Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 6 Issue 3(2014), Pages 1-30.
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
(COMMUNICATED BY H. M. SRIVASTAVA)
SAMI GHAZOUANI, FETHI BOUZAFFOUR
Abstract. A new fractional version of the Dunkl transform for real order α
is obtained. An integral representation, a Bochner type identity and a Master
formula for this transform are derived.
1. Introduction
Recently, various works have been published that develop the theory of fractional
powers of operators. We mention particularly the fractional versions of classical
integral transform such that Fourier transform and Hankel transform (see[15, 13, 14,
24]). Dunkl theory generalizes classical Fourier analysis on RN . It started twenty
years ago with Dunkl’s seminal work [4] and was further developed by several
mathematicians (see[2, 6, 8, 17]). In this paper, we consider the Dunkl operators
Ti , i, . . . , N , associated with an arbitrary finite reflection group G and a nonnegative
multiplicity function k. These operators are very important and they provide a
useful tool in the study of special functions with root systems. The Dunkl kernel
Ek has been introduced by C.F. Dunkl in [5]. It generalizes the usual exponential
function in many respects, and can be characterized as the solution of a joint
eigenvalue problem for the associated Dunkl operators. For a family of weight
functions ωk invariant under a reflection group G, Dunkl [6] introduced an integral
transform associated with the kernel Ek and proved the Plancherel theorem. In
[2], de Jeu studied the Dunkl transform by completely different method and proved
the inversion formula and the Plancherel theorem. The Dunkl transform, initially
defined on L1 (RN , ωk (x)dx) by
Z
ck
f (y)Ek (−ix, y)ωk (y)dy,
Dk f (x) = γ+N/2
2
RN
extends to an isometry of L2 (RN , ωk (x)dx) and commutes with the reflection group
G. In the setting of general Dunkl’s theory Rösler [17] constructed
systems of natu
rally associated multivariable generalized Hermite polynomials Hν ; ν = (ν1 , . . . , νN ) ∈ ZN
+
2000 Mathematics Subject Classification. 42B10, 42C05, 47D06.
Key words and phrases. Fractional Fourier transform, Dunkl transform, Generalized Hermite
polynomials and functions, semigroups of operators.
c
2014
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted February 21, 2014. Published May 16, 2014.
1
2
S. GHAZOUANI, F. BOUZAFFOUR
and Hermite functions hν ; ν = (ν1 , . . . , νN ) ∈ ZN
+ . He proved that the generalized Hermite functions {hν }ν∈ZN form an orthonormal basis of eigenfunctions for
+
the Dunkl operator on L2 (RN , ωk (x)dx) with Dk (hν ) = (−i)|ν| hν .
This paper deals with the constraction of a fractional power of the Dunkl transform called the fractional Dunkl transform (FDT), using the multivariable generalized Hermite function introduced by Rösler [17]. The resulting family of operators
{Dkα }α∈R was proved to be a C0 -group of unitary operators on L2 RN , ωk (x) dx ,
with infinitesimal generator T. The spectral properties of T is studied using the
semigroup techniques. The FDT given in this paper has an integral representation
which used with the analogue of the Funk-Hecke formula for k-spherical harmonics
[23] to derive a Bochner type identity for the FDT. The Master formula of the FDT
is proved and founded to be generalizing the one given by Rösler [17] in Proposition
3.10. This Master formula is used to develop a new proof of the statement (2) of
the 3.4 Rösler’s theorem [17].
The contents of the present paper are as follows. In section 2, some basic definitions and results about harmonic analysis associated with Dunkl operators are
collected. In section 3, the fractional Dunkl transform definition is given and then
some elementary properties of this transformation are listed. In section 4, the
spectral properties of T is studied. In section 5, the integral representation of the
fractional Dunkl transform as well as the Bochner type identity and Master formula
are given. In section 6, we find a subspace of L2 (RN , ωk (x)dx) in which we define
T explicitly.
2. Background: Dunkl theory
In this section, we recall some notations and results on Dunkl operators, Dunkl
transform, and generalized Hermite functions (see, [4, 5, 2, 16, 19]).
