Sonine Transform Associated to the Dunkl Kernel on the Real Line
We consider the Dunkl intertwining operator Vα and its dual tVα, we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian Δα and establish inversion formulas for the Dunkl Sonine operator Sα,β and its dual tSα,β. Also, we give a Planch...
Gespeichert in:
| Datum: | 2008 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2008
|
| Schriftenreihe: | Symmetry, Integrability and Geometry: Methods and Applications |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/147997 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Sonine Transform Associated to the Dunkl Kernel on the Real Line / F. Soltani // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 20 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-147997 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1479972019-02-17T01:26:47Z Sonine Transform Associated to the Dunkl Kernel on the Real Line Soltani, F. We consider the Dunkl intertwining operator Vα and its dual tVα, we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian Δα and establish inversion formulas for the Dunkl Sonine operator Sα,β and its dual tSα,β. Also, we give a Plancherel formula for the operator tSα,β. 2008 Article Sonine Transform Associated to the Dunkl Kernel on the Real Line / F. Soltani // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 20 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 43A62; 43A15; 43A32 http://dspace.nbuv.gov.ua/handle/123456789/147997 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We consider the Dunkl intertwining operator Vα and its dual tVα, we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian Δα and establish inversion formulas for the Dunkl Sonine operator Sα,β and its dual tSα,β. Also, we give a Plancherel formula for the operator tSα,β. |
| format |
Article |
| author |
Soltani, F. |
| spellingShingle |
Soltani, F. Sonine Transform Associated to the Dunkl Kernel on the Real Line Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Soltani, F. |
| author_sort |
Soltani, F. |
| title |
Sonine Transform Associated to the Dunkl Kernel on the Real Line |
| title_short |
Sonine Transform Associated to the Dunkl Kernel on the Real Line |
| title_full |
Sonine Transform Associated to the Dunkl Kernel on the Real Line |
| title_fullStr |
Sonine Transform Associated to the Dunkl Kernel on the Real Line |
| title_full_unstemmed |
Sonine Transform Associated to the Dunkl Kernel on the Real Line |
| title_sort |
sonine transform associated to the dunkl kernel on the real line |
| publisher |
Інститут математики НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/147997 |
| citation_txt |
Sonine Transform Associated to the Dunkl Kernel on the Real Line / F. Soltani // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 20 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT soltanif soninetransformassociatedtothedunklkernelontherealline |
| first_indexed |
2025-07-11T03:46:20Z |
| last_indexed |
2025-07-11T03:46:20Z |
| _version_ |
1837320751684780032 |
| fulltext |
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: Fethi.Soltani@fst.rnu.tn
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 tVα, 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 tSα,β . Also, we give a Plancherel formula for the operator tSα,β .
Key words: Dunkl intertwining operator; Dunkl transform; Dunkl Sonine transform; comp-
lex 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)
∫ 1
−1
f(xt)(1− t2)β−α−1(1 + t)|t|2α+1dt,
and its dual tSα,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 = Λ2
α.
– We give inversion formulas for Sα,β and tSα,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 tSα,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 tSα,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
mailto:Fethi.Soltani@fst.rnu.tn
http://www.emis.de/journals/SIGMA/2008/092/
http://www.emis.de/journals/SIGMA/Dunkl_operators.html
2 F. Soltani
In Section 3, we consider the tempered distribution |x|λ for λ ∈ C\{−(`+ 1), ` ∈ N} defined
by
〈|x|λ, ϕ〉 :=
∫
R
|x|λϕ(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,βK1(tSα,β)(g), f = (tSα,β)K2Sα,β(f),
where
K1(f) =
cβ
cα
(−∆α)β−αf, K2(f) =
cβ
cα
(−∆β)β−αf and cα =
1
[2α+1Γ(α+ 1)]2
.
Next, we give the following Plancherel formula for the operator tSα,β:∫
R
|f(x)|2|x|2β+1dx =
∫
R
|K3(tSα,β(f))(y)|2|x|2α+1dy,
where
K3(f) =
√
cβ
cα
(−∆α)(β−α)/2f.
