sigma08-092. 247KB Jun 04 2011 12:10:21 AM
Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 4 (2008), 092, 14 pages
Sonine Transform Associated to the Dunkl Kernel
on the Real Line⋆
Fethi SOLTANI
Department of Mathematics, Faculty of Sciences of Tunis,
Tunis-El Manar University, 2092 Tunis, Tunisia
E-mail: [email protected]
Received June 19, 2008, in final form December 19, 2008; Published online December 26, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/092/
Abstract. We consider the Dunkl intertwining operator Vα and its dual t Vα , we define and
study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of
the Dunkl Laplacian ∆α and establish inversion formulas for the Dunkl Sonine operator Sα,β
and its dual t Sα,β . Also, we give a Plancherel formula for the operator t Sα,β .
Key words: Dunkl intertwining operator; Dunkl transform; Dunkl Sonine transform; complex powers of the Dunkl Laplacian
2000 Mathematics Subject Classification: 43A62; 43A15; 43A32
1
Introduction
In this paper, we consider the Dunkl operator Λα , α > −1/2, associated with the reflection
group Z2 on R. The operators were in general dimension introduced by Dunkl in [2] in connection
with a generalization of the classical theory of spherical harmonics; they play a major role in
various fields of mathematics [3, 4, 5] and also in physical applications [6].
The Dunkl analysis with respect to α ≥ −1/2 concerns the Dunkl operator Λα , the Dunkl
transform Fα and the Dunkl convolution ∗α on R. In the limit case (α = −1/2); Λα , Fα and ∗α
agree with the operator d/dx , the Fourier transform and the standard convolution respectively.
First, we study the Dunkl Sonine operator Sα,β , β > α:
Sα,β (f )(x) :=
Γ(β + 1)
Γ(β − α)Γ(α + 1)
Z
1
−1
f (xt)(1 − t2 )β−α−1 (1 + t)|t|2α+1 dt,
and its dual t Sα,β connected with these operators. Next, we establish for them the same results
as those given in [8, 14] for the Radon transform and its dual; and in [9] for the spherical mean
operator and its dual on R. Especially:
– We define and study the complex powers for the Dunkl Laplacian ∆α = Λ2α .
– We give inversion formulas for Sα,β and t Sα,β associated with integro-differential and
integro-differential-difference operators when applied to some Lizorkin spaces of functions
(see [9, 1, 13]).
– We establish a Plancherel formula for the operator t Sα,β .
The content of this work is the following. In Section 2, we recall some results about the
Dunkl operators. In particular, we give some properties of the operators Sα,β and t Sα,β .
⋆
This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collection
is available at http://www.emis.de/journals/SIGMA/Dunkl operators.html
2
by
F. Soltani
In Section 3, we consider the tempered distribution |x|λ for λ ∈ C\{−(ℓ + 1), ℓ ∈ N} defined
λ
h|x| , ϕi :=
Z
R
|x|λ ϕ(x)dx.
Also we study the complex powers of the Dunkl Laplacian (−∆α )λ , for some complex number λ.
In the classical case when α = −1/2, the complex powers of the usual Laplacian are given in [16].
In Section 4, we give the following inversion formulas:
g = Sα,β K1 (t Sα,β )(g),
f = (t Sα,β )K2 Sα,β (f ),
where
K1 (f ) =
cβ
(−∆α )β−α f,
cα
K2 (f ) =
cβ
(−∆β )β−α f
cα
and
cα =
1
[2α+1 Γ(α
+ 1)]2
.
Next, we give the following Plancherel formula for the operator t Sα,β :
Z
Z
2
2β+1
|f (x)| |x|
dx =
|K3 (t Sα,β (f ))(y)|2 |x|2α+1 dy,
R
R
where
K3 (f ) =
2
r
cβ
(−∆α )(β−α)/2 f.
cα
The Dunkl intertwining operator and its dual
We consider the Dunkl operator Λα , α ≥ −1/2, associated with the reflection group Z2 on R:
2α + 1 f (x) − f (−x)
d
f (x) +
.
(1)
Λα f (x) :=
dx
x
2
For α ≥ −1/2 and λ ∈ C, the initial problem:
Λα f (x) = λf (x),
f (0) = 1,
has a unique analytic solution Eα (λx) called Dunkl kernel [3, 5] given by
Eα (λx) = ℑα (λx) +
λx
ℑα+1 (λx),
2(α + 1)
where
ℑα (λx) := Γ(α + 1)
∞
X
n=0
(λx/2)2n
,
n! Γ(n + α + 1)
is the modified spherical Bessel function of order α.
Notice that in the case α = −1/2, we have
Λ−1/2 = d/dx
and
E−1/2 (λx) = eλx .
For λ ∈ C and x ∈ R, the Dunkl kernel Eα has the following Bochner-type representation
(see [3, 11]):
Z 1
α−1/2
eλxt 1 − t2
Eα (λx) = aα
(1 + t)dt,
−1
Sonine Transform Associated to the Dunkl Kernel on the Real Line
3
where
aα = √
Γ(α + 1)
,
π Γ(α + 1/2)
which can be written as:
−(2α+1)
Eα (λx) = aα sgn(x) |x|
|x|
Z
−|x|
Eα (0) = 1.
eλy x2 − y 2
α−1/2
(x + y)dy,
x 6= 0,
We notice that, the Dunkl kernel Eα (λx) can be also expanded in a power series [10] in the
form:
Eα (λx) =
∞
X
(λx)n
n=0
bn (α)
,
(2)
where
b2n (α) =
22n n!
Γ(n + α + 1),
Γ(α + 1)
b2n+1 (α) = 2(α + 1)b2n (α + 1).
Let α > −1/2 and we define the Dunkl intertwining operator Vα on E(R) (the space of
on R), by
C ∞ -functions
Vα (f )(x) := aα
1
Z
−1
f (xt) 1 − t2
which can be written as:
−(2α+1)
Vα (f )(x) = aα sgn(x) |x|
α−1/2
Z
|x|
−|x|
Vα (f )(0) = f (0).
(1 + t)dt,
f (y) x2 − y 2
α−1/2
(x + y)dy,
x 6= 0,
Remark 1. For α > −1/2, we have
Eα (λ.) = Vα (eλ. ),
λ ∈ C.
Proposition 1 (see [18], Theorem 6.3). The operator Vα is a topological automorphism
of E(R), and satisfies the transmutation relation:
d
f ,
f ∈ E(R).
Λα (Vα (f )) = Vα
dx
Let α > −1/2 and we define the dual Dunkl intertwining operator t Vα on S(R) (the Schwartz
space on R), by
Z
α−1/2
t
sgn(y) y 2 − x2
(x + y)f (y)dy,
Vα (f )(x) := aα
|y|≥|x|
which can be written as:
t
2α+1
Vα (f )(x) = aα sgn(x)|x|
Z
|t|≥1
α−1/2
sgn(t) t2 − 1
(1 + t)f (xt)dt.
4
F. Soltani
Proposition 2 (see [19], Theorems 3.2, 3.3).
(i) The operator t Vα is a topological automorphism of S(R), and satisfies the transmutation
relation:
t
Vα (Λα f ) =
d t
( Vα (f )),
dx
f ∈ S(R).
(ii) For all f ∈ E(R) and g ∈ S(R), we have
Z
Z
2α+1
Vα (f )(x)g(x)|x|
dx =
f (x) t Vα (g)(x)dx.
R
R
Remark 2 (see [15]).
(i) For α > −1/2 and f ∈ E(R), we can write
Vα (f )(x) = ℜα (fe )(|x|) +
1
ℜα (M fo )(|x|),
x
where
1
fe (x) = (f (x) + f (−x)),
2
1
fo (x) = (f (x) − f (−x)),
2
M fo (x) = xfo (x),
and ℜα is the Riemann–Liouville transform (see [17], page 75) given by
Z 1
α−1/2
fe (xt) 1 − t2
dt,
x ≥ 0.
