The class of Clifford-Fourier transforms

0.32 MB·English

by  H. De Bie

THE CLASS OF CLIFFORD-FOURIER TRANSFORMS
HENDRIKDEBIE,NELEDESCHEPPER,ANDFRANKSOMMEN
1
1
0 Abstract. Recently, there has been an increasing interest in the study of
2 hypercomplex signalsandtheirFouriertransforms. Thispaperaimstostudy
n suchintegraltransformsfromgeneralprinciples,using4differentyetequivalent
a definitions of the classical Fourier transform. This is applied to the so-called
J Clifford-Fourier transform (see [F. Brackx et al., The Clifford-Fourier trans-
form. J.FourierAnal. Appl. 11(2005),669–681]). Theintegralkernelofthis
0
transformisaparticularsolutionofasystemofPDEsinaCliffordalgebra,but
1
is, contrary to the classical Fourier transform, not the unique solution. Here

wedetermineanentireclassofsolutionsofthissystemofPDEs,undercertain
]
A constraints. Foreachsolution,seriesexpressionsintermsofGegenbauerpoly-
nomials and Bessel functions are obtained. Thisallows to compute explicitly
C
theeigenvalues oftheassociatedintegraltransforms. Intheeven-dimensional
. case, this also yields the inverse transform for each of the solutions. Finally,
h
severalpropertiesoftheentireclassofsolutionsareproven.
t
a
m
[

1. Introduction
1
v
Thelastdecades,therehasbeenanincreasinginterestinthetheoryofhypercom-
3
plex signals (i.e. functions taking values in a Clifford algebra) and the possibility
9
7 of defining and using Fourier transforms that interact with the Clifford algebra
1 structure. This has been investigated from a practical engineering point of view
. (see e.g. [8, 14, 15, 16, 18]) but also from a purely mathematical point of view
1
0 (see e.g. [23, 24, 25, 26]) using the function theory of Clifford analysis established
1 in the books [2, 12]. For more references, we refer the reader to the reviewpaper
1 [5]. Also in applications, there is an increasing interest in having available a good
:
v hypercomplex Fourier transform (e.g. in GIS research,see [31]).
i There are severaldrawbacksto most of the kernelsproposedso far in the litera-
X
ture. First,severalauthorsworkonlyinlowdimensions(dimension3or4,enabling
r
a them to use quaternions insteadof a full Cliffordalgebra)which is usually because
they have a specific application in mind in these dimensions. Second, and more
importantly, most authors use ad hoc formulations for the kernel function of their
transforms: they propose very specific kernels, where e.g. the complex unit I is
replaced by a generator of the Clifford algebra. Once the kernel is defined, they
studyindetailallthepropertiesoftherelatedtransform. Fromourperspective,one
should work the other way round, namely start from a list of properties or general
mathematicalprinciples one wants the transformto satisfy, andthen determine all
kernels that satisfy these properties.
Date:January11,2011.
1991 Mathematics Subject Classification. 30G35,42B10.
Key words and phrases. Clifford analysis, Fourier transform, hypercomplex signals, Bessel-
Gegenbauer series.
H.DeBieisaPostdoctoral FellowoftheResearchFoundation-Flanders(FWO).
1
2 HENDRIKDEBIE,NELEDESCHEPPER,ANDFRANKSOMMEN
Forthatreason,themainaimofthispaperistwofold. Firstofall,wewanttouse
general ideas on Fourier transforms borrowedfrom other fields of mathematics (in
casuthe theory of Dunkl operators(see [13]) and double affine Hecke algebras,the
theoryofminimalrepresentations)togivea morestructuralapproachto the study
of hypercomplex Fourier kernels. We do this by formulating 4 different possible
definitions of the classical Fourier transform, and by generalizing these definitions
to the Clifford analysis context.