Notation: We denote by Z+ the set of non-negative integers. For a multi-index
ν = (ν1 , . . . , νN ) ∈ ZN
+ , we write |ν| = ν1 + · · · + νN . The C-algebra of polynomial
functions on RN is denoted by P = C[RN ]. It has a natural grading
M
P=
Pn ,
n≥0
where Pn is the subspace of homogenous polynomials of (total) degree n. S(RN ) is
the Schwartz space of rapidly decreasing functions on RN and C0 (RN ) is the space
of continuous functions on RN vanishing at infinity.
2.1. Dunkl operators and Dunkl Kernel. In RN , we consider the standard
inner product
N
X
hx, yi =
xk yk .
k=1
We shall use p
the same notation for its bilinear extension to CN × CN . For x ∈ RN ,
denote |x| = hx, xi.
For u ∈ RN \{0}, let σu be the reflection in the hyperplane (Ru)⊥ orthogonal to u
σu (x) = x − 2
hu, xi
u,
|u|2
x ∈ RN .
(2.1)
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
3
A root system is a finite spanning set R ⊂ RN of nonzero vectors such that, for
every u ∈ R, σu preserves R. We shall always assume that R is reduced, i.e.
R ∩ Ru = ±u, for all u ∈ R. Each root system can be written as a disjoint union
R = R+ ∪(−R+ ), where R+ and (−R+ ) are separated by a hyperplane through the
origin. The subgroup G ⊂ O(N ) generated by the reflections {σu ; u ∈ R} is called
the finite reflection group associated with R. Henceforth, we shall normalize R so
that hu, ui = 2 for all u ∈ R. This simplifies formulas, without loss of generality for
our purposes. We refer to [12] for more details on the theory of root systems and
reflection groups.
A multiplicity function on R is a G-invariant function k : R → C, i.e. k(σu) =
k(u), for fall u ∈ R and σ ∈ G. The C-vector space of multiplicity functions on R
is denoted by K. The dimension of K is equal to the number of G-orbits in R. We
set K+ to be the set of multiplicity functions k such that k(u) ≥ 0 for all u ∈ R.
For ξ ∈ CN and k ∈ K, C. Dunkl [4] defined a family of first order differentialdifference operators Tξ (k) that play the role of the usual partial differentiation.
Dunkl’s operators are defined by
X
f (x) − f (ση x)
, f ∈ C 1 (RN ). (2.2)
Tξ (k)f (x) := ∂ξ f (x) +
k(η) < η, ξ >
hη,
xi
+
η∈R
Here ∂ξ denotes the derivative in the direction of ξ. Thanks to the G-invariance of
the function k, this definition is independent of the choice of the positive subsystem
R+ . The operators Tξ (k) are homogeneous of degree (−1). Moreover, by the Ginvariance of the multiplicity function k, the Dunkl operators satisfy
h ◦ Tξ (k) ◦ h−1 = Thξ (k),
∀ h ∈ G,
where h.f (x) = f (h−1 x). The most striking property of Dunkl operators Tξ (k),
which is the foundation for rich analytic structures with them, is the following
Theorem 2.1. For fixed k, Tξ (k) ◦ Tη (k) = Tη (k) ◦ Tξ (k),
∀ξ, η ∈ RN .
This result was obtained in [4] by a clever direct argumentation. An alternative
proof, relying on Koszul complex ideas, is given in [7].
For k ∈ K+ , there exists a generalization of the usual exponential kernel eh.,.i by
means of the Dunkl system of differential equations.
Theorem 2.2. Assume that k ∈ K+ .
(i) (Cf. [5, 16].) There exists a unique holomorphic function Ek on CN × CN
characterized by
Tξ (k)Ek (z, w) = hξ, wiEk (z, w), ∀ ξ ∈ CN ,
(2.3)
Ek (0, w) = 1,
Further, the Dunkl kernel Ek is symmetric in its arguments and satisfies
Ek (λz, w) = Ek (z, λw),
Ek (z, w) = Ek (z, w)
and
Ek (gz, gw) = Ek (z, w) (2.4)
N
for all z, w ∈ C , λ ∈ C and g ∈ G.
(ii) (Cf. [20].) For all x ∈ RN , y ∈ CN and all multi-indices ν ∈ ZN
+,
|∂yν Ek (x, y)| ≤ |x||ν| max eRehgx,yi .
g∈G
In particular,
|∂yν Ek (x, y)| ≤ |x||ν| e|x||Rey| ,
(2.5)
4
S. GHAZOUANI, F. BOUZAFFOUR
and for all x, y ∈ RN :
|Ek (ix, y)| ≤ 1.