ℜα (fe )(x) := 2aα
0
Thus, we obtain
Vα−1 (f )(x) = ℜ−1
α (fe )(|x|) +
1 −1
ℜ (M fo )(|x|).
x α
Therefore (see also [20], Proposition 2.2), we get
Z 1
d r
d
2r+1
2 r−α−1/2 2α+1
−1
x
fe (xt) 1 − t
t
dt ,
Vα (fe )(x) = dα
dx xdx
0
Z 1
d r+1
−1
2 r−α−1/2 2α+2
2r+2
Vα (fo )(x) = dα
fo (xt) 1 − t
t
dt ,
x
xdx
0
where r = [α + 1/2] denote the integer part of α + 1/2, and dα =
(ii) For α > −1/2 and f ∈ S(R), we can write
t
2−r π
Γ(α+1)Γ(r−α+1/2) .
Vα (f )(x) = Wα (fe )(|x|) + xWα (M −1 fo )(|x|),
where
M −1 fo (x) =
1
(f (x) − f (−x)),
2x
and Wα is the Weyl integral transform (see [17, page 85]) given by
Z ∞
α−1/2
2α+1
fe (xt) t2 − 1
tdt,
x ≥ 0.
Wα (fe )(x) := 2aα x
1
Thus, we obtain
(t Vα )−1 f (x) = Wα−1 (fe )(|x|) + xWα−1 (M −1 fo )(|x|).
Sonine Transform Associated to the Dunkl Kernel on the Real Line
5
The Dunkl kernel gives rise to an integral transform, called Dunkl transform on R, which
was introduced by Dunkl in [4], where already many basic properties were established. Dunkl’s
results were completed and extended later on by de Jeu in [5].
The Dunkl transform of a function f ∈ S(R), is given by
Z
Fα (f )(λ) :=
Eα (−iλx)f (x)|x|2α+1 dx,
λ ∈ R.
R
We notice that F−1/2 agrees with the Fourier transform F that is given by:
F(f )(λ) :=
Z
e−iλx f (x) dx,
λ ∈ R.
R
Proposition 3 (see [5]).
(i) For all f ∈ S(R), we have
Fα (Λα f )(λ) = iλ Fα (f )(λ),
λ ∈ R,
where Λα is the Dunkl operator given by (1).
(ii) Fα possesses on S(R) the following decomposition:
Fα (f ) = F ◦ t Vα (f ),
f ∈ S(R).
(iii) Fα is a topological automorphism of S(R), and for f ∈ S(R) we have
Z
Eα (iλx)Fα (f )(λ)|λ|2α+1 dλ,
f (x) = cα
R
where
cα =
1
[2α+1 Γ(α
+ 1)]2
.
√
(iv) The normalized Dunkl transform cα Fα extends uniquely to an isometric isomorphism
of L2 (R, |x|2α+1 dx) onto itself. In particular,
Z
Z
2
2α+1
|Fα (f )(λ)|2 |λ|2α+1 dλ.
|f (x)| |x|
dx = cα
R
R
For T ∈ S ′ (R), we define the Dunkl transform Fα (T ) of T , by
hFα (T ), ϕi := hT, Fα (ϕ)i,
ϕ ∈ S(R).
Thus the transform Fα extends to a topological automorphism on S ′ (R).
In [19], the author defines:
• The Dunkl translation operators τx , x ∈ R, on E(R), by
y ∈ R.
τx f (y) := (Vα )x ⊗ (Vα )y (Vα )−1 (f )(x + y) ,
These operators satisfy for x, y ∈ R and λ ∈ C the following properties:
Eα (λx)Eα (λy) = τx (Eα (λ.))(y),
and
Fα (τx f )(λ) = Ek (iλx)Fα (f )(λ),
f ∈ S(R).
(3)
6
F. Soltani
Proposition 4 (see [11]). If f ∈ C(R) (the space of continuous functions on R) and x, y ∈ R
such that (x, y) 6= (0, 0), then
Z π
x+y
fe ((x, y)θ ) + fo ((x, y)θ )
τx f (y) = aα
[1 − sgn(xy) cos θ] sin2α θdθ,
(x,
y)
θ
0
1
1
fo (z) = 2 (f (z) − f (−z)),
fe (z) = 2 (f (z) + f (−z)),
p
2
2
(x, y)θ = x + y − 2|xy| cos θ.
• The Dunkl convolution product ∗α of two functions f and g in S(R), by
Z
f ∗α g(x) :=
τx f (−y)g(y)|y|2α+1 dy,
x ∈ R.
R
This convolution is associative, commutative in S(R) and satisfies (see [19, Theorem 7.2]):
Fα (f ∗α g) = Fα (f )Fα (g).
For T ∈ S ′ (R) and f ∈ S(R), we define the Dunkl convolution product T ∗α f , by
T ∗α f (x) := hT (y), τx f (−y)i,
x ∈ R.
(4)
Note that ∗−1/2 agrees with the standard convolution ∗:
T ∗ f (x) := hT (y), f (x − y)i.
3
The Dunkl Sonine transform
In this section we study the Dunkl Sonine transform, which also studied by Y. Xu on polynomials
in [20]. For thus we consider the following identity, which is a consequence of Xu’s result when
we extend the result of Lemma 2.1 on E(R).
Proposition 5. Let α, β ∈ ]−1/2, ∞[, such that β > α. Then
Z 1
β−α−1
Eα (λxt) 1 − t2
(1 + t)|t|2α+1 dt,
Eβ (λx) = aα,β
−1
where
aα,β =
Γ(β + 1)
.
Γ(β − α)Γ(α + 1)
Proof . From (2), we have
Z
1
−1
Eα (λxt) 1 − t
2 β−α−1
2α+1
(1 + t)|t|
dt =
∞
X
(λx)n
n=0
bn (α)
In (α, β),
where
In (α, β) =
Z
1
t n 1 − t2
−1
or
I2n (α, β) = 2
Z
0
1
1−t
β−α−1
(1 + t)|t|2α+1 dt,
2 β−α−1 2n+2α+1
t
dt =
Z
0
1
(1 − y)β−α−1 y n+α dy
(5)
Sonine Transform Associated to the Dunkl Kernel on the Real Line
=
7
Γ(β − α)Γ(n + α + 1)
,
Γ(n + β + 1)
and
I2n+1 (α, β) = 2
Z
1
1 − t2
0
Thus
Z
1
−1
Eα (λxt) 1 − t2
β−α−1
β−α−1
t2n+2α+3 dt = I2n (α + 1, β + 1).
(1 + t)|t|2α+1 dt =
Γ(β − α)Γ(α + 1)
Eβ (λx),
Γ(β + 1)
which gives the desired result.
Remark 3. We can write the formula (5) by the following
−(2β+1)
Eβ (λx) = aα,β sgn(x) |x|
Z
|x|
−|x|
Eα (λy) x2 − y 2
β−α−1
(x + y)|y|2α+1 dy,
x 6= 0.
Definition 1. Let α, β ∈ ]−1/2, ∞[, such that β > α. We define the Dunkl Sonine transform Sα,β on E(R), by
Sα,β (f )(x) := aα,β
Z
1
−1
f (xt) 1 − t2
which can be written as:
−(2β+1)
Sα,β (f )(x) = aα,β sgn(x) |x|
β−α−1
Z
|x|
−|x|
Sα,β (f )(0) = f (0).
(1 + t)|t|2α+1 dt,
f (y) x2 − y 2
β−α−1
(x + y)|y|2α+1 dy,
x 6= 0,
Remark 4. For α, β ∈ ]−1/2, ∞[, such that β > α, we have
Eβ (λ.) = Sα,β (Eα (λ.)),
λ ∈ C.
(6)
Definition 2. Let α, β ∈ ]−1/2, ∞[, such that β > α. We define the dual Dunkl Sonine
transform t Sα,β on S(R), by
Z
β−α−1
t
sgn(y) y 2 − x2
(x + y)f (y)dy,
Sα,β (f )(x) := aα,β
|y|≥|x|
which can be written as:
t
Sα,β (f )(x) = aα,β sgn(x)|x|2(β−α)
Z
|t|≥1
sgn(t) t2 − 1
β−α−1
Proposition 6.