2 The Dunkl intertwining operator and its dual
We consider the Dunkl operator Λα, α ≥ −1/2, associated with the reflection group Z2 on R:
Λαf(x) :=
d
dx
f(x) +
2α+ 1
x
[
f(x)− f(−x)
2
]
. (1)
For α ≥ −1/2 and λ ∈ C, the initial problem:
Λαf(x) = λf(x), f(0) = 1,
has a unique analytic solution Eα(λx) called Dunkl kernel [3, 5] given by
Eα(λx) = =α(λx) +
λx
2(α+ 1)
=α+1(λx),
where
=α(λx) := Γ(α+ 1)
∞∑
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]):
Eα(λx) = aα
∫ 1
−1
eλxt
(
1− t2
)α−1/2(1 + t)dt,
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:
Eα(λx) = aα sgn(x) |x|−(2α+1)
∫ |x|
−|x|
eλy
(
x2 − y2
)α−1/2(x+ y)dy, x 6= 0,
Eα(0) = 1.
We notice that, the Dunkl kernel Eα(λx) can be also expanded in a power series [10] in the
form:
Eα(λx) =
∞∑
n=0
(λx)n
bn(α)
, (2)
where
b2n(α) =
22nn!
Γ(α+ 1)
Γ(n+ 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
C∞-functions on R), by
Vα(f)(x) := aα
∫ 1
−1
f(xt)
(
1− t2
)α−1/2(1 + t)dt,
which can be written as:
Vα(f)(x) = aα sgn(x) |x|−(2α+1)
∫ |x|
−|x|
f(y)
(
x2 − y2
)α−1/2(x+ y)dy, x 6= 0,
Vα(f)(0) = f(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:
Λα(Vα(f)) = Vα
(
d
dx
f
)
, f ∈ E(R).
Let α > −1/2 and we define the dual Dunkl intertwining operator tVα on S(R) (the Schwartz
space on R), by
tVα(f)(x) := aα
∫
|y|≥|x|
sgn(y)
(
y2 − x2
)α−1/2(x+ y)f(y)dy,
which can be written as:
tVα(f)(x) = aα sgn(x)|x|2α+1
∫
|t|≥1
sgn(t)
(
t2 − 1
)α−1/2(1 + t)f(xt)dt.
4 F. Soltani
Proposition 2 (see [19], Theorems 3.2, 3.3).
(i) The operator tVα is a topological automorphism of S(R), and satisfies the transmutation
relation:
tVα(Λαf) =
d
dx
(tVα(f)), f ∈ S(R).
(ii) For all f ∈ E(R) and g ∈ S(R), we have∫
R
Vα(f)(x)g(x)|x|2α+1dx =
∫
R
f(x) tVα(g)(x)dx.
Remark 2 (see [15]).
(i) For α > −1/2 and f ∈ E(R), we can write
Vα(f)(x) = <α(fe)(|x|) +
1
x
<α(Mfo)(|x|),
where
fe(x) =
1
2
(f(x) + f(−x)), fo(x) =
1
2
(f(x)− f(−x)), Mfo(x) = xfo(x),
and <α is the Riemann–Liouville transform (see [17], page 75) given by
<α(fe)(x) := 2aα
∫ 1
0
fe(xt)
(
1− t2
)α−1/2
dt, x ≥ 0.
Thus, we obtain
V −1
α (f)(x) = <−1
α (fe)(|x|) +
1
x
<−1
α (Mfo)(|x|).
Therefore (see also [20], Proposition 2.2), we get
V −1
α (fe)(x) = dα
d
dx
(
d
xdx
)r {
x2r+1
∫ 1
0
fe(xt)
(
1− t2
)r−α−1/2
t2α+1dt
}
,
V −1
α (fo)(x) = dα
(
d
xdx
)r+1 {
x2r+2
∫ 1
0
fo(xt)
(
1− t2
)r−α−1/2
t2α+2dt
}
,
where r = [α+ 1/2] denote the integer part of α+ 1/2, and dα = 2−rπ
Γ(α+1)Γ(r−α+1/2) .
(ii) For α > −1/2 and f ∈ S(R), we can write
tVα(f)(x) = Wα(fe)(|x|) + xWα(M−1fo)(|x|),
where
M−1fo(x) =
1
2x
(f(x)− f(−x)),
and Wα is the Weyl integral transform (see [17, page 85]) given by
Wα(fe)(x) := 2aαx
2α+1
∫ ∞
1
fe(xt)
(
t2 − 1
)α−1/2
tdt, x ≥ 0.