Secondly,wewanttoapplytheseideastotheso-calledClifford-Fouriertransform
(introduced in [3]). This transform was already based on a Lie algebraic approach
to the classical Fourier transform, although until recently (see [10]) its kernel was
not known in closed form. However, studying this transform using the 4 different
definitions mentioned above provides much more insight in this specific transform,
and allows us to expand it to a whole class of transforms, all of which will satisfy
similar properties (see section 6).
Let us now first give 4 different definitions of the classical Fourier transform,
after which we discuss where they appear in the literature (in different fields of
mathematics) and what their implications are.
The classicalFourier transformin Rm canbe defined in many ways. In its most
basic formulation, it is given by the integral transform
1
F1 [f](y)= e−Ihx,yi f(x) dV(x)
F (2π)m/2 ZRm
withI thecomplexunit, x,y thestandardinnerproductanddV(x)theLebesgue
measure on Rm. Alternathivelyi, one can rewrite the transform as
1
F2 [f](y)= K(x,y) f(x) dV(x)
F (2π)m/2 ZRm
whereK(x,y)is,upto amultiplicativeconstant,the uniquesolutionofthe system
of PDEs
∂ K(x,y)= Ix K(x,y), j =1,...,m.
yj − j
Yet another formulation is given by
F3 =eIπ4meI4π(∆−|x|2)
F
with ∆ the Laplacian in Rm. This expression connects the Fourier transform with
the Lie algebra sl generated by ∆ and x2 and with the theory of the quantum
2
| |
harmonic oscillator. Finally, the kernel can also be expressed as an infinite series
in terms of special functions as (see [32, Section 11.5])
∞
F4 K(x,y)=2λΓ(λ) (k+λ)( I)k(x y )−λJ (x y )Cλ( ξ,η ),
− | || | k+λ | || | k h i
kX=0
where ξ =x/x, η =y/y and λ=(m 2)/2. Here, J is the Bessel function and
ν
| | | | −
Cλ the Gegenbauer polynomial.
k
Eachformulationhasits specific advantagesanduses. The classicalformulation
F1allowstoimmediatelycomputeaboundofthekernelandishenceidealtostudy
the transform on L spaces or more general function spaces.
1
Formulation F2 yields the calculus properties of the transform, and allows to
generalize the transform to e.g. the so-called Dunkl transform (see [11]). This
formulation (defining the kernel as the solution of an eigenvalue problem) is also
THE CLASS OF CLIFFORD-FOURIER TRANSFORMS 3
frequently used in the context of double affine Hecke algebras (see e.g. [9, 29]), an
algebraic generalization of Dunkl operators.
FormulationF3emphasizesthestructural(Liealgebraic)propertiesoftheFourier
transformandalsoallowstocomputeitseigenfunctionsandspectrum. Thisformu-
lation stems from representation theory (see [19, 20]) and has been used in recent
work on minimal representations (see [21, 22] and further generalizations in [1]).
Finally, F4 connects the Fourier transform with the theory of special functions,
and is the ideal formulation to obtain e.g. the Bochner identities (which are a
specialcase of the subsequent Proposition3.1). Similar series representations have
alsobeenusedinthecontextofDunkloperatorsandhaveapplicationsinthestudy
of generalized translation operators (see e.g. [27, 10]).
In [3], F3 was adapted to the case of functions taking values in the Clifford
algebra l to define a couple of Fourier transforms in Clifford analysis by
0,m
C
± =eIπ4me∓I2πΓeI4π(∆−|x|2) =eIπ4meI4π(∆−|x|2∓2Γ)
F
with2Γ=(∂ x x∂ )+m. Here,∂ istheDiracoperatorandxthevectorvariable.
x x x
−
The exponential now contains the generators of the Lie superalgebra osp(12). For
|
several years, the problem remained open to write this Clifford-Fourier transform
as an integral transform and to determine explicitly its kernel. A breakthrough
was obtained in [10], where the kernel was determined in all even dimensions, and
the problem for odd dimensions was reduced to dimension 3 (where an integral
representationof the kernel was obtained).
