Dunkl wavelets and applications to inversion of the Dunkl intertwining operator and its dual

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PII. S0161171204212285 http://ijmms.hindawi.com © Hindawi Publishing Corp.

DUNKL WAVELETS AND APPLICATIONS TO INVERSION

OF THE DUNKL INTERTWINING OPERATOR

AND ITS DUAL

ABDELLATIF JOUINI

Received 28 December 2002

We deﬁne and study Dunkl wavelets and the corresponding Dunkl wavelets transforms, and we prove for these transforms Plancherel and reconstruction formulas. We give as applica-tion the inversion of the Dunkl intertwining operator and its dual.

2000 Mathematics Subject Classiﬁcation: 42C15, 44A15.

1. Introduction. We consider the differential-difference operatorΛαonRintroduced by Dunkl [1] and called in the literature Dunkl operator onRof index(α+1/2) associ-ated with the reflection groupZ2, given by

Λαu(x)=du(x)

dx +

α+1/2 x

u(x)−u(−x), α >−1

2. (1.1)

These operators are very important in mathematics and physics. They allow the devel-opment of generalized wavelets from generalized continuous classical wavelet analysis. Moreover, we have proved in [2] that the generalized two-scale equation associated with the Dunkl operator has a solution and then we can deﬁne continuous multiresolution analysis.

Dunkl has proved in [1] that there exists a unique isomorphismVα, called the Dunkl intertwining operator, from the space of polynomials onR of degree nonto itself, satisfying the transmutation relation

ΛαVα=Vα

d

dx, (1.2)

Vα(1)=1. (1.3)

Rösler has proved in [3] that for eachx∈Rthere exists a probability measureµxon Rwith support in the interval[−x, x], such that for all polynomialsponR, we have

Vα(p)(x)=

Rp(y)dµx(y). (1.4)

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that

Vα

−1_{(f )(x)}₌_η x, f

, f∈Ᏹ(R). (1.5)

He has shown also in [5] that the transposed operatort_V

αof the operatorVαhas the integral representation

∀y∈R, tVα(f )(y)=

Rf (x)dνy(x), f∈Ᏸ(R), (1.6)

whereνyis a positive measure onRwith support in the set{x∈R:|x| ≥ |y|}andfin Ᏸ(R), the space ofC∞-functions onRwith compact support, andt_V

αis an isomorphism fromᏰ(R)onto itself, satisfying the relations

∀y∈R, tVαΛαf(y)=

d dy

t_V

α(f )(y), (1.7)

and for eachy∈R, there exists a unique distributionZyin᏿(R), the space of tempered distributions onR, such that

t_V α

−1

(f )(y)=Zy, f

, f∈Ᏸ(R). (1.8)

In this paper, we are interested in Dunkl wavelets and associated Dunkl continuous wavelet transforms. More precisely, we give here a general construction allowing inverse formulas for the Dunkl intertwining operator and its dual.

The contents of this paper are as follows. InSection 2, we deﬁne and study the Dunkl intertwining operator and its dual.Section 3is devoted to Dunkl wavelets and associ-ated Dunkl wavelet transforms. In the last section, we give as application of the previous results inverse formulas for the Dunkl intertwining operator and its dual.

2. The Dunkl intertwining operator onRand its dual. We deﬁne and study in this section the Dunkl intertwining operator onRand its dual and we give their properties.

Notation2.1. We have the following notations:

(i) Da(R)is the space ofC∞-functions onRwith support in the interval[−a, a]; (ii) S(R)is the space ofC∞-functions onR, rapidly decreasing together with their

derivatives;

(iii) ᏿◦(R)is the subspace of᏿(R)consisting of functionsfsuch that

∀n∈N,

Rf (x)x

n_dx₌_0; _(2.1)

(iv) ᏿α

◦(R)is the subspace of᏿(R)consisting of functionsfsuch that

∀n∈N,

Rf (x)mn(x)|x|

2α+1_dx₌₀_, _(2.2)

where

∀x∈R, mn(x)=Vα

_un

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(v) µαis the measure deﬁned onRby

dµα(x)= |

x|2α+1

2α+1_Γ_(α₊₁₎dx; (2.4)

(vi) Lr

α(R), 1≤r≤ +∞, is the space of measurable functionsf onRsuch that

fα,r=

R

_{f (x)}r

dµα(x)

1/r

<+∞,

fα,∞=ess sup x∈R

f (x)<+∞.