(2.6)
Remark 2.1.
• When k = 0, we have E0 (z, w) = ehz,wi for z, w ∈ CN .
• For complex-valued k, there is a detailed investigation of (2.3) by Opdam [16].
Theorem 2.2 (i) is a weak version of Opdam’s result.
• M. de Jeu had already an estimate
on Ek with slightly weaker bounds in [2],
p
differing by an additional factor |G|.
The counterpart of the usual Laplacian is the Dunkl-Laplacian operator deN
X
fined by ∆k :=
Tξi (k)2 , where {ξ1 , . . . , ξN } is an arbitrary orthonormal basis of
i=0
(RN , h., , i). It is homogeneous of degree −2. By the normalization hu, ui = 2, we
can rewrite ∆k as
X
h∇f (x), ηi f (x) − f (ση x)
−
,
(2.7)
∆k f (x) = ∆f (x) + 2
k(η)
hη, xi
hη, xi2
+
η∈R
where ∆ and ∇ denote the usual Laplacian and gradient operators, respectively
(cf.[4]).
2.2. Dunkl transform. For fixed k ∈ K+ , let ωk be the weight function on RN
defined by
Y
ωk (x) =
|hη, xi|2k(η) .
η∈R+
It is G-invariant and homogeneous of degree 2γ, with the index
X
γ = γ(k) =
k(η).
η∈R+
Let dx be the Lebesgue measure corresponding to h., .i and set Lpk (RN ) the space
of measurable functions on RN such that
Z
p1
p
< +∞, if 1 ≤ p < +∞.
kf kp =
|f (x)| ωk (x) dx
RN
Following Dunkl [6], we define the Dunkl transform on the space L1k (RN ) by
Z
ck
Dk f (x) = γ+N/2
f (y)Ek (−ix, y)ωk (y)dy,
2
RN
Z
−1
−|x|2
where ck denotes the Mehta-type constant ck =
e
wk (x)dx
. Many
RN
properties of the Euclidean Fourier transform carry over to the Dunkl transform.
In particular:
Theorem 2.3. (Cf. [6, 2].)
a) (Riemann-Lebesgue lemma) For all f ∈ L1k (RN ), the Dunkl transform Dk f
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
5
belongs to C0 (RN ).
b) (L1 -inversion) For all f ∈ L1k (RN ) with Dk f ∈ L1k (RN ),
Dk2 f = fˇ, a.e, where fˇ(x) = f (−x).
(2.8)
c) The Dunkl transform f → Dk f is an automorphism of S(RN ).
d) (Plancherel Theorem)
i) If f ∈ L1k (RN ) ∩ L2k (RN ), then Dk f ∈ L2k (RN ) and kDk f k2 = kf k2 .
ii) The Dunkl transform has a unique extension to an isometric isomorphism of
L2k (RN ). The extension is also denoted by f → Dk f.
We conclude this subsection with two important reproducing properties for the
Dunkl kernel due to [6].
Theorem 2.4.
(Cf. [6].) For all p ∈ P and y, z ∈ CN ,
Z
2
ck
e−∆k /2 p(x) Ek (x, y) ωk (x) e−|x| /2 dx = el(y)/2 p(y).
(1) 2γ+N/2
N
ZR
2
ck
(2) 2γ+N/2
Ek (x, y)Ek (x, z) ωk (x) e−|x| /2 dx = e(l(y)+l(z))/2 p(y).
RN
2.3. Generalized Hermite functions. For an arbitrary finite reflection group G
and for any non-negative multiplicity function k, Rösler [17] introduced a complete
2
systems of orthogonal polynomials with respect to the weight function ωk (x) e−|x| dx,
called generalised Hermite polynomials. The key to their definition is the following
bilinear form on P, which was introduced in [5]:
[p, q]k := (p(T )q)(0)
for p, q ∈ P.
The homogeneity of the Dunkl operators implies that Pn ⊥ Pm for n 6= m. Moreover, if p, q ∈ Pn , then
Z
2
[p, q]k = 2n ck
e−∆k /4 p(x) e−∆k /4 q(x) ωk (x) e−|x| dx.
