), we have}

FΛ(Λf)(λ) =iλFΛ(f)(λ). (14)

(ii) For all f inS(R_{), we have}

FΛ(f)(λ) =F∆(fe)(λ) +iλF∆(J fo)(λ), (15) whereF_∆stands for the Fourier transform related to the differential operator ∆,defined onSe(R)by

F∆(h)(λ) =

Z

R

h(x)ϕλ(x)A(x)d x, λ∈R,

Proof.

(i) Let f ∈ S(R_{). By (5), (10) and (13),}

FΛ(Λf)(λ) =

Z

R

‚

f_o′(x) +A

′_(x)

A(x) fo(x)

Œ

ϕλ(x)A(x)d x

− 1

iλ Z

R

f_e′(x)ϕ_λ′(x)A(x)d x

= κ₁₋κ2

iλ. By integrating by parts we get

κ1= Z

R

(A(x)f_o(x))′ϕλ(x)d x=−

Z

R

f_o(x)ϕ_λ′(x)A(x)d x

and

κ2 = Z

R

f_e′(x)ϕ_λ′(x)A(x)d x

= −

Z

R

f_e(x)(A(x)ϕ′_λ(x))′d x

= ₋

Z

R

fe(x)∆ϕλ(x)A(x)d x

= λ2 Z

R

f_e(x)ϕλ(x)A(x)d x

by virtue of (11). Hence

κ1− κ2

iλ = iλ Z

R

‚

fe(x)ϕλ(x)−fo(x) ϕ′_λ(x)

iλ Œ

A(x)d x

= iλ Z

R

f(x)Φ_−λ(x)A(x)d x.

This clearly yields (14).

(ii) If f ∈ Se(R), identity (15) is obvious. Assume f ∈ So(R). By using (10), (11), (13) and by integrating by parts we obtain

FΛ(f)(λ) = −

1 iλ

Z

R

f(x)ϕ_λ′(x)A(x)d x

= 1

iλ Z

R

= 1

iλ Z

R

J f(x)∆ϕλ(x)A(x)d x

= iλ Z

R

J f(x)ϕλ(x)A(x)d x

= iλ_F∆J f(λ),

which completes the proof.

Theorem 2. The generalized Fourier transformF_Λis a topological isomorphism - fromS(R_{)onto itself;}

- fromB(R)ontoH(R).

Proof. By[13]we know that the transformF∆is a topological isomorphism

- fromS_e(R_{)onto itself;} - fromBe(R)ontoHe(R).

The result follows then from (15), Lemma 2 and the fact that the operatorλ7→λf is a topological isomorphism

- fromSe(R)ontoSo(R); - fromH_e(R_)ontoH_o(R_).

Proposition 2.

(i) For all f ∈ S(R_),

FΛ(f) =Fu◦ tV(f), (16) whereFu denotes the usual Fourier transform onRgiven by

Fu(f)(λ) = Z

R

f(x)e−iλxd x.

(ii) For all f ∈ S(R_),

d d x

t_{V f =} t_{VΛf. (17)}

Proof. Assertion (i) follows by applying the usual Fourier transformFuto both sides of (6) and by using the identity

F∆h(λ) =Fu tXh(λ), h∈ Se(R),

Theorem 3. The intertwining operatortV is a topological isomorphism - fromS(R_{)onto itself;}

- fromB(R_)ontoW₍R_).

Proof. We deduce the result from (16), Theorem 2 and the fact that the usual Fourier transform_Fuis a topological isomorphism

- fromS(R_{)onto itself;} - fromW(R)ontoH(R).

Definition 2.

(i) The generalized translation operators Tx, x∈R_{, are defined on} L2(R_{,A(x)d x)by the relation}

FΛ(Txf)(λ) = Ψλ(x)FΛ(f)(λ). (18)

(ii) The generalized convolution product of two functions f and g in L2(R_{,A(x)d x)is defined} by

f#g(x) =

Z

R

Txf(₋y)g(y)A(y)d y. (19)

Remark 5. Let f and g be in L2(R_{,A(x)d x). Then} (i) By (18), Lemma 1 and Theorem 1, we deduce that

Txf_2,A≤f_2,A (20) for any x∈R_.