Thus, we obtain
(tVα)−1f(x) = W−1
α (fe)(|x|) + xW−1
α (M−1fo)(|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
Fα(f)(λ) :=
∫
R
Eα(−iλx)f(x)|x|2α+1dx, λ ∈ R.
We notice that F−1/2 agrees with the Fourier transform F that is given by:
F(f)(λ) :=
∫
R
e−iλxf(x) dx, λ ∈ R.
Proposition 3 (see [5]).
(i) For all f ∈ S(R), we have
Fα(Λαf)(λ) = iλFα(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 ◦ tVα(f), f ∈ S(R).
(iii) Fα is a topological automorphism of S(R), and for f ∈ S(R) we have
f(x) = cα
∫
R
Eα(iλx)Fα(f)(λ)|λ|2α+1dλ,
where
cα =
1
[2α+1Γ(α+ 1)]2
.
(iv) The normalized Dunkl transform
√
cαFα extends uniquely to an isometric isomorphism
of L2(R, |x|2α+1dx) onto itself. In particular,∫
R
|f(x)|2|x|2α+1dx = cα
∫
R
|Fα(f)(λ)|2|λ|2α+1dλ.
For T ∈ S ′(R), we define the Dunkl transform Fα(T ) of T , by
〈Fα(T ), ϕ〉 := 〈T,Fα(ϕ)〉, ϕ ∈ S(R). (3)
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
τxf(y) := (Vα)x ⊗ (Vα)y
[
(Vα)−1(f)(x+ y)
]
, y ∈ R.
These operators satisfy for x, y ∈ R and λ ∈ C the following properties:
Eα(λx)Eα(λy) = τx(Eα(λ.))(y), and
Fα(τxf)(λ) = Ek(iλx)Fα(f)(λ), f ∈ S(R).
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
τxf(y) = aα
∫ π
0
[
fe((x, y)θ) + fo((x, y)θ)
x+ y
(x, y)θ
]
[1− sgn(xy) cos θ] sin2α θdθ,
fe(z) = 1
2(f(z) + f(−z)), fo(z) = 1
2(f(z)− f(−z)),
(x, y)θ =
√
x2 + y2 − 2|xy| cos θ.
• The Dunkl convolution product ∗α of two functions f and g in S(R), by
f ∗α g(x) :=
∫
R
τxf(−y)g(y)|y|2α+1dy, x ∈ 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) := 〈T (y), τxf(−y)〉, x ∈ R. (4)
Note that ∗−1/2 agrees with the standard convolution ∗:
T ∗ f(x) := 〈T (y), f(x− y)〉.
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
Eβ(λx) = aα,β
∫ 1
−1
Eα(λxt)
(
1− t2
)β−α−1(1 + t)|t|2α+1dt, (5)
where
aα,β =
Γ(β + 1)
Γ(β − α)Γ(α+ 1)
.
Proof. From (2), we have∫ 1
−1
Eα(λxt)
(
1− t2
)β−α−1(1 + t)|t|2α+1dt =
∞∑
n=0
(λx)n
bn(α)
In(α, β),
where
In(α, β) =
∫ 1
−1
tn
(
1− t2
)β−α−1(1 + t)|t|2α+1dt,
or
I2n(α, β) = 2
∫ 1
0
(
1− t2
)β−α−1
t2n+2α+1dt =
∫ 1
0
(1− y)β−α−1yn+αdy
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
∫ 1
0
(
1− t2
)β−α−1
t2n+2α+3dt = I2n(α+ 1, β + 1).
Thus ∫ 1
−1
Eα(λxt)
(
1− t2
)β−α−1(1 + t)|t|2α+1dt =
Γ(β − α)Γ(α+ 1)
Γ(β + 1)
Eβ(λx),
which gives the desired result. �
Remark 3. We can write the formula (5) by the following
Eβ(λx) = aα,β sgn(x) |x|−(2β+1)
∫ |x|
−|x|
Eα(λy)
(
x2 − y2
)β−α−1(x+ y)|y|2α+1dy, x 6= 0.