RN
This is obtained from Theorem 3.10 of [5] by rescaling, see lemma (2.1) in [17]. So
in particular, [., .]k is a scalar product on the vector space PR = R[x1 , . . . , xN ].
Now let {ϕν , ν ∈ ZN
+ } be an (arbitrary) orthonormal basis of PR with respect
to [., .]k such that ϕν ∈ P|ν| (For details concerning the construction and canonical
choices of such a basis, we refer to [17]). Then the generalised Hermite polynomials
N
{Hν , ν ∈ ZN
+ } and the (normalised) generalised Hermite functions {hν , ν ∈ Z+ }
associated with G, k and {ϕν } are defined by
√
2
ck
Hν (x) := 2|ν| e−∆k /4 ϕν (x) and hν (x) := |ν|/2 e−|x| /2 Hν (x) (x ∈ RN ). (2.9)
2
We list some standard properties of generalised Hermite functions that we shall use
in this article.
Theorem 2.5. (Cf. [17].) Let {Hν } and {hν } be the Hermite polynomials and
Hermite functions associated with the basis {ϕν } on RN and let x, y ∈ RN . Then
(1) The hν satisfy hν (−x) = (−1)|ν| hν (x).
2
N
(2) {hν , ν ∈ ZN
+ } is an orthonormal basis of Lk (R ).
(3) The hν are eigenfunctions of the Dunkl transform on L2k (RN ), with Dk hν =
(−i)|ν| hν .
6
S. GHAZOUANI, F. BOUZAFFOUR
(4) (Mehler formula) For r ∈ C with |r| < 1,
r 2 (|x|2 +|y|2 )
−
X Hν (x)Hν (y)
1−r 2
e
2zx
|ν|
r =
Ek
,y .
1 − z2
2|ν|
(1 − r2 )γ+(N/2)
N
ν∈Z+
Throughout this paper, R denotes a root system in RN , R+ a fixed positive
subsystem of R and k a nonnegative multiplicity function defined on R.
3. The fractional Dunkl transforms
n×n
If A ∈ R
is a square diagonalizable matrix A then we may write its eigenvalue
decomposition A = P DP −1 . Clearly for any integer a it holds that
Aa = P Da P −1 .
So it is a natural generalization to use the same formula as a definition if a is not
integer. Exactly the same idea can be used for a linear operator A on a linear space
if it has a sequence of eigenvectors that is complete in the whole space [14, 24]. Let
{λk , ek }∞
k=0 be the sequence of eigenvalues and corresponding eigenvectors. Since
the set of eigenvectors is complete, we can associate with each element f in the
Hilbert space a unique set of coordinates and conversely. These mappings are called
the analysis and the synthesis operators respectively. They are adjoint operators.
If E is the synthesis operator and E ∗ the analysis operator, which for a given set of
basis vectors {ek } are defined by
E : {ck }∞
k=0 7−→ f =
∞
X
ck ek
and E ∗ : f 7−→ {ck }∞
k=0 ,
k=0
then we can write
A = EΛE ∗
where Λ is the simple diagonal scaling operator
∞
Λ : {ck }∞
k=0 7−→ {λk ck }k=0 .
Its fractional power is then clearly Aa = EΛa E ∗ .
3.1. Definition and properties. In order to construct a fractional power of the
Dunkl transform, we use the idea developed in the above by restricting ourselves
to the Hilbert space L2k (RN ) with the inner product given by:
Z
hf, gik =
f (x)g(x)ωk (x)dx.
RN
Let l2 (ZN
such that
+ ) be the space of complex sequences u = (uν )ν∈ZN
+
X
ν∈ZN
+
∞. This is a Hilbert space for the inner product
X
hu, vi =
uν v ν , u = (uν )ν∈ZN
, v = (vν )ν∈ZN
∈ l2 (ZN
+ ).
+
+
ν∈ZN
+
|uν |2 <
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
7
Define the analysis and the synthesis operators associated to the generalized Hermite functions {hν , ν ∈ ZN
+ } respectively by
E : l2 (ZN
+)
(uν )ν∈ZN
+
N
L2k (RX
)
uν hν
f=
−→
7−→
ν∈ZN
+
and
E∗ :
L2k (RN )
f
−→
7−→
l2 (ZN
+)
(hf, hν ik )ν∈ZN .