(2.5)

Definition2.2. Deﬁne the Dunkl intertwining operatorVαonᏱ(R)by

∀x∈R, Vαf (x)=

      

|x|

−|x|kα(x, y)f (y)dy ifx≠0,

f (0) ifx=0,

(2.6)

where

kα(x, y)=Γ(α+1)|x| −2α−1

√

πΓ(α+1/2)

x2−y2α−1/2|x|+yχ]−|x|,|x|[(y), (2.7)

withχ]−|x|,|x|[the characteristic function of the interval]−|x|,|x|[. Theorem2.3. (i)For allf inᏱ(R),

∀x∈R, Vα(f )(x)=Rαfe(x)+ d

dxRαI

f0(x), (2.8)

wherefe(resp.,f0) is the even (resp., odd) part off,Rαis the Riemann-Liouville integral

operator defined in[5], andIis the operator given by

∀x∈R, If0(x)=

|x|

0 f0(t)dt. (2.9)

(ii) The transformVα is the unique topological isomorphism fromᏱ(R) onto itself,

satisfying

Vα

_d

dyf =ΛαVα(f ), f∈Ᏹ(R), Vα(f )(0)=f (0).

(2.10)

The inverse transformVα−1is given by

∀x∈R, Vα−1(f )(x)=R−α1

fe(x)+ d

dxR

−1 α I

f0(x), (2.11)

whereR−1

α is the inverse operator ofRα.

Letfbe inᏱ(R)andginD(R). The operatort_V

α, deﬁned onD(R)by the relation

RVα(f )(x)g(x)dµα(x)=

Rf (y)

t_V

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is given by

∀y∈R, tVα(f )(y)=

|x|≥|y|kα(x, y)f (x)dµα(x), (2.13)

wherekαis the kernel given by relation (2.7). It is called the dual Dunkl intertwining operator. It has the following properties.

Theorem2.4. For allf inD(R),

∀y∈R, t_V

α(f )(y)=tRαfe(y)+

d dy

t_R

αJf0(y), (2.14)

wherefe (resp.,f0) is the even (resp., odd) part off,tRα is the Weyl integral operator

defined in[4], andJis the operator given by

Jf0

(x)=

x

−∞f0(y)dy, x∈R. (2.15)

Theorem2.5. (i)The transformt_V

αis a topological isomorphism fromD(R)(resp.,

S(R)) onto itself. Moreover,

f∈Da(R)⇐⇒tVα(f )∈Da(R). (2.16)

The inverse transform(t_V

α)−1is given by

∀y∈R, t_V

α−1(f )(y)=tRα−1fe(y)+

d dy

t_R

α−1Jf0(y), (2.17)

where(t_R

α)−1is the inverse operator oftRα.

(ii)The transformt_V

αsatisfies the transmutation relation

∀y∈R, tVα

Λαf(y)=_dyd t_V

α(f )(y), f∈D(R). (2.18)

3. Classical continuous wavelets and Dunkl wavelets

3.1. Classical continuous wavelets onR. We say that a measurable functiongonR is a classical continuous wavelet onRif it satisﬁes, for almost allx∈R, the condition

0< Cc g=

∞

0

_Ᏺ(g)(λx)2dλ

λ <+∞, (3.1)

whereᏲis the classical Fourier transform.