(ii) It follows from (19), (20) and Schwarz inequality that f#g∈L∞(R_)and

f#g_∞≤f_2,Ag_2,A. (21) (iii) By virtue of (18), (19) and Theorem 1, f#g may be rewritten as

f#g(x) =

Z

R

FΛ(f)(λ)FΛ(g)(λ)Ψλ(x)dσ(λ). (22)

Proposition 3. Let f ∈L2(R_{,A(x)d x)and g} ∈_L1∩_L2₍R_{,A(x)d x). Then f#g}∈_L2₍R_{,A(x)d x),}

f#g_2,A≤f_2,Ag_1,A, (23) and

Proof. By Schwarz inequality,F_Λ(f)F_Λ(g)∈L1(R_{,dσ). Moreover, by Remark 3,}

FΛ(f)FΛ(g) ∈ L2(R,dσ) and

_FΛ(f)_FΛ(g)_{2,σ ≤ FΛ}(f)_2,σg_1,A. The result follows then by combining (22) and Theorem 1.

Proposition 4. If f,g∈ S(R_{), then f#g}∈ S₍R_)and

t_{V(f#g) =} t_{V f ∗} t_{V g, (25)}

where∗denotes the usual convolution onR_.

Proof. The fact that f#g_{∈ S}(R_{)follows from (24) and Theorem 2. Identity (25) follows} by applying the usual Fourier transform to both its sides and by using (16) and (24).

Remark 6. Notice by (24) and Theorem 2 thatB(R_)#S₍R₎⊂ B₍R_).

3. Generalized Wavelets

Definition 3. We say that a function g _∈ L2(R_{,A(x)d x)is a generalized wavelet if it satisfies} the admissibility condition :

0<Cg= Z ∞

0

|FΛg(aλ)|2

d a

a <∞, (26)

for almost allλ∈R_.

Remark 7.

(i) The admissibility condition (26) can also be written as

0<C_g=

Z ∞

0

|FΛ(g)(λ)|2

dλ λ =

Z ∞

0

|FΛ(g)(−λ)|2

dλ λ <∞.

(ii) If g is real-valued we haveF_Λ(g)(−λ) =F_Λ(g)(λ), so (26) reduces to

0<C_g=

Z ∞

0

|FΛ(g)(λ)|2

dλ λ <∞.

(iii) If0₆=g_∈L2(R_{,A(x)d x)is real-valued and satisfies}

∃η >0 such that FΛ(g)(λ)− FΛ(g)(0) =O(λη), asλ→0+,

then (26) is equivalent toFΛ(g)(0) =0.

Proposition 5.

(i) Let h∈L2(R_,dσ)and a>0. Then the functionλ_7→h(aλ)belongs to L2(R_,dσ)and we have

kh(a·)_k2,σ≤ k_p(a)

a khk2,σ, where

k(a) =sup

λ>0

|c(λ)_|

|c(λ/a)_|.

(ii) For every a>0, the dilatation operator H_a(f)(x) = p1

af

_x

a

‹

, x∈R_, is a topological automorphism of L2(R_,dσ).

Proof.

(i) Notice first that according to the properties of the function c(λ) given in Theorem 1, there exist two positive constantsm1 andm2 such that

m1

aα+1/2 ≤k(a)≤

m2

aα+1/2 for alla>0. We have

kh(a·)_k2_2,σ =

Z

R

|h(aλ)_|2 dλ

|c(_|λ_|)_|2

= 1

a

Z

R

|h(s)_|2 |c(|s|)|

2

|c(_|s|/a)_|2

ds

|c(_|s|)_|2

≤ k

2_(a)

a khk

2 2,σ

(ii) We deduce the result from (i).