(i) For all f ∈ E(R) and g ∈ S(R), we have
Z
Z
2β+1
Sα,β (f )(x)g(x)|x|
dx =
f (x) t Sα,β (g)(x)|x|2α+1 dx.
R
R
(ii) Fβ possesses on S(R) the following decomposition:
Fβ (f ) = Fα ◦ t Sα,β (f ),
f ∈ S(R).
(t + 1)f (xt)dt.
8
F. Soltani
Proof . Part (i) follows from Definition 1 by Fubini’s theorem. Then part (ii) follows from (i)
and (6) by taking f = Eα (−iλ.).
In [20, Lemma 2.1] Y. Xu proves the identity Sα,β = Vβ ◦ Vα−1 on polynomials. As the
intertwiner is a homeomorphism on E(R) and polynomials are dense in E(R), this gives the
identity also on E(R). In the following we give a second method to prove this identity.
Theorem 1.
(i) The operator t Sα,β is a topological automorphism of S(R), and satisfies the following
relations:
t
t
Sα,β (f ) = (t Vα )−1 ◦ t Vβ (f ),
f ∈ S(R),
t
Sα,β (Λβ f ) = Λα ( Sα,β (f )),
f ∈ S(R).
(ii) The operator Sα,β is a topological automorphism of E(R), and satisfies the following
relations:
Sα,β (f ) = Vβ ◦ Vα−1 (f ),
Λβ (Sα,β (f )) = Sα,β (Λα f ),
f ∈ E(R),
f ∈ E(R).
Proof . (i) From Proposition 6 (ii), we have
t
Sα,β (f ) = (Fα )−1 ◦ Fβ (f ).
(7)
Using Proposition 3 (ii), we obtain
t
Sα,β (f ) = (t Vα )−1 ◦ t Vβ (f ),
f ∈ S(R).
(8)
Thus from Proposition 2 (i),
t
t
−1
Sα,β (Λβ f ) = ( Vα )
t
t
−1
◦ Vβ (Λβ f ) = ( Vα )
d t
Vβ (f ) .
dx
Using the fact that
t
d t
( Vα (f )) ⇐⇒ Λα (t Vα )−1 (f ) = (t Vα )−1
Vα (Λα f ) =
dx
d
f
dx
,
we obtain
t
Sα,β (Λβ f ) = Λα (t Vα )−1 ( t Vβ (f )) = Λα (t Sα,β (f )).
(ii) From Proposition 2 (ii), we have
Z
Z
t
f (x) Vβ (g)(x)dx =
Vβ (f )(x)g(x)|x|2β+1 dx.
R
R
On other hand, from (8), Proposition 2 (ii) and Proposition 6 (i) we have
Z
Z
Z
t
t
t
f (x) Vβ (g)(x)dx =
f (x) Vα ◦ Sα,β (g)(x)dx =
Vα (f )(x) t Sα,β (g)(x)|x|2α+1 dx
R
R
ZR
2β+1
=
Sα,β ◦ Vα (f )(x)g(x)|x|
dx.
R
Then
Sα,β ◦ Vα (f ) = Vβ (f ).
Sonine Transform Associated to the Dunkl Kernel on the Real Line
9
Hence from Proposition 1,
Λβ (Sα,β (f )) =
Λβ Vβ (Vα−1 (f ))
= Vβ
d −1
V (f ) .
dx α
Using the fact that
Λα (Vα (f )) = Vα
d
f
dx
⇐⇒ Vα−1 (Λα f ) =
d −1
V (f ),
dx α
we obtain
Λβ (Sα,β (f )) = Vβ ◦ Vα−1 (Λα f ) = Sα,β (Λα f ),
which completes the proof of the theorem.
4
Complex powers of ∆α
For λ ∈ C, Re(λ) > −1, we denote by |x|λ the tempered distribution defined by
λ
h|x| , ϕi :=
Z
R
|x|λ ϕ(x)dx,
ϕ ∈ S(R).
(9)
We write
λ
h|x| , ϕi =
Z
∞
xλ [ϕ(x) + ϕ(−x)]dx,
0
ϕ ∈ S(R),
then from [1], we obtain the following result.
Lemma 1. Let ϕ ∈ S(R). The mapping g : λ → h|x|λ , ϕi is complex-valued function and has
an analytic extension to C\{−(1 + 2ℓ), ℓ ∈ N}, with simple poles −(2ℓ + 1), ℓ ∈ N and
Res(g, −1 − 2ℓ) = 2
ϕ(2ℓ) (0)
.
(2ℓ)!
Proposition 7. Let ϕ ∈ S(R).
(i) The function λ → h|x|λ+2α+1 , ϕi is analytic on C\{−(2α + 2ℓ + 2), ℓ ∈ N}, with simple
poles −(2α + 2ℓ + 2), ℓ ∈ N.
22α+λ+2 Γ(α+1)Γ( 2α+λ+2 )
2
(ii) The function λ →
h|x|−(λ+1) , ϕi is analytic on C\{−(2α + 2ℓ + 2),
Γ(−λ/2)
ℓ ∈ N}, with simple poles −(2α + 2ℓ + 2), ℓ ∈ N.
(iii) For λ ∈ C\{−(2α + 2ℓ + 2), ℓ ∈ N} we have
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
) −(λ+1)
2
|x|
,
Fα |x|λ+2α+1 =
Γ(−λ/2)
in S ′ -sense.
(iv) For λ ∈ C\{−(2α + 2ℓ + 2), ℓ ∈ N} we have
|x|λ+2α+1 =
2λ Γ( 2α+λ+2
)
2
Fα (|x|−(λ+1) ),
Γ(α + 1)Γ(−λ/2)
in S ′ -sense.
10
F. Soltani
Proof . (i) Follows directly from Lemma 1.
(ii) From [7, pages 2 and 8] the function λ → Γ( 2α+λ+2
) has an analytic extension to
2
1
C\{−(2α+2ℓ+2), ℓ ∈ N}, with simple poles −(2α+2ℓ+2), ℓ ∈ N, and the function λ → Γ(−λ/2)
has zeros 2ℓ, ℓ ∈ N. Thus from Lemma 1 we see that
λ→
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
)
2
h|x|−(λ+1) , ϕi
Γ(−λ/2)
is analytic on C\{−(2α + 2ℓ + 2), ℓ ∈ N}, with simple poles −(2α + 2ℓ + 2), ℓ ∈ N.
2
(iii) Let determine the value of Fα (|x|λ+2α+1 ) in the S ′ -sense. We put ψt (x) := e−tx , t > 0.
Then ψt ∈ S(R), and from [12]:
Fα (ψt )(x) = Γ(α + 1)t−(α+1) e−x
2 /4t
,
x ∈ R.
Furthermore, for ϕ ∈ S(R) we have
Z
Z
2
2α+1
Fα (ϕ)(x)ψt (x)|x|
dx = Γ(α + 1) ϕ(x)t−(α+1) e−x /4t |x|2α+1 dx.
R
R
Multiplying both sides by t−λ/2−1 and integrating over (0, ∞), we obtain for Re(λ) ∈ ]−(2α +
2), 0[:
Z
Z
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
)
2
Fα (ϕ)(x)|x|λ+2α+1 dx =
ϕ(x)|x|−(λ+1) dx.
Γ(−λ/2)
R
R
This and from (3) we get for Re(λ) ∈ ]−(2α + 2), 0[:
Fα (|x|λ+2α+1 ) =
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
) −(λ+1)
2
|x|
.
Γ(−λ/2)
The result follows by analytic continuation.
(iv) From (iii) we have
|x|λ+2α+1 =
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
) −1
2
Fα |x|−(λ+1) .
Γ(−λ/2)
Using the fact that
hFα−1 |x|−(λ+1) , ϕi = h|x|−(λ+1) , Fα−1 (ϕ)i,
ϕ ∈ S(R).
By applying (9) and Proposition 3 (iii), we obtain
Z
−1
−(λ+1)
|x|−(λ+1) Fα (ϕ)(−x)dx,
hFα (|x|
), ϕi = cα
R
ϕ ∈ S(R).