As a by-product, it was also obtained that the kernel K (x,y) of the integral
+
transform satisfies a system of PDEs, namely
+
F
∂ [K (x,y)]=( I)m K (x, y) c x
y + +
(1.1) − −
[K (x,y)]∂ =( I)m y(cid:0) K (x, y(cid:1)) c,
+ x +
− −
(cid:0) (cid:1)
where c denotes the complex conjugation.
The main aim of this paper is to study this system of PDEs. We will show
that, contrary to the classical formulation F2, this system does not have a unique
solution,butinsteadm 1linearlyindependentsolutionsKi (whenwerestrictto
− +,m
a special subclass of solutions satisfying nice symmetries). Each of these solutions
gives rise to an associated integral transform
1
i [f(x)](y)= Ki (x,y) f(x) dV(x)
F+,m (2π)m/2 ZRm +,m
and we study each of these transforms in-depth. In particular, we determine series
representationsoftheformF4forallrelevantsolutionsof(1.1). Thisinturnallows
us to obtain the spectrum for the associated integral transforms and allows us to
prove the surprising fact that in case of m even
i m−2−i =id.
F+,mF+,m
In other words, for m even we find a complete class of integral transforms, where
the inverseofeachelementis againanelementofthe class. We alsoobtainbounds
onthekernelsKi ,whichallowsustodefinethebroadestfunctionspaceonwhich
+,m
the associated transform is defined (compare with F1). Finally, we prove several
important properties for all the kernels obtained.
One of the strengths ofour results is thatfor the kernels obtainedin this paper,
we obtain always both the F4 and F1 formulation. This is much more than in,
4 HENDRIKDEBIE,NELEDESCHEPPER,ANDFRANKSOMMEN
say, the Dunkl case, where for almost all finite reflection groups the formulation
F1is missingandonehasto use differentandcomplicatedtechniquesto provee.g.
boundedness of the transform.
Thepaperisorganizedasfollows. Insection2werepeatbasicnotionsofClifford
algebras and related differential operators. We give the explicit expression of the
kernel of the Clifford-Fourier transform and of the Fourier-Bessel transform. In
section 3 we prove some generalstatements for kernels expressed as series of prod-
ucts of Gegenbauer polynomials and Bessel functions. In section 4 we study the
Clifford-Fourier system (1.1) in even dimension. We determine an interesting class
of solutions, find recursion relations between these solutions and obtain series ex-
pansions. Wealsodeterminetheeigenvaluesforeachsolution. Insection5wetreat
the case of odd dimension. We omit most proofs in this section, because they are
similarasintheevendimensionalcase. Nevertheless,thiscasehastobeconsidered
separately, because the solutions will now be complex instead of real. Finally, in
section 6, we collect some important properties of the new class of Clifford-Fourier
transformsandprovethe importantfactthatinthe evendimensionalcasealsothe
inverse of each transform is again an element of the same class.
2. Preliminaries
2.1. Cliffordanalysisandspecialfunctions. Cliffordanalysis(seee.g. [12])isa
theorythatoffersanaturalgeneralizationofcomplexanalysistohigherdimensions.
ToRm,theEuclideanspaceinmdimensions,wefirstassociatetheCliffordalgebra
l , generated by the canonical basis e , i = 1,...,m. These generators satisfy
0,m i
C
the multiplication rules e e +e e = 2δ .
i j j i ij
−
The Clifford algebra l can be decomposed as l = m lk with lk
C 0,m C 0,m ⊕k=0C 0,m C 0,m
the space of k-vectors defined by
lk =span e =e ...e ,i <...<i .
C 0,m { i1...ik i1 ik 1 k}
Moreprecisely,we havethatthe space of1-vectorsis givenby l1 =span e ,i=
1,...,m and it is obvious that this space is isomorphic withCR0,mm. The sp{aice of
}
so-called bivectors is given explicitly by l2 =span e =e e ,i<j .