Proposition 6. Let g∈L2(R_,A(x)d x)and a>0. Then there exists a function g_a∈L2(R_,A(x)d x)

(and only one) such that

FΛ(ga)(λ) =FΛ(g)(aλ) (27)

for almost everyλ_∈R_{. This function is given by the relation}

g_a= _p1

aF −1

Λ ◦Ha−1◦ F_Λ(g) (28)

and satisfies

g_a2,A≤ k_p(a) a

Proof. The result follows by combining Theorem 1 and Proposition 5. Remark 8. For A(x) =_|x|2α+1,α >₋1/2, the function ga, a>0, is given by

g_a(x) = 1

a2α+2 g

_x

a

‹

, x_∈R_.

Proposition 7. Let g be inS(R_{). Then for all a>0, the function ga belongs to}S₍R_{)and we} have the relation

g_a= _p1

a

t_V−1_◦H

a◦tV(g). (29)

Proof. The result follows from (16), (28), Theorem 2, and the fact thatF_u◦H_a=H_a−1◦F_u.

Notation. For a functiongin L2(R_{,A(x)d x)and for(a,b)}∈_]0,∞_[×R_{we write}

g_a,b(x):=pa T−bg_a(x), (30) where T−bare the generalized translation operators given by (18).

Definition 4. Let g ∈ L2(R_{,A(x)d x) be a generalized wavelet. The generalized continuous} wavelet transformΦ_gis defined for regular functions f onR_{by :}

Φ_g(f)(a,b) =

Z

R

f(x)g_a,b(x)A(x)d x.

This transform can also be written in the form

Φ_g(f)(a,b) =pa f#_fg_a(b), (31) where#is the generalized convolution product given by (19), and_ega(x) =ga(−x), x∈R. Lemma 3. For all f,g∈L2(R_{,A(x)d x)and all h}∈ S₍R_{)we have the identity}

Z

R

f#g(x)_FΛ−1(h)(x)A(x)d x=

Z

R

FΛ(f)(λ)FΛ(g)(λ)h−(λ)dσ(λ)

where h−(λ) =h(₋λ),λ_∈R_.

Proof. Fixg∈L2(R_{,A(x)d x)andh}∈ S₍R_{). For f} ∈_L2₍R_{,A(x)d x)put}

S₁(f) =

Z

R

f#g(x)F_Λ−1(h)(x)A(x)d x

and

S₂(f) =

Z

R

In view of Proposition 3 and Theorem 1, we see thatS₁(f) =S₂(f)for each

f ∈L1∩L2(R_{,A(x)d x). Moreover, by using (21), Schwarz inequality and Theorem 1 we get}

|S₁(f)| ≤f#g_∞F_Λ−1(h)_1,A≤f_2,Ag_2,AF_Λ−1(h)_1,A

and

|S2(f)| ≤

_FΛ(f)_FΛ(g)_1,σkhk_∞

≤ FΛ(f)

2,σ

_FΛ(g)_2,σkhk_∞

≤ f_2,Ag_2,Akhk_∞,

which shows that the linear functionalsS₁andS₂are bounded on L2(R_,A(x)d x). Therefore S₁≡S₂, and the lemma is proved.

Lemma 4. Let f₁,f₂∈L2(R_,A(x)d x). Then f₁#f₂∈L2(R_,A(x)d x)if and only if

FΛ(f1)FΛ(f2)∈L2(R,dσ)and we have

FΛ(f1#f2) =FΛ(f1)FΛ(f2)

in the L2−case.

Proof. Suppose f1#f2 ∈ L2(R,A(x)d x). By Lemma 3 and Theorem 1, we have for any

h∈ S(R), Z

R

FΛ(f1)(λ)FΛ(f2)(λ)h(λ)dσ(λ) = Z

R

f1#f2(x)FΛ−1(h−)(x)A(x)d x

=

Z

R

f₁#f₂(x)F_Λ−1€hŠ(x)A(x)d x

=

Z

R

FΛ(f1#f2)(λ)h(λ)dσ(λ),

which shows that F_Λ(f₁)F_Λ(f₂) = F_Λ(f₁#f₂). Conversely, if F_Λ(f₁)F_Λ(f₂) ∈ L2(R_,dσ), then by Lemma 3 and Theorem 1, we have for anyh∈ S(R_),