Definition 1. Let α, β ∈ ]−1/2,∞[, such that β > α. We define the Dunkl Sonine trans-
form Sα,β on E(R), by
Sα,β(f)(x) := aα,β
∫ 1
−1
f(xt)
(
1− t2
)β−α−1(1 + t)|t|2α+1dt,
which can be written as:
Sα,β(f)(x) = aα,β sgn(x) |x|−(2β+1)
∫ |x|
−|x|
f(y)
(
x2 − y2
)β−α−1(x+ y)|y|2α+1dy, x 6= 0,
Sα,β(f)(0) = f(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 tSα,β on S(R), by
tSα,β(f)(x) := aα,β
∫
|y|≥|x|
sgn(y)
(
y2 − x2
)β−α−1(x+ y)f(y)dy,
which can be written as:
tSα,β(f)(x) = aα,β sgn(x)|x|2(β−α)
∫
|t|≥1
sgn(t)
(
t2 − 1
)β−α−1(t+ 1)f(xt)dt.
Proposition 6.
(i) For all f ∈ E(R) and g ∈ S(R), we have∫
R
Sα,β(f)(x)g(x)|x|2β+1dx =
∫
R
f(x) tSα,β(g)(x)|x|2α+1dx.
(ii) Fβ possesses on S(R) the following decomposition:
Fβ(f) = Fα ◦ tSα,β(f), f ∈ S(R).
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 ◦ V −1
α 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 tSα,β is a topological automorphism of S(R), and satisfies the following
relations:
tSα,β(f) = (tVα)−1 ◦ tVβ(f), f ∈ S(R),
tSα,β(Λβf) = Λα(tSα,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 ◦ V −1
α (f), f ∈ E(R),
Λβ(Sα,β(f)) = Sα,β(Λαf), f ∈ E(R).
Proof. (i) From Proposition 6 (ii), we have
tSα,β(f) = (Fα)−1 ◦ Fβ(f). (7)
Using Proposition 3 (ii), we obtain
tSα,β(f) = (tVα)−1 ◦ tVβ(f), f ∈ S(R). (8)
Thus from Proposition 2 (i),
tSα,β(Λβf) = (tVα)−1 ◦ tVβ(Λβf) = (tVα)−1
(
d
dx
tVβ(f)
)
.
Using the fact that
tVα(Λαf) =
d
dx
(tVα(f)) ⇐⇒ Λα(tVα)−1(f) = (tVα)−1
(
d
dx
f
)
,
we obtain
tSα,β(Λβf) = Λα(tVα)−1( tVβ(f)) = Λα(tSα,β(f)).
(ii) From Proposition 2 (ii), we have∫
R
f(x) tVβ(g)(x)dx =
∫
R
Vβ(f)(x)g(x)|x|2β+1dx.
On other hand, from (8), Proposition 2 (ii) and Proposition 6 (i) we have∫
R
f(x) tVβ(g)(x)dx =
∫
R
f(x) tVα ◦ tSα,β(g)(x)dx =
∫
R
Vα(f)(x) tSα,β(g)(x)|x|2α+1dx
=
∫
R
Sα,β ◦ Vα(f)(x)g(x)|x|2β+1dx.
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)) = ΛβVβ(V −1
α (f)) = Vβ
(
d
dx
V −1
α (f)
)
.
Using the fact that
Λα(Vα(f)) = Vα
(
d
dx
f
)
⇐⇒ V −1
α (Λαf) =
d
dx
V −1
α (f),
we obtain
Λβ(Sα,β(f)) = Vβ ◦ V −1
α (Λαf) = Sα,β(Λαf),
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
〈|x|λ, ϕ〉 :=
∫
R
|x|λϕ(x)dx, ϕ ∈ S(R). (9)
We write
〈|x|λ, ϕ〉 =
∫ ∞
0
xλ[ϕ(x) + ϕ(−x)]dx, ϕ ∈ S(R),
then from [1], we obtain the following result.
Lemma 1. Let ϕ ∈ S(R). The mapping g : λ → 〈|x|λ, ϕ〉 is complex-valued function and has
an analytic extension to C\{−(1 + 2`), ` ∈ N}, with simple poles −(2`+ 1), ` ∈ N and
Res(g,−1− 2`) = 2
ϕ(2`)(0)
(2`)!
.
Proposition 7. Let ϕ ∈ S(R).
(i) The function λ → 〈|x|λ+2α+1, ϕ〉 is analytic on C\{−(2α + 2` + 2), ` ∈ N}, with simple
poles −(2α+ 2`+ 2), ` ∈ N.