Then
Fα−1 |x|−(λ+1) = cα Fα |x|−(λ+1) ,
which gives the result.
Definition 3. For λ ∈ C\{−(α + ℓ + 1), ℓ ∈ N}, the complex powers of the Dunkl Laplacian ∆α
are defined for f ∈ S(R) by
(−∆α )λ f (x) :=
22λ Γ(α + λ + 1) −(2λ+1)
|x|
∗α f (x),
Γ(α + 1)Γ(−λ)
where ∗α is the Dunkl convolution product given by (4).
Sonine Transform Associated to the Dunkl Kernel on the Real Line
11
In the next part of this section we use Definition 3 and Proposition 7 (iv) to establish the
following result:
Fα (−∆α )λ f (x) = |x|2λ Fα (f )(x).
Proposition 8. For λ ∈ C\{−(α + ℓ + 1), ℓ ∈ N} and f ∈ S(R),
λ
(−∆α ) f (x) = bα (λ)
Z "Z
R
π
(1 + sgn(xy) cos θ)
2(λ+α+1)
(x, y)θ
0
sin
2α
#
θdθ f (y)|y|2α+1 dy,
where
bα (λ) = √
22λ Γ(α + λ + 1)
,
π Γ(α + 1/2)Γ(−λ)
(x, y)θ =
p
x2 + y 2 − 2|xy| cos θ.
Proof . From Definition 3, (4) and (9), we have
22λ Γ(α + λ + 1)
h|y|−(2λ+1) , τx f (−y)i
Γ(α + 1)Γ(−λ)
Z
22λ Γ(α + λ + 1)
=
|y|−2(λ+α+1) τx f (−y)|y|2α+1 dy.
Γ(α + 1)Γ(−λ) R
(−∆α )λ f (x) =
So
λ
(−∆α ) f (x) =
Z
τx (|y|−2(λ+α+1) )(−y)f (y)|y|2α+1 dy.
R
Then the result follows from Proposition 4.
Note 1. We denote by
• Ψ the subspace of S(R) consisting of functions f , such that
f (k) (0) = 0,
∀ k ∈ N.
• Φα the subspace of S(R) consisting of functions f , such that
Z
R
f (y) y k |y|2α+1 dy = 0,
∀ k ∈ N.
The spaces Ψ and Φ−1/2 are well-known in the literature as Lizorkin spaces (see [1, 9, 13]).
Lemma 2 (see [1]). The multiplication operator Mλ : f → |x|λ f , λ ∈ C, is a topological
automorphism of Ψ. Its inverse operator is (Mλ )−1 = M−λ .
Theorem 2.
(i) The Dunkl transform Fα is a topological isomorphism from Φα onto Ψ.
(ii) The operator t Sα,β is a topological isomorphism from Φβ onto Φα .
(iii) For λ ∈ C\{−(α + ℓ + 1), ℓ ∈ N} and f ∈ Φα , the function (−∆α )λ f belongs to ∈ Φα ,
and
Fα ((−∆α )λ f )(x) = |x|2λ Fα (f )(x).
(10)
12
F. Soltani
Proof . (i) Let f ∈ Φα , then
(k)
(Fα (f ))
k!
(0) = (−i)
bk (α)
k
Z
R
f (x) xk |x|2α+1 dy = 0,
∀ k ∈ N.
Hence Fα (f ) ∈ Ψ.
Conversely, let g ∈ Ψ. Since Fα is a topological automorphism of S(R). There exists
f ∈ S(R), such that Fα (f ) = g. Thus
Z
(k)
k k!
g (0) = (−i)
f (x) xk |x|2α+1 dy = 0,
∀ k ∈ N.
bk (α) R
So f ∈ Φα and Fα (f ) = g.
(ii) follows directly from (i) and (7).
(iii) Similarly to the standard convolution if f ∈ S(R) and S ∈ S ′ (R), then S ∗α f ∈ E(R)
and T|x|2α+1 S∗α f ∈ S ′ (R). Moreover
Fα (T|x|2α+1 S∗α f ) = Fα (f )Fα (S).
Let f ∈ Φα and λ ∈ C\{−(α + ℓ + 1), ℓ ∈ N}. Consequently, from Definition 3, Proposition 7 (iv) and (9) we have
Fα (T|x|2α+1 (−∆α )λ f ) = |x|2λ+2α+1 Fα (f ) = T|x|2λ+2α+1 Fα (f ) .
(11)
On the other hand from (3),
Fα (T|x|2α+1 (−∆α )λ f ) = T|x|2α+1 Fα ((−∆α )λ f ) .
(12)
From (11) and (12), we obtain
Fα ((−∆α )λ f ) = |x|2λ Fα (f ).
Then by Lemma 2 and (i) we deduce that (−∆α )λ f ∈ Φα .
5
Inversion formulas for Sα,β and t Sα,β
In this section, we establish inversion formulas for the Dunkl Sonine transform and its dual.
Definition 4. We define the operators K1 , K2 and K3 , by
cβ
cβ −1
Fα |λ|2(β−α) Fα (f ) =
(−∆α )β−α f,
f ∈ Φα ,
cα
cα
cβ
cβ −1
K2 (f ) :=
Fβ |λ|2(β−α) Fβ (f ) =
(−∆β )β−α f,
f ∈ Φβ ,
cα
cα
r
r
cβ −1
cβ
K3 (f ) :=
Fα |λ|β−α Fα (f ) =
(−∆α )(β−α)/2 f,
f ∈ Φα .
cα
cα
K1 (f ) :=
Lemma 3. For all g ∈ Φβ , we have
K1 (t Sα,β )(g) = (t Sα,β )K2 (g).
(13)
Proof . Let g ∈ Φβ . Using Proposition 6 (ii),
K1 (t Sα,β )(g) =
cβ −1
Fα |λ|2(β−α) Fβ (g) = (t Sα,β )K2 (g).
cα
Sonine Transform Associated to the Dunkl Kernel on the Real Line
13
Theorem 3.
(i) Inversion formulas: For all f ∈ Φα and g ∈ Φβ , we have the inversions formulas:
(a) g = Sα,β K1 (t Sα,β )(g),
(b) f = (t Sα,β )K2 Sα,β (f ).
(ii) Plancherel formula: For all f ∈ Φβ we have
Z
Z
|f (x)|2 |x|2β+1 dx =
|K3 (t Sα,β (f ))(x)|2 |x|2α+1 dx.
R
R
Proof . (i) Let g ∈ Φβ . From Proposition 3 (iii), (6) and Proposition 6 (ii), we obtain
Z
g = cβ
Sα,β (Eα (iλ.)) Fβ (g)(λ)|λ|2β+1 dλ
R
Z
t
2β+1
Eα (iλ.) Fα ◦ Sα,β (g)(λ)|λ|
dλ
= cβ Sα,β
R
h
i
cβ
Sα,β Fα−1 (|λ|2(β−α) Fα ◦ t Sα,β (g)) .
=
cα
Thus
g = Sα,β K1 (t Sα,β )(g),
g ∈ Φβ .
From the previous relation and (13), we deduce the relation:
f = (t Sα,β )K2 Sα,β (f ),
f ∈ Φα .
(ii) Let f ∈ Φβ . From Proposition 3 (iv) and Proposition 6 (ii), we deduce that
Z
Z
2
β−α
2
2β+1
|λ|
Fα (t Sα,β (f ))(λ) |λ|2α+1 dλ.
|f (x)| |x|
dx = cβ
R
R
Thus we obtain
Z
Z
2
2β+1
Fα K3 (t Sα,β (f )) (λ) 2 |λ|2α+1 dλ.
|f (x)| |x|
dx = cα
R
R
Then the result follows from this identity by applying Proposition 3 (iv).
Remark 5. Let f ∈ Φα and g ∈ Φβ . By writing (a) and (b) respectively for the functions
Sα,β (f ) and t Sα,β (g), we obtain
(c) f = K1 (t Sα,β )Sα,β (f ),
(d) g = K2 Sα,β (t Sα,β )(g).
Acknowledgements
The author is very grateful to the referees and editors for many critical comments on this paper.