Leta∈]0,+∞[and letgbe a classical wavelet onRinL2₍_R₎_{. We consider the family}

ga,x,x∈R, of classical wavelets onRinL2(R)deﬁned by

ga,x(y)=Ha(g)(x−y), (3.2)

whereHais the dilation operator given by the relation

Ha(f )(x)= 1 af

_x

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We deﬁne, for regular functionsfonR, the classical wavelet transformTgonRby

Tg(f )(a, x)=

Rf (y)ga,x(y)dy ∀x∈R. (3.4)

This transform can also be written in the form

Tg(f )(a, x)=f∗Ha(g)(x), (3.5)

where∗is the classical convolution product.

The transformTghas been studied in [5]. Several properties are given; in particular, if we consider a classical waveletgonRinL2₍_R₎_{, we have the following results.}

(i) Plancherel formula. For allf inL2₍_R₎_{, we have}

R

_{f (x)}2

dx= 1 Cgc

∞

0

R

_T_g_{(f )(a, x)}2

dxda

a . (3.6)

(ii) Inversion formula. For allfinL1₍_R₎_{such that}_Ᏺ_{(f )}_{belongs to}_L1₍_R₎_{, we have}

f (x)= 1 Cgc

∞

0

RTg(f )(a, y)ga,x(y)dy

da

a , a.e., (3.7)

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

3.2. Dunkl wavelets onR. Using the Dunkl transformᏲDand the Dunkl translation operatorsτx,x∈R, we deﬁne and study in this section Dunkl wavelets onRand the Dunkl continuous wavelet transform onR, and we prove for this transform Plancherel and inversion formulas.

Notation3.1. We have the following notations:

(i) σ (x, y, z),ρ(x, y, z), andWα(x, y, z)are the functions deﬁned for allx, y, z∈ R\{0}by

σ (x, y, z)=     

1 2xy

x2₊_y2₋_z2 _if_{x, y}_≠₀_,

0 otherwise,

ρ(x, y, z)= ₁

2(1−σ (x, y, z)+σ (z, x, y)+σ (z, y, x) , Wα(x, y, z)=2α+1Γ(α+1)Kα

|x|,|y|,|z|ρ(x, y, z),

(3.8)

whereKαis the Bessel kernel; (ii) for allx, y∈R,µα

x,yis the measure onRgiven by

dµα x,y(z)=

        

Wα(x, y, z)dµα(z) ifx, y≠0,

δx ify=0,

δy ifx=0,

(3.9)

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(iii) the Dunkl translation operatorsτx,x∈R, are deﬁned onᏱ(R)by

∀y∈R, τxf (y)=

Rf (z)dµ

α

x,y. (3.10)

Definition3.2. A Dunkl wavelet onRis a measurable functiongonRsatisfying, for almost allx∈R, the condition

0< Cg=

∞

0

_Ᏺ_D(g)(λx)2dλ

λ <+∞, (3.11)

where

ᏲD(f )(λ)=

Rf (x)ψ

α

λ(x)dµα(x), λ∈R, (3.12)

andψα_λ(z)is the Dunkl kernel given by

ψα_λ(z)=√ Γ(α+1) πΓ(α+1/2)

1

−1e

−iλzt₍₁₋_t)α−1/2₍₁₊_t)α+1/2_dt, _{λ, z}_∈_C_. _(3.13)

Example3.3. The functionαt,t >0, deﬁned by

∀x∈R, αt=

Ck

(4t)α+1/2e

−x2_/4t

, (3.14)

satisﬁes

∀y∈R, ᏲDαt(y)=e−ty 2

. (3.15)

The functiong(x)= −(d/dt)αt(x)is a Dunkl wavelet onRin᏿(R), andCg=1/8t2.

Proposition3.4. A functiongis a Dunkl wavelet onRin᏿(R)(resp.,᏿◦_◦(R)) if and

only if the functiont_V

α(g)is a classical wavelet onRin᏿(R)(resp.,᏿◦(R)), and

Ct_V_α_(g)=Cg. (3.16)

Proof. The transformᏲDis a topological isomorphism from᏿(R)onto itself, from ᏿α

◦(R)onto᏿0(R). We deduce then these results fromTheorem 2.4.