We identify the point (x ,...,x ) inCR0,mmwith th{e ivjectori vjariable}x given by
1 m
x= m x e . The Clifford product of two vectors splits into a scalar part and a
j=1 j j
bivecPtor part:
xy =x.y+x y,
∧
with
m
1
x.y = x,y = x y = (xy+yx)
j j
−h i − 2
Xj=1
and
1
x y = e (x y x y )= (xy yx).
jk j k k j
∧ − 2 −
Xj<k
It is interesting to note that the square of a vector variable x is scalar-valued and
equals the norm squared up to a minus sign: x2 = x,x = x2. Similarly, we
−h i −| |
introduce a first order vector differential operator by
m
∂ = ∂ e .
x xj j
Xj=1
THE CLASS OF CLIFFORD-FOURIER TRANSFORMS 5
Thisoperatoristheso-calledDiracoperator. Itssquareequals,uptoaminussign,
the Laplace operator in Rm: ∂2 = ∆. A function f defined in some open domain
x −
Ω Rm with values in the Clifford algebra l is called monogenic if ∂ (f)=0.
0,m x
⊂ C
AnotherimportantoperatorinCliffordanalysisistheso-calledGammaoperator,
defined by
Γ = x ∂ = e (x ∂ x ∂ ).
x − ∧ x − jk j xk − k xj
Xj<k
This operator is bivector-valued.
Abasis ψ forthespace (Rm) l ,where (Rm)denotestheSchwartz
j,k,ℓ 0,m
{ } S ⊗C S
space, is given by (see [28])
ψ (x):=Lm2+k−1(x2) M(ℓ)(x) e−|x|2/2,
(2.1) 2j,k,ℓ j | | k
ψ (x):=Lm2+k(x2) x M(ℓ)(x) e−|x|2/2,
2j+1,k,ℓ j | | k
wherej,k N,LαaretheLaguerrepolynomialsand M(ℓ) ,(ℓ=1,2,...,dim( ))
∈ j { k } Mk
is a basis for the space . is the space of spherical monogenics of degree k,
k k
M M
i.e. homogeneous polynomial null-solutions of the Dirac operator of degree k.
In the sequel we will frequently need the following well-known properties of
Gegenbauer polynomials (see e.g. [30]):
λ+n
(2.2) Cλ(w)=Cλ+1(w) Cλ+1(w)
λ n n − n−2
and
n n+2λ
(2.3) w Cλ+1(w)= Cλ+1(w)+ Cλ+1(w),
n−1 2(n+λ) n 2(n+λ) n−2
as well as the Bessel function identity
z
(2.4) J (z)= (J (z)+J (z)).
ν ν+1 ν−1
2ν
2.2. The Clifford-Fourier transform. Several attempts have been made to in-
troduce a generalization of the classical Fourier transform F1 to the setting of
Cliffordanalysis(see theintroductionand[5]forareview). Wewillconcentrateon
the so-called Clifford-Fourier transform introduced in [3] by an operator exponen-
tial, similar as the F3 representation of the classical Fourier transform:
± =eIπ4me∓I2πΓeI4π(∆−|x|2).
F
ThisFouriertypetransformcanequivalentlybewrittenasanintegraltransform
1
[f](y)= K (x,y) f(x) dV(x),
F± (2π)m/2 ZRm ±
where the kernel function K±(x,y) is given by the operator exponential e∓Iπ2Γ
acting on the classical Fourier kernel, i.e.
(2.5) K (x,y)=e∓Iπ2Γy e−Ihx,yi .