References
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SIGMA 4 (2008), 092, 14 pages
Sonine Transform Associated to the Dunkl Kernel
on the Real Line⋆
Fethi SOLTANI
Department of Mathematics, Faculty of Sciences of Tunis,
Tunis-El Manar University, 2092 Tunis, Tunisia
E-mail: [email protected]
Received June 19, 2008, in final form December 19, 2008; Published online December 26, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/092/
Abstract. We consider the Dunkl intertwining operator Vα and its dual t Vα , we define and
study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of
the Dunkl Laplacian ∆α and establish inversion formulas for the Dunkl Sonine operator Sα,β
and its dual t Sα,β . Also, we give a Plancherel formula for the operator t Sα,β .
Key words: Dunkl intertwining operator; Dunkl transform; Dunkl Sonine transform; complex powers of the Dunkl Laplacian
2000 Mathematics Subject Classification: 43A62; 43A15; 43A32
1
Introduction
In this paper, we consider the Dunkl operator Λα , α > −1/2, associated with the reflection
group Z2 on R. The operators were in general dimension introduced by Dunkl in [2] in connection
with a generalization of the classical theory of spherical harmonics; they play a major role in
various fields of mathematics [3, 4, 5] and also in physical applications [6].
The Dunkl analysis with respect to α ≥ −1/2 concerns the Dunkl operator Λα , the Dunkl
transform Fα and the Dunkl convolution ∗α on R. In the limit case (α = −1/2); Λα , Fα and ∗α
agree with the operator d/dx , the Fourier transform and the standard convolution respectively.
First, we study the Dunkl Sonine operator Sα,β , β > α:
Sα,β (f )(x) :=
Γ(β + 1)
Γ(β − α)Γ(α + 1)
Z
1
−1
f (xt)(1 − t2 )β−α−1 (1 + t)|t|2α+1 dt,
and its dual t Sα,β connected with these operators. Next, we establish for them the same results
as those given in [8, 14] for the Radon transform and its dual; and in [9] for the spherical mean
operator and its dual on R. Especially:
– We define and study the complex powers for the Dunkl Laplacian ∆α = Λ2α .
– We give inversion formulas for Sα,β and t Sα,β associated with integro-differential and
integro-differential-difference operators when applied to some Lizorkin spaces of functions
(see [9, 1, 13]).
– We establish a Plancherel formula for the operator t Sα,β .
The content of this work is the following. In Section 2, we recall some results about the
Dunkl operators. In particular, we give some properties of the operators Sα,β and t Sα,β .
⋆
This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collection
is available at http://www.emis.de/journals/SIGMA/Dunkl operators.html
2
by
F. Soltani
In Section 3, we consider the tempered distribution |x|λ for λ ∈ C\{−(ℓ + 1), ℓ ∈ N} defined
λ
h|x| , ϕi :=
Z
R
|x|λ ϕ(x)dx.
Also we study the complex powers of the Dunkl Laplacian (−∆α )λ , for some complex number λ.
In the classical case when α = −1/2, the complex powers of the usual Laplacian are given in [16].
In Section 4, we give the following inversion formulas:
g = Sα,β K1 (t Sα,β )(g),
f = (t Sα,β )K2 Sα,β (f ),
where
K1 (f ) =
cβ
(−∆α )β−α f,
cα
K2 (f ) =
cβ
(−∆β )β−α f
cα
and
cα =
1
[2α+1 Γ(α
+ 1)]2
.
Next, we give the following Plancherel formula for the operator t Sα,β :
Z
Z
2
2β+1
|f (x)| |x|
dx =
|K3 (t Sα,β (f ))(y)|2 |x|2α+1 dy,
R
R
where
K3 (f ) =
2
r
cβ
(−∆α )(β−α)/2 f.
cα
The Dunkl intertwining operator and its dual
We consider the Dunkl operator Λα , α ≥ −1/2, associated with the reflection group Z2 on R:
2α + 1 f (x) − f (−x)
d
f (x) +
.
(1)
Λα f (x) :=
dx
x
2
For α ≥ −1/2 and λ ∈ C, the initial problem:
Λα f (x) = λf (x),
f (0) = 1,
has a unique analytic solution Eα (λx) called Dunkl kernel [3, 5] given by
Eα (λx) = ℑα (λx) +
λx
ℑα+1 (λx),
2(α + 1)
where
ℑα (λx) := Γ(α + 1)
∞
X
n=0
(λx/2)2n
,
n! Γ(n + α + 1)
is the modified spherical Bessel function of order α.
Notice that in the case α = −1/2, we have
Λ−1/2 = d/dx
and
E−1/2 (λx) = eλx .
For λ ∈ C and x ∈ R, the Dunkl kernel Eα has the following Bochner-type representation
(see [3, 11]):
Z 1
α−1/2
eλxt 1 − t2
Eα (λx) = aα
(1 + t)dt,
−1
Sonine Transform Associated to the Dunkl Kernel on the Real Line
3
where
aα = √
Γ(α + 1)
,
π Γ(α + 1/2)
which can be written as:
−(2α+1)
Eα (λx) = aα sgn(x) |x|
|x|
Z
−|x|
Eα (0) = 1.
eλy x2 − y 2
α−1/2
(x + y)dy,
x 6= 0,
We notice that, the Dunkl kernel Eα (λx) can be also expanded in a power series [10] in the
form:
Eα (λx) =
∞
X
(λx)n
n=0
bn (α)
,
(2)
where
b2n (α) =
22n n!
Γ(n + α + 1),
Γ(α + 1)
b2n+1 (α) = 2(α + 1)b2n (α + 1).
Let α > −1/2 and we define the Dunkl intertwining operator Vα on E(R) (the space of
on R), by
C ∞ -functions
Vα (f )(x) := aα
1
Z
−1
f (xt) 1 − t2
which can be written as:
−(2α+1)
Vα (f )(x) = aα sgn(x) |x|
α−1/2
Z
|x|
−|x|
Vα (f )(0) = f (0).
(1 + t)dt,
f (y) x2 − y 2
α−1/2
(x + y)dy,
x 6= 0,
Remark 1. For α > −1/2, we have
Eα (λ.) = Vα (eλ. ),
λ ∈ C.
Proposition 1 (see [18], Theorem 6.3). The operator Vα is a topological automorphism
of E(R), and satisfies the transmutation relation:
d
f ,
f ∈ E(R).
Λα (Vα (f )) = Vα
dx
Let α > −1/2 and we define the dual Dunkl intertwining operator t Vα on S(R) (the Schwartz
space on R), by
Z
α−1/2
t
sgn(y) y 2 − x2
(x + y)f (y)dy,
Vα (f )(x) := aα
|y|≥|x|
which can be written as:
t
2α+1
Vα (f )(x) = aα sgn(x)|x|
Z
|t|≥1
α−1/2
sgn(t) t2 − 1
(1 + t)f (xt)dt.
4
F. Soltani
Proposition 2 (see [19], Theorems 3.2, 3.3).
(i) The operator t Vα is a topological automorphism of S(R), and satisfies the transmutation
relation:
t
Vα (Λα f ) =
d t
( Vα (f )),
dx
f ∈ S(R).
(ii) For all f ∈ E(R) and g ∈ S(R), we have
Z
Z
2α+1
Vα (f )(x)g(x)|x|
dx =
f (x) t Vα (g)(x)dx.
R
R
Remark 2 (see [15]).
(i) For α > −1/2 and f ∈ E(R), we can write
Vα (f )(x) = ℜα (fe )(|x|) +
1
ℜα (M fo )(|x|),
x
where
1
fe (x) = (f (x) + f (−x)),
2
1
fo (x) = (f (x) − f (−x)),
2
M fo (x) = xfo (x),
and ℜα is the Riemann–Liouville transform (see [17], page 75) given by
Z 1
α−1/2
fe (xt) 1 − t2
dt,
x ≥ 0.