Leta∈]0,+∞[and letg be a regular function onR. We consider the functionga deﬁned by

∀x∈R, ga(x)= 1 a2α+1g

_x

a . (3.17)

It satisﬁes the following properties: (i) forginL2

α(R), the functiongabelongs toL2α(R)and we have

ᏲD

ga

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(ii) forgin᏿(R)(resp.,᏿α

◦(R)), the functiongabelongs to᏿(R)(resp.,᏿α_◦(R)) and we have

ga=

t_V α

−1

oHaotVα(g). (3.19)

Letgbe a Dunkl wavelet onRinL2

α(R). We consider the familyga,x, x∈R, of Dunkl wavelets onRinL2

α(R)deﬁned by

ga,x(y)=τxga(−y), y∈R, (3.20)

whereτx,x∈R, are the Dunkl translation operators.

Using (3.15), we deduce that the familyga,x,x∈R, given by

∀y∈R, ga,x(y)= −τx

_d

dtαt (−y), (3.21)

is a family of Dunkl wavelets onRin᏿(R).

Definition3.5. Letg be a Dunkl wavelet on Rin L2

α(R). The Dunkl continuous wavelet transformSD

g onRis deﬁned for regular functionsf onRby

SgD(f )(a, x)=

Rf (y)ga,x(y)|y|

2α+1_dy, _{a >}₀_{, x}_∈_R_. _(3.22)

This transform can also be written in the form

SD

g(f )(a, x)=f∗Dga(x), (3.23)

where∗Dis the Dunkl convolution product deﬁned by

∀x∈R, f∗Dg(x)=

Rτxf (−y)g(y)dµα(y). (3.24)

Theorem3.6(Plancherel formula forSD

g). Letgbe a Dunkl wavelet onRinL2α(R).

For allfinL2

α(R),

R

_{f (x)}2

|x|2α+1_dx₌ 1

Cg

_∞

0

R

_SD

g(f )(a, x) 2

|x|2α+1_dxda

a . (3.25)

Proof. The functionf∗Dgsatisﬁes the relationᏲD(f∗Dg)=ᏲD(f )·ᏲD(g). Using Fubini-Tonnelli’s theorem and relations (3.23) and (3.18), we obtain

1 Cg

∞

0

R

SD

g(f )(a, x) 2

|x|2α+1_dxda

a

=_C1

g

∞

0

R

f∗Dga(x) 2

|x|2α+1_dx da

a ,

= 1 Cg

∞

0

R

_Ᏺ_D(f )(y)2ᏲDga(y)2|y|2α+1dy

da a ,

=

R

_Ᏺ_D(f )(x)2

₁

Cg

∞

0

_Ᏺ_D(g)(ay)2da a |y|

2α+1_dy.

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But fromDeﬁnition 3.2, we have for almost ally∈R,

1 Cg

∞

0

_Ᏺ_D(g)(ay)2da

a =1, (3.27)

then

1 Cg

∞

0

R

SD

g(f )(a, x) 2

x2α+1dxda a =

R

_Ᏺ_D(f )(y)2y2α+1dy. (3.28)

We then deduce the relation (3.25).

The following theorem gives an inversion formula for the transformSD g.

Theorem3.7. Letgbe a Dunkl wavelet onRinL2

α(R). ForfinL1α(R)(resp.,L2α(R))

such thatᏲD(f )belongs toL1α(R)(resp.,L1α(R)∩L∞α(R)),

f (x)= 1 Cg

∞

0

RS

D

g(f )(a, y)ga,x(y)|y|2α+1dy

da

a , a.e., (3.29)

where for eachx∈R, both the inner and the outer integrals are absolutely convergent,

but possibly not the double integral.