Z

R

f₁#f₂(x)F_Λ−1(h)(x)A(x)d x =

Z

R

FΛ(f1)(λ)FΛ(f2)(λ)eh(λ)dσ(λ)

=

Z

R

F−1

Λ [FΛ(f1)FΛ(f2)](x)FΛ−1(h)(x)A(x)d x,

which shows, in view of Theorem 2, that f₁#f₂ = F_Λ−1[F_Λ(f₁)F_Λ(f₂)]. This achieves the proof.

Lemma 5. Let f₁,f₂∈L2(R_{,A(x)d x). Then} Z

R

|f₁#f₂(x)|2A(x)d x=

Z

R

|FΛ(f1)(λ)|2|FΛ(f2)(λ)|2dσ(λ)

where both sides are finite or infinite.

Theorem 4. Let g∈ L2(R_{,A(x)d x)be a generalized wavelet. Then for all f} ∈_L2₍R_{,A(x)d x),} we have the Plancherel formula

Z

R

|f(x)|2A(x)d x= 1

Cg Z ∞

0 Z

R

|Φ_g(f)(a,b)|2A(b)d bd a a2.

Proof. Using (26), (27), (31), Fubini’s Theorem and Lemma 5, we have 1 Cg Z ∞ 0 Z R

|Φ_g(f)(a,b)|2A(b)d bd a a2 =

= 1 C_g Z ∞ 0 ‚Z R

|f#_eg_a(b)_|2A(b)d b

Œ d a a = 1 C_g Z ∞ 0 ‚Z R

|FΛ(f)(λ)|2|FΛ(g)(aλ)|2dσ(λ)

Πd a a = Z R

|FΛ(f)(λ)|2

‚ 1 C_g

Z ∞

0

|FΛ(g)(aλ)|2

d a a

Œ

dσ(λ)

=

Z

R

|FΛ(f)(λ)|2dσ(λ)

The result is now a direct consequence of Theorem 1.

Theorem 5. Let g∈L2(R_{,A(x)d x)be a generalized wavelet. Then for f} ∈_L1∩_L2₍R_{,A(x)d x)} such thatF_Λ(f)∈L1(R_{,dσ), we have}

f(x) = 1

Cg Z ∞

0 ‚Z

R

Φ_g(f)(a,b)g_a,b(x)A(b)d b

Œ

d a

a2, a.e.,

where, for each x∈R_{, both the inner integral and the outer integral are absolutely convergent,} but possibly not the double integral.

Proof. Put

I(a,x) =

Z

R

Φ_g(f)(a,b)g_a,b(x)A(b)d b and

J(x) = 1

C_g

Z ∞

0

I(a,x)d a

By (30) and (31) we have

I(a,x) =a

Z

R

f#˜g_a(b)T−x_eg

a(b)A(b)d b.

From (20), (23) and Schwarz inequality we deduce that the integral I(a,x) is absolutely convergent. On the other hand, by (18), (24) and (27),

FΛ(f#ega)(λ) =FΛ(f)(λ)FΛ(g)(aλ)

and

FΛ(T−x˜ga)(λ) = Ψλ(−x)FΛ(g)(aλ).

So using Theorem 1 we obtain

I(a,x) =a

Z

R

FΛ(f)(λ)Ψλ(x)|FΛ(g)(aλ)|2dσ(λ).

In particular, this implies that 1

C_g

Z ∞

0

|I(a,x)_|d a

a2 ≤

Z

R

|FΛ(f)(λ)|

‚ 1 C_g

Z ∞

0

|FΛ(g)(aλ)|2

d a a

Œ

dσ(λ)

= F_Λ(f)_1,σ<∞,

that is, the integralJ(x)is absolutely convergent. Finally, using Fubini’s theorem we get

J(x) = 1

C_g

Z ∞

0 ‚Z

R

FΛ(f)(λ)|FΛ(g)(aλ)|2Ψλ(x)dσ(λ)

Œ

d a a

=

Z

R

FΛ(f)(λ)

‚ 1 C_g

Z ∞

0

|FΛ(g)(aλ)|2

d a a

Œ

Ψ_λ(x)dσ(λ)

=

Z

R

FΛ(f)(λ)Ψλ(x)dσ(λ),

which ends the proof in view of Theorem 1.