ℜα (fe )(x) := 2aα
0
Thus, we obtain
Vα−1 (f )(x) = ℜ−1
α (fe )(|x|) +
1 −1
ℜ (M fo )(|x|).
x α
Therefore (see also [20], Proposition 2.2), we get
Z 1
d r
d
2r+1
2 r−α−1/2 2α+1
−1
x
fe (xt) 1 − t
t
dt ,
Vα (fe )(x) = dα
dx xdx
0
Z 1
d r+1
−1
2 r−α−1/2 2α+2
2r+2
Vα (fo )(x) = dα
fo (xt) 1 − t
t
dt ,
x
xdx
0
where r = [α + 1/2] denote the integer part of α + 1/2, and dα =
(ii) For α > −1/2 and f ∈ S(R), we can write
t
2−r π
Γ(α+1)Γ(r−α+1/2) .
Vα (f )(x) = Wα (fe )(|x|) + xWα (M −1 fo )(|x|),
where
M −1 fo (x) =
1
(f (x) − f (−x)),
2x
and Wα is the Weyl integral transform (see [17, page 85]) given by
Z ∞
α−1/2
2α+1
fe (xt) t2 − 1
tdt,
x ≥ 0.
Wα (fe )(x) := 2aα x
1
Thus, we obtain
(t Vα )−1 f (x) = Wα−1 (fe )(|x|) + xWα−1 (M −1 fo )(|x|).
Sonine Transform Associated to the Dunkl Kernel on the Real Line
5
The Dunkl kernel gives rise to an integral transform, called Dunkl transform on R, which
was introduced by Dunkl in [4], where already many basic properties were established. Dunkl’s
results were completed and extended later on by de Jeu in [5].
The Dunkl transform of a function f ∈ S(R), is given by
Z
Fα (f )(λ) :=
Eα (−iλx)f (x)|x|2α+1 dx,
λ ∈ R.
R
We notice that F−1/2 agrees with the Fourier transform F that is given by:
F(f )(λ) :=
Z
e−iλx f (x) dx,
λ ∈ R.
R
Proposition 3 (see [5]).
(i) For all f ∈ S(R), we have
Fα (Λα f )(λ) = iλ Fα (f )(λ),
λ ∈ R,
where Λα is the Dunkl operator given by (1).
(ii) Fα possesses on S(R) the following decomposition:
Fα (f ) = F ◦ t Vα (f ),
f ∈ S(R).
(iii) Fα is a topological automorphism of S(R), and for f ∈ S(R) we have
Z
Eα (iλx)Fα (f )(λ)|λ|2α+1 dλ,
f (x) = cα
R
where
cα =
1
[2α+1 Γ(α
+ 1)]2
.
√
(iv) The normalized Dunkl transform cα Fα extends uniquely to an isometric isomorphism
of L2 (R, |x|2α+1 dx) onto itself. In particular,
Z
Z
2
2α+1
|Fα (f )(λ)|2 |λ|2α+1 dλ.
|f (x)| |x|
dx = cα
R
R
For T ∈ S ′ (R), we define the Dunkl transform Fα (T ) of T , by
hFα (T ), ϕi := hT, Fα (ϕ)i,
ϕ ∈ S(R).
Thus the transform Fα extends to a topological automorphism on S ′ (R).
In [19], the author defines:
• The Dunkl translation operators τx , x ∈ R, on E(R), by
y ∈ R.
τx f (y) := (Vα )x ⊗ (Vα )y (Vα )−1 (f )(x + y) ,
These operators satisfy for x, y ∈ R and λ ∈ C the following properties:
Eα (λx)Eα (λy) = τx (Eα (λ.))(y),
and
Fα (τx f )(λ) = Ek (iλx)Fα (f )(λ),
f ∈ S(R).
(3)
6
F. Soltani
Proposition 4 (see [11]). If f ∈ C(R) (the space of continuous functions on R) and x, y ∈ R
such that (x, y) 6= (0, 0), then
Z π
x+y
fe ((x, y)θ ) + fo ((x, y)θ )
τx f (y) = aα
[1 − sgn(xy) cos θ] sin2α θdθ,
(x,
y)
θ
0
1
1
fo (z) = 2 (f (z) − f (−z)),
fe (z) = 2 (f (z) + f (−z)),
p
2
2
(x, y)θ = x + y − 2|xy| cos θ.
• The Dunkl convolution product ∗α of two functions f and g in S(R), by
Z
f ∗α g(x) :=
τx f (−y)g(y)|y|2α+1 dy,
x ∈ R.
R
This convolution is associative, commutative in S(R) and satisfies (see [19, Theorem 7.2]):
Fα (f ∗α g) = Fα (f )Fα (g).
For T ∈ S ′ (R) and f ∈ S(R), we define the Dunkl convolution product T ∗α f , by
T ∗α f (x) := hT (y), τx f (−y)i,
x ∈ R.
(4)
Note that ∗−1/2 agrees with the standard convolution ∗:
T ∗ f (x) := hT (y), f (x − y)i.
3
The Dunkl Sonine transform
In this section we study the Dunkl Sonine transform, which also studied by Y. Xu on polynomials
in [20]. For thus we consider the following identity, which is a consequence of Xu’s result when
we extend the result of Lemma 2.1 on E(R).
Proposition 5. Let α, β ∈ ]−1/2, ∞[, such that β > α. Then
Z 1
β−α−1
Eα (λxt) 1 − t2
(1 + t)|t|2α+1 dt,
Eβ (λx) = aα,β
−1
where
aα,β =
Γ(β + 1)
.
Γ(β − α)Γ(α + 1)
Proof . From (2), we have
Z
1
−1
Eα (λxt) 1 − t
2 β−α−1
2α+1
(1 + t)|t|
dt =
∞
X
(λx)n
n=0
bn (α)
In (α, β),
where
In (α, β) =
Z
1
t n 1 − t2
−1
or
I2n (α, β) = 2
Z
0
1
1−t
β−α−1
(1 + t)|t|2α+1 dt,
2 β−α−1 2n+2α+1
t
dt =
Z
0
1
(1 − y)β−α−1 y n+α dy
(5)
Sonine Transform Associated to the Dunkl Kernel on the Real Line
=
7
Γ(β − α)Γ(n + α + 1)
,
Γ(n + β + 1)
and
I2n+1 (α, β) = 2
Z
1
1 − t2
0
Thus
Z
1
−1
Eα (λxt) 1 − t2
β−α−1
β−α−1
t2n+2α+3 dt = I2n (α + 1, β + 1).
(1 + t)|t|2α+1 dt =
Γ(β − α)Γ(α + 1)
Eβ (λx),
Γ(β + 1)
which gives the desired result.
Remark 3. We can write the formula (5) by the following
−(2β+1)
Eβ (λx) = aα,β sgn(x) |x|
Z
|x|
−|x|
Eα (λy) x2 − y 2
β−α−1
(x + y)|y|2α+1 dy,
x 6= 0.
Definition 1. Let α, β ∈ ]−1/2, ∞[, such that β > α. We define the Dunkl Sonine transform Sα,β on E(R), by
Sα,β (f )(x) := aα,β
Z
1
−1
f (xt) 1 − t2
which can be written as:
−(2β+1)
Sα,β (f )(x) = aα,β sgn(x) |x|
β−α−1
Z
|x|
−|x|
Sα,β (f )(0) = f (0).
(1 + t)|t|2α+1 dt,
f (y) x2 − y 2
β−α−1
(x + y)|y|2α+1 dy,
x 6= 0,
Remark 4. For α, β ∈ ]−1/2, ∞[, such that β > α, we have
Eβ (λ.) = Sα,β (Eα (λ.)),
λ ∈ C.
(6)
Definition 2. Let α, β ∈ ]−1/2, ∞[, such that β > α. We define the dual Dunkl Sonine
transform t Sα,β on S(R), by
Z
β−α−1
t
sgn(y) y 2 − x2
(x + y)f (y)dy,
Sα,β (f )(x) := aα,β
|y|≥|x|
which can be written as:
t
Sα,β (f )(x) = aα,β sgn(x)|x|2(β−α)
Z
|t|≥1
sgn(t) t2 − 1
β−α−1
Proposition 6.