±
(cid:16) (cid:17)
Similartotheclassicalcase,theClifford-Fouriertransformsatisfiessomecalculus
rules, which translates to the following system of equations satisfied by the kernel:
∂ [K (x,y)]= ( I)m K (x,y) x
y ∓ ±
(2.6) ∓ ±
[K (x,y)]∂ = ( I)m y K (x,y),
± x ∓
± ∓
6 HENDRIKDEBIE,NELEDESCHEPPER,ANDFRANKSOMMEN
where
m
[K (x,y)]∂ = ∂ K (x,y) e
± x xi ± i
Xi=1(cid:0) (cid:1)
denotes the action of the Dirac operator on the right. The system of PDEs (2.6)
should be compared with the formulation F2 of the classical Fourier transform.
Explicit computation of (2.5) is a hard problem. Until recently, the Clifford-
Fourierkernelwasknownexplicitlyonlyinthecasem=2(see[4]);forhighereven
dimensions,acomplicatediterativeprocedureforconstructingthekernelwasgiven
in [7], which could only be used practically in low dimensions. A breakthrough
was obtained in [10]. In this paper it is found that for m even the kernel can be
expressed as follows in terms of a finite sum of Bessel functions:
π 1/2
(2.7) K (x,y)= A(s,t)+B(s,t)+(x y) C(s,t)
+
(cid:16)2(cid:17) (cid:0) ∧ (cid:1)
with
⌊m4−34⌋ 1 Γ m
A(s,t)= sm/2−2−2ℓ 2 J (t)
2ℓℓ!Γ m (cid:0)2ℓ(cid:1) 1 (m−2ℓ−3)/2
Xℓ=0 2 − −
(cid:0) (cid:1) e
⌊m4−12⌋ 1 Γ m
(2.8) B(s,t)= sm/2−1−2ℓ 2 J (t)
2ℓℓ!Γ m(cid:0) (cid:1)2ℓ (m−2ℓ−3)/2
Xℓ=0 2 −
(cid:0) (cid:1) e
⌊m4−12⌋ 1 Γ m
C(s,t)= sm/2−1−2ℓ 2 J (t).
− 2ℓℓ!Γ m(cid:0) (cid:1)2ℓ (m−2ℓ−1)/2
Xℓ=0 2 −
(cid:0) (cid:1) e
Here ℓ denotesthelargestn Nwhichsatisfiesn ℓandthenotationss= x,y ,
⌊ ⌋ ∈ ≤ h i
t = x y = x2 y 2 x,y 2 and J (t) = t−αJ (t) are used. Moreover, it is
α α
| ∧ | | | | | −h i
q
shown that
e
c
(2.9) K (x,y)= K (x, y)
+ −
−
(cid:0) (cid:1)
holds and also that in the case m even, K (x,y) is real-valued, hence in this case
−
the complex conjugation in the above relation can be omitted. Note that the
Clifford-Fourierkernelis parabivector-valued,i.e. it takes the form of a scalarplus
a bivector.
For m odd, the question of determining the kernel explicitly was reduced to the
case of m =3. There, a more or less complicated integral expression of the kernel
was obtained (see [10, Lemma 4.5]). A simple expression as in formula (2.8) is not
known in this case.
Finally, let us mention the action of the Clifford-Fourier transform on the basis
elements (2.1) (see [3]):
[ψ ](y)=( 1)p+k ( 1)k ψ (y)
± 2p,k,ℓ 2p,k,ℓ
(2.10) F − ∓
[ψ ](y)=Im ( 1)p+1 ( 1)k+m−1 ψ (y).
± 2p+1,k,ℓ 2p+1,k,ℓ
F − ∓
2.3. TheFourier-Besseltransform. In[6]anothernewintegraltransformwithin
the Clifford analysis setting was devised, the so-called Fourier-Bessel transform
given by
1
Bessel[f](y)= KBessel(x,y) f(x) dV(x).
F (2π)m/2 ZRm
THE CLASS OF CLIFFORD-FOURIER TRANSFORMS 7
Its integral kernel takes the form
π
(2.11) KBessel(x,y)= ( 1)m/2 J (t)+(x y) J (t) ,
r2 (cid:16) − (m−3)/2 ∧ (m−1)/2 (cid:17)
e e
where we use a different normalization as in [6].