(ii) The function λ→ 22α+λ+2Γ(α+1)Γ( 2α+λ+2
2
)
Γ(−λ/2) 〈|x|−(λ+1), ϕ〉 is analytic on C\{−(2α+ 2`+ 2),
` ∈ N}, with simple poles −(2α+ 2`+ 2), ` ∈ N.
(iii) For λ ∈ C\{−(2α+ 2`+ 2), ` ∈ N} we have
Fα
(
|x|λ+2α+1
)
=
22α+λ+2Γ(α+ 1)Γ(2α+λ+2
2 )
Γ(−λ/2)
|x|−(λ+1), in S ′-sense.
(iv) For λ ∈ C\{−(2α+ 2`+ 2), ` ∈ N} we have
|x|λ+2α+1 =
2λ Γ(2α+λ+2
2 )
Γ(α+ 1)Γ(−λ/2)
Fα(|x|−(λ+1)), 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
2 ) has an analytic extension to
C\{−(2α+2`+2), ` ∈ N}, with simple poles −(2α+2`+2), ` ∈ N, and the function λ→ 1
Γ(−λ/2)
has zeros 2`, ` ∈ N. Thus from Lemma 1 we see that
λ→
22α+λ+2Γ(α+ 1)Γ(2α+λ+2
2 )
Γ(−λ/2)
〈|x|−(λ+1), ϕ〉
is analytic on C\{−(2α+ 2`+ 2), ` ∈ N}, with simple poles −(2α+ 2`+ 2), ` ∈ N.
(iii) Let determine the value of Fα(|x|λ+2α+1) in the S ′-sense. We put ψt(x) := e−tx2
, t > 0.
Then ψt ∈ S(R), and from [12]:
Fα(ψt)(x) = Γ(α+ 1)t−(α+1)e−x2/4t, x ∈ R.
Furthermore, for ϕ ∈ S(R) we have∫
R
Fα(ϕ)(x)ψt(x)|x|2α+1dx = Γ(α+ 1)
∫
R
ϕ(x)t−(α+1)e−x2/4t|x|2α+1dx.
Multiplying both sides by t−λ/2−1 and integrating over (0,∞), we obtain for Re(λ) ∈ ]−(2α +
2), 0[:∫
R
Fα(ϕ)(x)|x|λ+2α+1dx =
22α+λ+2Γ(α+ 1)Γ(2α+λ+2
2 )
Γ(−λ/2)
∫
R
ϕ(x)|x|−(λ+1)dx.
This and from (3) we get for Re(λ) ∈ ]−(2α+ 2), 0[:
Fα(|x|λ+2α+1) =
22α+λ+2Γ(α+ 1)Γ(2α+λ+2
2 )
Γ(−λ/2)
|x|−(λ+1).
The result follows by analytic continuation.
(iv) From (iii) we have
|x|λ+2α+1 =
22α+λ+2Γ(α+ 1)Γ(2α+λ+2
2 )
Γ(−λ/2)
F−1
α
(
|x|−(λ+1)
)
.
Using the fact that
〈F−1
α
(
|x|−(λ+1)
)
, ϕ〉 = 〈|x|−(λ+1),F−1
α (ϕ)〉, ϕ ∈ S(R).
By applying (9) and Proposition 3 (iii), we obtain
〈F−1
α (|x|−(λ+1)), ϕ〉 = cα
∫
R
|x|−(λ+1)Fα(ϕ)(−x)dx, ϕ ∈ S(R).
Then
F−1
α
(
|x|−(λ+1)
)
= cαFα
(
|x|−(λ+1)
)
,
which gives the result. �
Definition 3. For λ ∈ C\{−(α+`+1), ` ∈ N}, the complex powers of the Dunkl Laplacian ∆α
are defined for f ∈ S(R) by
(−∆α)λf(x) :=
22λΓ(α+ λ+ 1)
Γ(α+ 1)Γ(−λ)
|x|−(2λ+1) ∗α f(x),
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)λf
)
(x) = |x|2λFα(f)(x).