Proof. We obtain (3.29) by using an analogous proof as for [5, Theorem 6.III.3, page 199].

4. Inversion of the Dunkl intertwining operator and of its dual by using Dunkl wavelets. Using the inversion formulas for the Dunkl continuous wavelet transform SD

g and classical continuous transformSg, we deduce relations which give the inverse operators of the Dunkl intertwining operatorVαand of its dualtVα.

Theorem4.1. (i)Letgbe a Dunkl wavelet onRinᏰ(R)(resp.,᏿(R)). Then for allf

in the same space asg,

∀x∈R, SD

g(f )(a, x)=

t_V

α−1StVα(g) t_V

α(f )(a,·)(x). (4.1)

(ii)Letgbe a Dunkl wavelet onRin᏿◦_◦(R). Then for allfin᏿◦(R),

∀x∈R, St_V_α_(g)(f )(a, x)=V_α−1S_gDVα(f )

(a,·)(x). (4.2)

Proof. We deduce these results from relations (2.8), (3.22), and properties of the Dunkl convolution product.

Theorem4.2. Letgbe a Dunkl wavelet onRin᏿α

◦(R). Then,

(i) for allfin᏿◦_◦(R),

∀x∈R, SgD(f )(a, x)=a−2αVαS_᏷(tVα(g)) t_V

α(f )(a,·)(x), (4.3)

where᏷is the operator given by the relation

∀x∈R, ᏷(f )(x)=Ᏺ−1_π |x|2α+1 2α_Γ_(α₊₁₎Ᏺ(f )

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(ii) for allfin᏿◦(R),

∀x∈R, St_V_α_(g)(f )(a, x)=a−2αtVα

SD

᏷D(g)

Vα(f )

(a,·)(x), (4.5)

where᏷Dis the operator given by the relation

∀x∈R, ᏷D(f )(x)=Ᏺ−1D

π |x|

2α+1

2α_Γ_(α₊₁₎ᏲD(f )

(x). (4.6)

Proof. We obtain these relations fromTheorem 4.1and the fact that

᏷t_V α(g)0a

=a−2α᏷t_V α(g)

0 a, ᏷D

ga

=a−2α᏷D(g)

a.

(4.7)

Theorem4.3. Letgbe a Dunkl wavelet onRin᏿α

◦(R). Then,

(i) for allfin᏿◦_◦(R),

∀x∈R, t_V α

−1_{(f )(x)}

=_C1

g

∞

0

RVα

S_᏷(t_V_α_(g))(f )(a,·)(y)ga,x(y)|y|2α+1dy

da a2α+1;

(4.8)

(ii) for allfin᏿◦(R),

∀x∈R, V_α−1(f )(x)

= 1

C0

tVk(g) ∞

0

R

t_V α

SD

᏷D(g)(f )(a,·)

(y)t_V

α(g)a,x(y)dy

da a2α+1.

(4.9)

Proof. We deduce (4.8) and (4.9) from Theorems4.2,2.4and relation (2.9). Remark4.4. We can establish in a similar way without major changes the results given above for the Dunkl intertwining operator and its dual in the multidimensional case.

References

[1] C. F. Dunkl,Integral kernels with reflection group invariance, Canad. J. Math.43(1991), no. 6, 1213–1227.

[2] A. Jouini,Continuous multiresolution analysis associated with the Dunkl operator onR, Math. Sci. Res. J.7(2003), no. 3, 79–98.

[3] M. Rösler,Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys.192(1998), no. 3, 519–542.

[4] E. M. Stein,Interpolation of linear operators, Trans. Amer. Math. Soc.83(1956), 482–492. [5] K. Trimèche,The Dunkl intertwining operator on spaces of functions and distributions and

integral representation of its dual, Integral Transform. Spec. Funct.12(2001), no. 4, 349–374.

Abdellatif Jouini: Department of Mathematics, Faculty of Sciences, University of Tunis II, 1060 Tunis, Tunisia