Note thatsimilar to the Clifford-Fourierkernel,it is parabivector-valued. More-
over, the basis elements (2.1) are also eigenfunctions of this transform. The eigen-
values are however quite a bit more complicated in this case. More precisely, we
have for k even
(k 1)!!
Bessel[ψ ](y)=( 1)m/2 ( 1)p − ψ (y)
2p,k,ℓ 2p,k,ℓ
F − − (k+m 3)!!
(2.12) −
(k 1)!!
Bessel[ψ ](y)=( 1)p − ψ (y),
2p+1,k,ℓ 2p+1,k,ℓ
F − (k+m 3)!!
−
while for k odd
k!!
Bessel[ψ ](y)=( 1)p+1 ψ (y)
2p,k,ℓ 2p,k,ℓ
F − (k+m 2)!!
(2.13) −
k!!
Bessel[ψ ](y)=( 1)m/2 ( 1)p ψ (y).
2p+1,k,ℓ 2p+1,k,ℓ
F − − (k+m 2)!!
−
Foru odd, u!!denotes the product: u!!=u(u 2)(u 4)...5 31, while foru even,
− −
u!! stands for the product: u!!=u(u 2)...6 4 2.
−
3. Series approach
In this section we consider a general kernel of the following form
(3.1) K (x,y)=A(w,z)+(x y) B(w,z)
−
∧
with
+∞
A(w,z)= α z−λJ (z)Cλ(w)
k k+λ k
Xk=0
(3.2)
+∞
B(w,z)= β z−λ−1J (z)Cλ+1(w)
k k+λ k−1
Xk=1
and α ,β C. Here, we have introduced the variables z = x y , w = ξ,η
k k
∈ | || | h i
(x = xξ, y = y η, ξ,η Sm−1) and use the notation λ = (m 2)/2. The kernel
| | | | ∈ −
c
K (x,y) is then obtained by the formula K (x,y)= K (x, y) .
+ + −
−
Note that the convergence of the series in (3.2) is n(cid:0)ever a prob(cid:1)lem for the coef-
ficients α and β we will consider. Indeed, we can e.g. estimate
k k
+∞ +∞
z −λ−k z k
α z−λJ (z)Cλ(w) 2−λ α J (z) Cλ(w)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Xk=0 k k+λ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)≤ Xk=0| k+|∞(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)2(cid:17) 1k+λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:16)z2(cid:17)k | k |
2−λλB(λ) α k2λ−1
k
≤ kX=0| |Γ(k+λ+1)(cid:16)2(cid:17)
where we used the estimate
z −λ−k 1
J (z)
(cid:12)(cid:12)(cid:16)2(cid:17) k+λ (cid:12)(cid:12)≤ Γ(k+λ+1)
(cid:12) (cid:12)
(cid:12) (cid:12)
8 HENDRIKDEBIE,NELEDESCHEPPER,ANDFRANKSOMMEN
which follows immediately from the integral representation of the Bessel function
(see [30, (1.71.6)]) and the fact that there exists a constant B(λ) such that
1
sup Cλ(w) B(λ)k2λ−1, k N,
(cid:12)λ k (cid:12)≤ ∀ ∈
−1≤w≤1(cid:12) (cid:12)
(cid:12) (cid:12)
see [1, Lemma 4.9]. We con(cid:12)clude th(cid:12)at if α is a fixed rational function of k (as
k
will be the case in Theorem 4.2 and 5.3), then the series converges absolutely and
uniformly on compacta because of the ratio test.
We define the following two integral transforms
Γ m
[f](y)= 2 K (x,y) f(x) dV(x).
F± 2π(cid:0)m/(cid:1)2 ZRm ±
Nowwecalculatetheactionofthesetransformsonthebasis(2.1)of (Rm) l .
0,m
S ⊗C
Westartwiththefollowingauxiliaryresult,whichisageneralizationoftheBochner
formulas for the classical Fourier transform.