Proposition 8. For λ ∈ C\{−(α+ `+ 1), ` ∈ N} and f ∈ S(R),
(−∆α)λf(x) = bα(λ)
∫
R
[∫ π
0
(1 + sgn(xy) cos θ)
(x, y)2(λ+α+1)
θ
sin2α θdθ
]
f(y)|y|2α+1dy,
where
bα(λ) =
22λΓ(α+ λ+ 1)√
π Γ(α+ 1/2)Γ(−λ)
, (x, y)θ =
√
x2 + y2 − 2|xy| cos θ.
Proof. From Definition 3, (4) and (9), we have
(−∆α)λf(x) =
22λΓ(α+ λ+ 1)
Γ(α+ 1)Γ(−λ)
〈|y|−(2λ+1), τxf(−y)〉
=
22λΓ(α+ λ+ 1)
Γ(α+ 1)Γ(−λ)
∫
R
|y|−2(λ+α+1)τxf(−y)|y|2α+1dy.
So
(−∆α)λf(x) =
∫
R
τx(|y|−2(λ+α+1))(−y)f(y)|y|2α+1dy.
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∫
R
f(y) yk|y|2α+1dy = 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|λf , 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 tSα,β is a topological isomorphism from Φβ onto Φα.
(iii) For λ ∈ C\{−(α+ `+ 1), ` ∈ N} and f ∈ Φα , the function (−∆α)λf belongs to ∈ Φα,
and
Fα((−∆α)λf)(x) = |x|2λFα(f)(x). (10)
12 F. Soltani
Proof. (i) Let f ∈ Φα, then
(Fα(f))(k)(0) = (−i)k k!
bk(α)
∫
R
f(x)xk|x|2α+1dy = 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
g(k)(0) = (−i)k k!
bk(α)
∫
R
f(x)xk|x|2α+1dy = 0, ∀ k ∈ N.
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∗αf ∈ S ′(R). Moreover
Fα(T|x|2α+1 S∗αf ) = Fα(f)Fα(S).
Let f ∈ Φα and λ ∈ C\{−(α + ` + 1), ` ∈ N}. Consequently, from Definition 3, Proposi-
tion 7 (iv) and (9) we have
Fα(T|x|2α+1(−∆α)λf ) = |x|2λ+2α+1Fα(f) = T|x|2λ+2α+1Fα(f). (11)
On the other hand from (3),
Fα(T|x|2α+1(−∆α)λf ) = T|x|2α+1Fα((−∆α)λf). (12)
From (11) and (12), we obtain
Fα((−∆α)λf) = |x|2λFα(f).
Then by Lemma 2 and (i) we deduce that (−∆α)λf ∈ Φα. �
5 Inversion formulas for Sα,β and tSα,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
K1(f) :=
cβ
cα
F−1
α
(
|λ|2(β−α)Fα(f)
)
=
cβ
cα
(−∆α)β−αf, f ∈ Φα,
K2(f) :=
cβ
cα
F−1
β
(
|λ|2(β−α)Fβ(f)
)
=
cβ
cα
(−∆β)β−αf, f ∈ Φβ ,
K3(f) :=
√
cβ
cα
F−1
α
(
|λ|β−αFα(f)
)
=
√
cβ
cα
(−∆α)(β−α)/2f, f ∈ Φα.
Lemma 3. For all g ∈ Φβ, we have
K1(tSα,β)(g) = (tSα,β)K2(g). (13)
Proof. Let g ∈ Φβ . Using Proposition 6 (ii),
K1(tSα,β)(g) =
cβ
cα
F−1
α
(
|λ|2(β−α)Fβ(g)
)
= (tSα,β)K2(g). �
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,βK1(tSα,β)(g), (b) f = (tSα,β)K2Sα,β(f).
(ii) Plancherel formula: For all f ∈ Φβ we have∫
R
|f(x)|2|x|2β+1dx =
∫
R
|K3(tSα,β(f))(x)|2|x|2α+1dx.
Proof. (i) Let g ∈ Φβ . From Proposition 3 (iii), (6) and Proposition 6 (ii), we obtain
g = cβ
∫
R
Sα,β(Eα(iλ.))Fβ(g)(λ)|λ|2β+1dλ
= cβ Sα,β
[∫
R
Eα(iλ.)Fα ◦ tSα,β(g)(λ)|λ|2β+1dλ
]
=
cβ
cα
Sα,β
[
F−1
α (|λ|2(β−α)Fα ◦ tSα,β(g))
]
.
Thus
g = Sα,βK1(tSα,β)(g), g ∈ Φβ.