(i) For all f ∈ E(R) and g ∈ S(R), we have
Z
Z
2β+1
Sα,β (f )(x)g(x)|x|
dx =
f (x) t Sα,β (g)(x)|x|2α+1 dx.
R
R
(ii) Fβ possesses on S(R) the following decomposition:
Fβ (f ) = Fα ◦ t Sα,β (f ),
f ∈ S(R).
(t + 1)f (xt)dt.
8
F. Soltani
Proof . Part (i) follows from Definition 1 by Fubini’s theorem. Then part (ii) follows from (i)
and (6) by taking f = Eα (−iλ.).
In [20, Lemma 2.1] Y. Xu proves the identity Sα,β = Vβ ◦ Vα−1 on polynomials. As the
intertwiner is a homeomorphism on E(R) and polynomials are dense in E(R), this gives the
identity also on E(R). In the following we give a second method to prove this identity.
Theorem 1.
(i) The operator t Sα,β is a topological automorphism of S(R), and satisfies the following
relations:
t
t
Sα,β (f ) = (t Vα )−1 ◦ t Vβ (f ),
f ∈ S(R),
t
Sα,β (Λβ f ) = Λα ( Sα,β (f )),
f ∈ S(R).
(ii) The operator Sα,β is a topological automorphism of E(R), and satisfies the following
relations:
Sα,β (f ) = Vβ ◦ Vα−1 (f ),
Λβ (Sα,β (f )) = Sα,β (Λα f ),
f ∈ E(R),
f ∈ E(R).
Proof . (i) From Proposition 6 (ii), we have
t
Sα,β (f ) = (Fα )−1 ◦ Fβ (f ).
(7)
Using Proposition 3 (ii), we obtain
t
Sα,β (f ) = (t Vα )−1 ◦ t Vβ (f ),
f ∈ S(R).
(8)
Thus from Proposition 2 (i),
t
t
−1
Sα,β (Λβ f ) = ( Vα )
t
t
−1
◦ Vβ (Λβ f ) = ( Vα )
d t
Vβ (f ) .
dx
Using the fact that
t
d t
( Vα (f )) ⇐⇒ Λα (t Vα )−1 (f ) = (t Vα )−1
Vα (Λα f ) =
dx
d
f
dx
,
we obtain
t
Sα,β (Λβ f ) = Λα (t Vα )−1 ( t Vβ (f )) = Λα (t Sα,β (f )).
(ii) From Proposition 2 (ii), we have
Z
Z
t
f (x) Vβ (g)(x)dx =
Vβ (f )(x)g(x)|x|2β+1 dx.
R
R
On other hand, from (8), Proposition 2 (ii) and Proposition 6 (i) we have
Z
Z
Z
t
t
t
f (x) Vβ (g)(x)dx =
f (x) Vα ◦ Sα,β (g)(x)dx =
Vα (f )(x) t Sα,β (g)(x)|x|2α+1 dx
R
R
ZR
2β+1
=
Sα,β ◦ Vα (f )(x)g(x)|x|
dx.
R
Then
Sα,β ◦ Vα (f ) = Vβ (f ).
Sonine Transform Associated to the Dunkl Kernel on the Real Line
9
Hence from Proposition 1,
Λβ (Sα,β (f )) =
Λβ Vβ (Vα−1 (f ))
= Vβ
d −1
V (f ) .
dx α
Using the fact that
Λα (Vα (f )) = Vα
d
f
dx
⇐⇒ Vα−1 (Λα f ) =
d −1
V (f ),
dx α
we obtain
Λβ (Sα,β (f )) = Vβ ◦ Vα−1 (Λα f ) = Sα,β (Λα f ),
which completes the proof of the theorem.
4
Complex powers of ∆α
For λ ∈ C, Re(λ) > −1, we denote by |x|λ the tempered distribution defined by
λ
h|x| , ϕi :=
Z
R
|x|λ ϕ(x)dx,
ϕ ∈ S(R).
(9)
We write
λ
h|x| , ϕi =
Z
∞
xλ [ϕ(x) + ϕ(−x)]dx,
0
ϕ ∈ S(R),
then from [1], we obtain the following result.
Lemma 1. Let ϕ ∈ S(R). The mapping g : λ → h|x|λ , ϕi is complex-valued function and has
an analytic extension to C\{−(1 + 2ℓ), ℓ ∈ N}, with simple poles −(2ℓ + 1), ℓ ∈ N and
Res(g, −1 − 2ℓ) = 2
ϕ(2ℓ) (0)
.
(2ℓ)!
Proposition 7. Let ϕ ∈ S(R).
(i) The function λ → h|x|λ+2α+1 , ϕi is analytic on C\{−(2α + 2ℓ + 2), ℓ ∈ N}, with simple
poles −(2α + 2ℓ + 2), ℓ ∈ N.
22α+λ+2 Γ(α+1)Γ( 2α+λ+2 )
2
(ii) The function λ →
h|x|−(λ+1) , ϕi is analytic on C\{−(2α + 2ℓ + 2),
Γ(−λ/2)
ℓ ∈ N}, with simple poles −(2α + 2ℓ + 2), ℓ ∈ N.
(iii) For λ ∈ C\{−(2α + 2ℓ + 2), ℓ ∈ N} we have
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
) −(λ+1)
2
|x|
,
Fα |x|λ+2α+1 =
Γ(−λ/2)
in S ′ -sense.
(iv) For λ ∈ C\{−(2α + 2ℓ + 2), ℓ ∈ N} we have
|x|λ+2α+1 =
2λ Γ( 2α+λ+2
)
2
Fα (|x|−(λ+1) ),
Γ(α + 1)Γ(−λ/2)
in S ′ -sense.
10
F. Soltani
Proof . (i) Follows directly from Lemma 1.
(ii) From [7, pages 2 and 8] the function λ → Γ( 2α+λ+2
) has an analytic extension to
2
1
C\{−(2α+2ℓ+2), ℓ ∈ N}, with simple poles −(2α+2ℓ+2), ℓ ∈ N, and the function λ → Γ(−λ/2)
has zeros 2ℓ, ℓ ∈ N. Thus from Lemma 1 we see that
λ→
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
)
2
h|x|−(λ+1) , ϕi
Γ(−λ/2)
is analytic on C\{−(2α + 2ℓ + 2), ℓ ∈ N}, with simple poles −(2α + 2ℓ + 2), ℓ ∈ N.
2
(iii) Let determine the value of Fα (|x|λ+2α+1 ) in the S ′ -sense. We put ψt (x) := e−tx , t > 0.
Then ψt ∈ S(R), and from [12]:
Fα (ψt )(x) = Γ(α + 1)t−(α+1) e−x
2 /4t
,
x ∈ R.
Furthermore, for ϕ ∈ S(R) we have
Z
Z
2
2α+1
Fα (ϕ)(x)ψt (x)|x|
dx = Γ(α + 1) ϕ(x)t−(α+1) e−x /4t |x|2α+1 dx.
R
R
Multiplying both sides by t−λ/2−1 and integrating over (0, ∞), we obtain for Re(λ) ∈ ]−(2α +
2), 0[:
Z
Z
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
)
2
Fα (ϕ)(x)|x|λ+2α+1 dx =
ϕ(x)|x|−(λ+1) dx.
Γ(−λ/2)
R
R
This and from (3) we get for Re(λ) ∈ ]−(2α + 2), 0[:
Fα (|x|λ+2α+1 ) =
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
) −(λ+1)
2
|x|
.
Γ(−λ/2)
The result follows by analytic continuation.
(iv) From (iii) we have
|x|λ+2α+1 =
22α+λ+2 Γ(α + 1)Γ( 2α+λ+2
) −1
2
Fα |x|−(λ+1) .
Γ(−λ/2)
Using the fact that
hFα−1 |x|−(λ+1) , ϕi = h|x|−(λ+1) , Fα−1 (ϕ)i,
ϕ ∈ S(R).
By applying (9) and Proposition 3 (iii), we obtain
Z
−1
−(λ+1)
|x|−(λ+1) Fα (ϕ)(−x)dx,
hFα (|x|
), ϕi = cα
R
ϕ ∈ S(R).