Proposition 3.1. Let M be a spherical monogenic of degree k. Let f(x)=
k k
f (x) be a real-valued radia∈lMfunction in (Rm). Further, put ξ =x/x, η =y/y
0
| | S | | | |
and r = x. Then one has
| |
λ k
[f(x)M (x)](y) = α β M (η)
− k k k k
F (cid:18)λ+k − 2(k+λ) (cid:19)
+∞
rm+k−1f (r)z−λJ (z)dr
0 k+λ
×Z
0
and
λ k+1+2λ
[f(x)xM (x)](y) = α + β η M (η)
− k k+1 k+1 k
F (cid:18)λ+k+1 2(k+1+λ) (cid:19)
+∞
rm+kf (r)z−λJ (z)dr
0 k+1+λ
×Z
0
with z =ry and λ=(m 2)/2.
| | −
Proof. The proof goes along similar lines as the proof of Theorem 6.4 in [10]. (cid:3)
We then have the following theorem.
Theorem 3.2. One has, putting β =0,
0
λ k
[ψ ](y)= α β ( 1)jψ (y)
− 2j,k,ℓ k k 2j,k,ℓ
F (cid:18)λ+k − 2(λ+k) (cid:19) −
(3.3)
λ k+1+2λ
[ψ ](y)= α + β ( 1)jψ (y)
− 2j+1,k,ℓ k+1 k+1 2j+1,k,ℓ
F (cid:18)λ+k+1 2(λ+k+1) (cid:19) −
and
λ k
[ψ ](y)= αc βc ( 1)j+kψ (y)
F+ 2j,k,ℓ (cid:18)λ+k k− 2(λ+k) k(cid:19) − 2j,k,ℓ
λ k+1+2λ
[ψ ](y)= αc + βc ( 1)j+k+1ψ (y).
F+ 2j+1,k,ℓ (cid:18)λ+k+1 k+1 2(λ+k+1) k+1(cid:19) − 2j+1,k,ℓ
THE CLASS OF CLIFFORD-FOURIER TRANSFORMS 9
Proof. This follows from the explicit expression (2.1) of the basis and the identity
(see e.g. [30, exercise 21, p. 371])
+∞
r2λ+1(rs)−λJ (rs)rkLk+λ(r2)e−r2/2dr =( 1)jskLk+λ(s2)e−s2/2.
Z k+λ j − j
0
(cid:3)
We are now able to construct the inverse of on the basis ψ . The
− j,k,ℓ
F { }
construction is similar for .
+
F
Theorem 3.3. The inverse of on the basis ψ is given by
− j,k,ℓ
F { }
Γ m ^
−1[f](y)= 2 K (x,y) f(x) dV(x)
F− 2π(cid:0)m/(cid:1)2 ZRm −
^
with K (x,y)=A(w,z)+(x y) B(w,z) given by
−
∧
+∞
1
A(w,z)= (α +β )z−λJ (z)Cλ(w)
N k k k+λ k
Xk=0
+∞
1
B(w,z)= β z−λ−1J (z)Cλ+1(w),
− N k k+λ k−1
Xk=1
where
λ k λ k+2λ
N = α β α + β .
k k k k
(cid:18)λ+k − 2(λ+k) (cid:19)(cid:18)λ+k 2(λ+k) (cid:19)
^
Proof. Put K (x,y)=A(w,z)+(x y) B(w,z) where
−
∧
+∞
A(w,z)= γ z−λJ (z)Cλ(w)
k k+λ k
Xk=0
+∞
B(w,z)= δ z−λ−1J (z)Cλ+1(w)
k k+λ k−1
Xk=1
and with γ ,δ C. We need to have that
k k
∈
−1 [f] = −1[f] =f.
F− F− F− F−
(cid:2) (cid:3) (cid:2) (cid:3)
Using Theorem 3.2 this condition is equivalent with the system of equations (k =
0,1,...)