4. Inversion of the Intertwining Operators Using Generalized Wavelets

In this section we suppose that the function|c(λ)|−2 isC∞on]0,∞[, and for alln∈N_: (i) dn/dλn|c(λ)|−26=0 on]0,∞[;

(ii) ∃p_n∈N_{andkn>0 such thatd}n_/dλn|_c(λ)|−2≤_knλpn _forλ≥_1;

Remark 9. These conditions are satisfied in the Dunkl operator case. Proposition 8. The operatorK (resp. M) defined by

K(f) =F_u−1”2π|c(|λ|)|−2F_u(f)— (32) €

resp.M(f) =_FΛ−1”2π_|c(_|λ_|)_|−2_FΛ(f)—Š (33) is a topological automorphism ofW(R_)(resp.B₍R_)).

Proof. Clearly, the mapping f 7→ 2π|c(λ)|−2f is a topological automorphism of H(R_), and its inverse is given by f 7−→ ₂1_π|c(_|λ_|)_|2f. We deduce the result from Theorem 2 and the fact that the usual Fourier transformF_u is a topological isomorphism fromW(R_)ontoH₍R_). Proposition 9. For f inB(R_{),we have}

M(f) =tV−1◦ K ◦tV(f). (34) Proof. By (16), (32) and (33),

M(f) = _FΛ−1”2π_|c(_|λ_|)_|−2_FΛ(f)—

= tV−1◦ F_u−1”2π_|c(_|λ_|)_|−2_Fu◦tV(f)— = tV−1◦ K ◦tV(f).

Proposition 10.

(i) For all f inW(R_{)and g in}S₍R_{),we have}

K(f ∗g) =K(f)∗g. (ii) For all f in_B(R_{)and g in}S₍R_{),we have}

M(f#g) =M(f)#g.

Proof. We have

K(f ∗g) = _Fu−1”2π_|c(_|λ_|)_|−2_Fu(f ∗g)— = F_u−1”2π|c(|λ|)|−2F_u(f)F_u(g)— = ¦_Fu−1”2π_|c(_|λ_|)_|−2_Fu(f)—©_∗g

= _K(f)_∗g

and

= _FΛ−1”2π_|c(_|λ_|)_|−2_FΛ(f)_FΛ(g)— = ¦F_Λ−1”2π|c(|λ|)|−2FΛ(f)

—© #g

= _M(f)#g, which ends the proof.

Theorem 6. 1. The intertwining operator V is a topological isomorphism from W(R) onto

B(R_).

2. We have the following inverse formulas for V andtV : (a) For f ∈ B(R),

f =VK tV(f); (35)

f =_MV tV(f). (36)

(b) For f ∈ W(R),

f =K tV V(f); (37)

f =tVMV(f). (38)

Proof. Let f _{∈ B}(R_{). From (12), (16) and Theorem 1 we have for all x} ∈R_, f(x) =

Z

R

FΛ(f)(λ) Ψλ(x)

dλ

|c(_|λ_|)_|2

= V

‚Z

R

FΛ(f)(λ)eiλ·

dλ

|c(|λ|)|2 Œ

(x)

= V

‚ 1 2π

Z

R

”

2π_|c(_|λ_|)_|−2_Fu◦tV(f)—eiλ·dλ Œ

(x)

= VK tV(f)(x).

This when combined with (34) yields formula (36). By replacing f respectively by V f and t_V−1_{f into formulas (35) and (36), we regain identities (37) and (38). From (35),} Propo-sition 8 and Theorem 3, we deduce that V is a topological isomorphism from W(R_{) onto}

B(R_).