Then
Fα−1 |x|−(λ+1) = cα Fα |x|−(λ+1) ,
which gives the result.
Definition 3. For λ ∈ C\{−(α + ℓ + 1), ℓ ∈ N}, the complex powers of the Dunkl Laplacian ∆α
are defined for f ∈ S(R) by
(−∆α )λ f (x) :=
22λ Γ(α + λ + 1) −(2λ+1)
|x|
∗α f (x),
Γ(α + 1)Γ(−λ)
where ∗α is the Dunkl convolution product given by (4).
Sonine Transform Associated to the Dunkl Kernel on the Real Line
11
In the next part of this section we use Definition 3 and Proposition 7 (iv) to establish the
following result:
Fα (−∆α )λ f (x) = |x|2λ Fα (f )(x).
Proposition 8. For λ ∈ C\{−(α + ℓ + 1), ℓ ∈ N} and f ∈ S(R),
λ
(−∆α ) f (x) = bα (λ)
Z "Z
R
π
(1 + sgn(xy) cos θ)
2(λ+α+1)
(x, y)θ
0
sin
2α
#
θdθ f (y)|y|2α+1 dy,
where
bα (λ) = √
22λ Γ(α + λ + 1)
,
π Γ(α + 1/2)Γ(−λ)
(x, y)θ =
p
x2 + y 2 − 2|xy| cos θ.
Proof . From Definition 3, (4) and (9), we have
22λ Γ(α + λ + 1)
h|y|−(2λ+1) , τx f (−y)i
Γ(α + 1)Γ(−λ)
Z
22λ Γ(α + λ + 1)
=
|y|−2(λ+α+1) τx f (−y)|y|2α+1 dy.
Γ(α + 1)Γ(−λ) R
(−∆α )λ f (x) =
So
λ
(−∆α ) f (x) =
Z
τx (|y|−2(λ+α+1) )(−y)f (y)|y|2α+1 dy.
R
Then the result follows from Proposition 4.
Note 1. We denote by
• Ψ the subspace of S(R) consisting of functions f , such that
f (k) (0) = 0,
∀ k ∈ N.
• Φα the subspace of S(R) consisting of functions f , such that
Z
R
f (y) y k |y|2α+1 dy = 0,
∀ k ∈ N.
The spaces Ψ and Φ−1/2 are well-known in the literature as Lizorkin spaces (see [1, 9, 13]).
Lemma 2 (see [1]). The multiplication operator Mλ : f → |x|λ f , λ ∈ C, is a topological
automorphism of Ψ. Its inverse operator is (Mλ )−1 = M−λ .
Theorem 2.
(i) The Dunkl transform Fα is a topological isomorphism from Φα onto Ψ.
(ii) The operator t Sα,β is a topological isomorphism from Φβ onto Φα .
(iii) For λ ∈ C\{−(α + ℓ + 1), ℓ ∈ N} and f ∈ Φα , the function (−∆α )λ f belongs to ∈ Φα ,
and
Fα ((−∆α )λ f )(x) = |x|2λ Fα (f )(x).
(10)
12
F. Soltani
Proof . (i) Let f ∈ Φα , then
(k)
(Fα (f ))
k!
(0) = (−i)
bk (α)
k
Z
R
f (x) xk |x|2α+1 dy = 0,
∀ k ∈ N.
Hence Fα (f ) ∈ Ψ.
Conversely, let g ∈ Ψ. Since Fα is a topological automorphism of S(R). There exists
f ∈ S(R), such that Fα (f ) = g. Thus
Z
(k)
k k!
g (0) = (−i)
f (x) xk |x|2α+1 dy = 0,
∀ k ∈ N.
bk (α) R
So f ∈ Φα and Fα (f ) = g.
(ii) follows directly from (i) and (7).
(iii) Similarly to the standard convolution if f ∈ S(R) and S ∈ S ′ (R), then S ∗α f ∈ E(R)
and T|x|2α+1 S∗α f ∈ S ′ (R). Moreover
Fα (T|x|2α+1 S∗α f ) = Fα (f )Fα (S).
Let f ∈ Φα and λ ∈ C\{−(α + ℓ + 1), ℓ ∈ N}. Consequently, from Definition 3, Proposition 7 (iv) and (9) we have
Fα (T|x|2α+1 (−∆α )λ f ) = |x|2λ+2α+1 Fα (f ) = T|x|2λ+2α+1 Fα (f ) .
(11)
On the other hand from (3),
Fα (T|x|2α+1 (−∆α )λ f ) = T|x|2α+1 Fα ((−∆α )λ f ) .
(12)
From (11) and (12), we obtain
Fα ((−∆α )λ f ) = |x|2λ Fα (f ).
Then by Lemma 2 and (i) we deduce that (−∆α )λ f ∈ Φα .
5
Inversion formulas for Sα,β and t Sα,β
In this section, we establish inversion formulas for the Dunkl Sonine transform and its dual.
Definition 4. We define the operators K1 , K2 and K3 , by
cβ
cβ −1
Fα |λ|2(β−α) Fα (f ) =
(−∆α )β−α f,
f ∈ Φα ,
cα
cα
cβ
cβ −1
K2 (f ) :=
Fβ |λ|2(β−α) Fβ (f ) =
(−∆β )β−α f,
f ∈ Φβ ,
cα
cα
r
r
cβ −1
cβ
K3 (f ) :=
Fα |λ|β−α Fα (f ) =
(−∆α )(β−α)/2 f,
f ∈ Φα .
cα
cα
K1 (f ) :=
Lemma 3. For all g ∈ Φβ , we have
K1 (t Sα,β )(g) = (t Sα,β )K2 (g).
(13)
Proof . Let g ∈ Φβ . Using Proposition 6 (ii),
K1 (t Sα,β )(g) =
cβ −1
Fα |λ|2(β−α) Fβ (g) = (t Sα,β )K2 (g).
cα
Sonine Transform Associated to the Dunkl Kernel on the Real Line
13
Theorem 3.
(i) Inversion formulas: For all f ∈ Φα and g ∈ Φβ , we have the inversions formulas:
(a) g = Sα,β K1 (t Sα,β )(g),
(b) f = (t Sα,β )K2 Sα,β (f ).
(ii) Plancherel formula: For all f ∈ Φβ we have
Z
Z
|f (x)|2 |x|2β+1 dx =
|K3 (t Sα,β (f ))(x)|2 |x|2α+1 dx.
R
R
Proof . (i) Let g ∈ Φβ . From Proposition 3 (iii), (6) and Proposition 6 (ii), we obtain
Z
g = cβ
Sα,β (Eα (iλ.)) Fβ (g)(λ)|λ|2β+1 dλ
R
Z
t
2β+1
Eα (iλ.) Fα ◦ Sα,β (g)(λ)|λ|
dλ
= cβ Sα,β
R
h
i
cβ
Sα,β Fα−1 (|λ|2(β−α) Fα ◦ t Sα,β (g)) .
=
cα
Thus
g = Sα,β K1 (t Sα,β )(g),
g ∈ Φβ .
From the previous relation and (13), we deduce the relation:
f = (t Sα,β )K2 Sα,β (f ),
f ∈ Φα .
(ii) Let f ∈ Φβ . From Proposition 3 (iv) and Proposition 6 (ii), we deduce that
Z
Z
2
β−α
2
2β+1
|λ|
Fα (t Sα,β (f ))(λ) |λ|2α+1 dλ.
|f (x)| |x|
dx = cβ
R
R
Thus we obtain
Z
Z
2
2β+1
Fα K3 (t Sα,β (f )) (λ) 2 |λ|2α+1 dλ.
|f (x)| |x|
dx = cα
R
R
Then the result follows from this identity by applying Proposition 3 (iv).
Remark 5. Let f ∈ Φα and g ∈ Φβ . By writing (a) and (b) respectively for the functions
Sα,β (f ) and t Sα,β (g), we obtain
(c) f = K1 (t Sα,β )Sα,β (f ),
(d) g = K2 Sα,β (t Sα,β )(g).
Acknowledgements
The author is very grateful to the referees and editors for many critical comments on this paper.
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