λ k λ k
α β γ δ =1
k k k k
(cid:18)λ+k − 2(λ+k) (cid:19)(cid:18)λ+k − 2(λ+k) (cid:19)
λ k+2λ λ k+2λ
α + β γ + δ =1.
k k k k
(cid:18)λ+k 2(λ+k) (cid:19)(cid:18)λ+k 2(λ+k) (cid:19)
Solving this system then yields the statement of the theorem. (cid:3)
Now our aim is to see what restrictionsshould be put on the coefficients α and
k
β such that satisfies the Clifford-Fourier system:
k ±
F
[xf](y)= ( I)m∂ [f](y)
± y ∓
F ∓ ∓ F
∂ [f] (y)= ( I)my (cid:2) [f](y) (cid:3)
± x ∓
F ∓ ∓ F
(cid:2) (cid:3)
10 HENDRIKDEBIE,NELEDESCHEPPER,ANDFRANKSOMMEN
and more specifically
(3.4) (∂ x)[f] (y)= ( I)m(∂ y) [f](y) .
± x y ∓
F − ± ∓ − F
(cid:2) (cid:3) (cid:2) (cid:3)
Now recall that (see [28])
ψ (x)= (−1)j 2−j (∂ x)j M(ℓ)(x) e−r2/2 .
j,k,ℓ j ! x− h k i
2
(cid:4) (cid:5)
We then have, on the one hand, using Theorem 3.2
λ k+1+2λ
[ψ ](y)= αc + βc ( 1)j+k+1 ψ (y)
F+ 2j+1,k,ℓ (cid:18)λ+k+1 k+1 2(λ+k+1) k+1(cid:19) − 2j+1,k,ℓ
and on the other hand, using (3.4)
1
[ψ ](y)= (∂ x)[ψ ] (y)
+ 2j+1,k,ℓ + x 2j,k,ℓ
F −2 F −
(cid:2) (cid:3)
1
= ( I)m (∂ y) [ψ ](y)
y − 2j,k,ℓ
−2 − − F
(cid:2) (cid:3)
λ k
=( I)m α β ( 1)j ψ (y).
k k 2j+1,k,ℓ
− (cid:18)λ+k − 2(λ+k) (cid:19) −
This leads to the following condition on α and β :
k k
k+1+2λ λ+k+1 k
λαc + βc =( I)m( 1)k+1 λα β .
(cid:18) k+1 2 k+1(cid:19) − − λ+k (cid:18) k− 2 k(cid:19)
4. New Clifford-Fourier transforms: the case m even
4.1. Parabivector-valuedsolutionsoftheClifford-Fouriersystem. Theaim
of this section is to solve the Clifford-Fourier system (2.6) in even dimension:
∂ [K+(x,y)]=a K−(x,y) x
y
(4.1)
[K+(x,y)]∂ =a y K−(x,y)
x
with a=( 1)m/2 and K−(x,y)=K+(x, y), see (2.9).
− −
As the even dimensional Clifford-Fourier transform is real-valued, we look for
real-valuedsolutions K+(x,y). Inspired by the expression (2.7), we want to deter-
mine parabivector-valuedsolutions of the form:
K+(x,y)=f(s,t)+(x y) g(s,t)
∧
K−(x,y)=f( s,t) (x y) g( s,t)
− − ∧ −
with s= x,y , t= x y and f and g real-valued functions.
h i | ∧ |
Taking into account that (see e.g. [6])
x(y x)
∂ [s]=x , ∂ [t]= ∧ and ∂ [x y]=(m 1)x,
y y y
t ∧ −
we obtain
∂ [K+(x,y)]=x ∂ [f(s,t)]+(m 1) g(s,t)+t∂ [g(s,t)]
y s t
−
(cid:0) (cid:1)
1
+x(x y) ∂ [g(s,t)] ∂ [f(s,t)] ,
s t
∧ (cid:18) − t (cid:19)