From the previous relation and (13), we deduce the relation:
f = (tSα,β)K2Sα,β(f), f ∈ Φα.
(ii) Let f ∈ Φβ. From Proposition 3 (iv) and Proposition 6 (ii), we deduce that∫
R
|f(x)|2|x|2β+1dx = cβ
∫
R
∣∣|λ|β−αFα(tSα,β(f))(λ)
∣∣2|λ|2α+1dλ.
Thus we obtain∫
R
|f(x)|2|x|2β+1dx = cα
∫
R
∣∣Fα
(
K3(tSα,β(f))
)
(λ)
∣∣2|λ|2α+1dλ.
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 tSα,β(g), we obtain
(c) f = K1(tSα,β)Sα,β(f), (d) g = K2Sα,β(tSα,β)(g).
Acknowledgements
The author is very grateful to the referees and editors for many critical comments on this paper.
References
[1] Baccar C., Hamadi N.B., Rachdi L.T., Inversion formulas for Riemann–Liouville transform and its dual
associated with singular partial differential operators, Int. J. Math. Math. Sci. 2006 (2006), Art. ID 86238,
26 pages.
[2] Dunkl C.F., Differential-difference operators associated with reflections groups, Trans. Amer. Math. Soc.
311 (1989), 167–183.
[3] Dunkl C.F., Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213–1227.
14 F. Soltani
[4] Dunkl C.F., Hankel transforms associated to finite reflection groups, Contemp. Math. 138 (1992), 123–138.
[5] de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147–162.
[6] Lapointe L., Vinet L., Exact operator solution of the Calogero–Sutherland model, Comm. Math. Phys. 178
(1996), 425–452, q-alg/9509003.
[7] Lebedev N.N., Special functions and their applications, Dover Publications, Inc., New York, 1972.
[8] Ludwig D., The Radon transform on Euclidean space, Comm. Pure. App. Math. 23 (1966), 49–81.
[9] Nessibi M.M., Rachdi L.T., Trimèche K., Ranges and inversion formulas for spherical mean operator and
its dual, J. Math. Anal. Appl. 196 (1995), 861–884.
[10] Rosenblum M., Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint
Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., Vol. 73, Birkhäuser, Basel,
1994, 369–396, math.CA/9307224.
[11] Rösler M., Bessel-type signed hypergroups on R, in Probability Measures on Groups and Related Struc-
tures, XI (Oberwolfach, 1994), Editors H. Heyer and A. Mukherjea, Oberwolfach, 1994, World Sci. Publ.,
River Edge, NJ, 1995, 292–304.
[12] Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys.
192 (1998), 519–542, q-alg/9703006.
[13] Samko S.G., Hypersingular integrals and their applications, Analytical Methods and Special Functions, Vol. 5,
Taylor & Francis, Ltd., London, 2002.
[14] Solmon D.C., Asymptotic formulas for the dual Radon transform and applications, Math. Z. 195 (1987),
321–343.
[15] Soltani F., Trimèche K., The Dunkl intertwining operator and its dual on R and applications, Preprint,
Faculty of Sciences of Tunis, Tunisia, 2000.
[16] Stein E.M., Singular integrals and differentiability properties of functions, Princeton University Press,
Princeton, 1970.
[17] Trimèche K., Transformation intégrale de Weyl et théorème de Paley–Wiener associés à un opérateur
différentiel singulier sur (0,∞), J. Math. Pures Appl. (9) 60 (1981), 51–98.
[18] Trimèche K., The Dunkl intertwining operator on spaces of functions and distributions and integral repre-
sentation of its dual, Integral Transform. Spec. Funct. 12 (2001), 349–374.
[19] Trimèche K., Paley–Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral
Transform. Spec. Funct. 13 (2002), 17–38.
[20] Xu Y., An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials, Adv. in Appl.
Math. 29 (2002), 328–343.
http://arxiv.org/abs/q-alg/9509003
http://arxiv.org/abs/math.CA/9307224
http://arxiv.org/abs/q-alg/9703006
1 Introduction
2 The Dunkl intertwining operator and its dual
3 The Dunkl Sonine transform
4 Complex powers of \Delta_\alpha
5 Inversion formulas for S_{\alpha,\beta} and ^tS_{\alpha,\beta}
References
|