In order to invert the intertwining operatorsV andtV we shall need some technical lemmas. Lemma 6. For all f inW(R_{)and g in}S₍R_{),we have}

V(f ∗ g) =V(f)#tV−1(g). (39) Proof. By using relations (25), (35), (37) and Proposition 10(i) we have

V−1”V(f)#tV−1(g)— = _K tV”V(f)#tV−1(g)— = _K tV V(f)_∗g

= _K tV V(f)_∗g

Definition 5. The classical continuous wavelet transform onR _{is defined for regular functions} by

S_g(f)(a,b) =

Z

R

f(x)g_a0_,b(x)d x, a>0, b∈R_, where

g0_a,b(x):= _p1

ag

_x−b

a

The function g is a classical wavelet onR_{, i.e., a function in L}2(R_{,d x})satisfying the admissibility condition:

0<C_g0=

Z ∞

0

|Fu(g)(aλ)|2 d a

a <∞, for almost allλ_∈R_.

A more complete and detailed discussion of the properties of the classical wavelet trans-form onR_{can be found in[3], from which we have the following inversion formula.}

Theorem 7. Let g ∈ L2(R_{,d x})be a classical wavelet. If both f and Fu(f) are in L1(R,d x) then we have

f(x) = 1

C_g0

Z ∞

0 ‚Z

R

Sg(f)(a,b)g0a,b(x)d b Œ

d a

a2

for almost every x∈R_.

Remark 10. According to (16) and Definitions 3, 5, g_{∈ S}(R_{)is a generalized wavelet, if and} only if,tV(g)is a classical wavelet and we have:

Ct0_V(g)=Cg. (40)

Lemma 7. Let g∈ W(R_{)be real-valued. Then for all f} ∈ S₍R_{)we have}

Φ_VKg(f)(a,b) =MV”S_g tV f(a,·)—(b).

Proof. Notice that VK g = t_V−1_{g by virtue of (35). Further, g is a classical wavelet} according to[3]. So it follows from Remark 10 that VKg∈ B(R_{) is a generalized wavelet} and

C_VKg=C0_g. (41)

Due to (25), (29), (31), (35), (38) and Definition 5 we have

Φ_VKg(f)(a,b) = pa f#(VKg)_ea(b)

= patV−1tV f ∗ tV(VKg)_ea(b) = tV−1tV f ∗Ha tV VK eg(b)

= _MVtV f ∗Ha(eg)(b)

Lemma 8. Let g∈ B(R_{)be real-valued. Then for all f} ∈ W₍R_{),we have} St_{V g}(f)(a,b) =K tV

”

Φ_g(V f)(a,·)—(b).

Proof. Observe that by Remarks 7(iv) and 10,tV g is a classical wavelet. Using (29), (31), (39) and Definition 5 we have

V€St_{V g}(f)(a,·)

Š

(b) = V f ∗Ha tVeg(b)

= pa V f ∗tV(_eg_a)(b)

= pa V(f)#_eg_a(b) = Φ_g(V f)(a,b). Thus

St_{V g}(f)(a,b) = V−1[Φ_g(V f)(a,·)](b)

= K tV[Φ_g(V f)(a,·)](b)

by virtue of (35).

We can now state our main result. Theorem 8.

(i) Let g∈ W(R_{)be real-valued. Then for all f} ∈ S₍R_{)we have}

t_V−1

f(x) = 1

C0 g

Z ∞

0 ‚Z

R

MV[Sg(f)(a,·)](b) (VKg)a,b(x)A(b)d b Œ

d a a2.

(ii) Let g∈ B(R_{)be real-valued. Then for all f} ∈ B₍R_{)we have}

V−1f(x) = 1

C_g

Z ∞

0 ‚Z

R

K t_V[Φ

g(f)(a,·)](b) tV g0a,b(x)d b Œ

d a

a2.

Proof. The result follows by combining Theorems 5, 7, Lemmas 7, 8 and identities (40), (41).

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