+
ZN
+}
As the generalized Hermite functions {hν , ν ∈
are a basis of eigenfunctions of
the Dunkl transform Dk on L2k (RN ), satisfying Dk (hν ) = e−iπ|ν|/2 hν , then we can
write
Dk = EΛE ∗ ,
(3.1)
where Λ is the diagonal scaling operator
Λ : l2 (ZN
+)
(uν )ν∈ZN
+
−→
7−→
l2 (ZN
+)
e−iπ|ν|/2 uν ν∈ZN .
+
Definition 3.1. Let α ∈ R, we define the fractional Dunkl transform Dkα on
L2k (RN ) by
Dkα = EΛα E ∗ ,
where Λα is a fractional power of the diagonal scaling operator given by
Λα : l2 (ZN
+)
(uν )ν∈ZN
+
More explicitly, if f ∈
L2k (RN ),
Dkα f
−→
7−→
l2 (ZN
+)
eiα|ν| uν ν∈ZN .
+
then
X
=
ei|ν|α hf, hν i hν .
(3.2)
ν∈ZN
+
We summarize the elementary properties of Dkα in the next Proposition.
Proposition 3.1. Let α, β ∈ R. The fractional Dunkl transform Dkα satisfies the
following properties:
1) Dk0 = I, which is the identity operator,
−π/2
2) Dk
= Dk ,
3) Dkα ◦ Dkβ = Dkα+β ,
4) Dkα+2π = Dkα ,
ˇ where If
ˇ (x) = f (−x),
5) Dkπ = I,
6) For all f and g ∈ L2 (RN , ωk (x)dx), hDkα f, gi = hf, Dk−α gi.
Proof 1), 2) and 4) follow immediately from (3.2).
3) From (3.2), we have
X
Dkα (Dkβ f ) =
ei|ν|α hDkβ f, hν i hν
ν∈ZN
+
=
X
ν∈ZN
+
ei|ν|(α+β) hf, hν i hν = Dkα+β f.
8
S. GHAZOUANI, F. BOUZAFFOUR
5) By (3.2) and Theorem 2.5, 1) , we have
X
Dk−π f =
e−i|ν|π hf, hν i hν
N
ν∈Z+
=
X
(−1)|ν| hf, hν i hν .
N
ν∈Z+
ˆ
= If.
6) Let f and g ∈ L2 (RN , ωk (x)dx). It is easy to check that
X
X
ei|ν|α hf, hν i hg, hν i =
hf, hν i e−i|ν|α hg, hν i
hDkα f, gi =
ν∈ZN
+
ν∈ZN
+
hf, Dk−α gi.
=
Theorem 3.1. The family of operators {Dkα }α∈R is a C0 -group of unitary operators
on L2k (RN ).
Proof From Proposition 3.1, we deduce that the family {Dkα }α∈R satisfies the
algebraic properties of a group:
Dk0 = I,
Dkα ◦ Dkβ = Dkα+β = Dkβ ◦ Dkα ;
α, β ∈ R.
L2k (RN ).
For the strong continuity, assume that f ∈
Then
X
2
2
kDkα f − f k2 =
|ei|ν|α − 1|2 |hf, hν i| .
ν∈ZN
+
For each ν ∈ ZN
+ , we have
2
=
2
≤ 4 |hf, hν i| .
lim |ei|ν|α − 1|2 |hf, hν i|
α→0
|ei|ν|α − 1|2 |hf, hν i|
0,
2
Since
X
2
|hf, hν i| = kf k22 < ∞,
ν∈ZN
+
then we can interchange limits and sum to get:
2
lim kDkα f − f k2 = 0.
α→0
Hence {Dkα }α∈R is a strongly continuous group of operators on L2k (RN ). In addition,
by Proposition 3.1, we have for all f, g ∈ L2k (RN ),
hDkα f, gi = hf, Dk−α gi,
and therefore (Dkα )∗ = Dk−α = (Dkα )−1 , establishing that each Dkα is unitary.
4. The infinitesimal generator of the C0 -group {Dkα }α∈R .
The infinitesimal generator T of the C0 -group {Dkα }α∈R is defined by
T : L2k (RN ) ⊇ D(T ) −→
f 7−→
L2k (RN ),
Tf
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
9
where
D(T )
=
n
o
f ∈ L2k (RN ) : lim (1/α)[Dkα f − f ] ∈ L2k (RN ) ,
Tf
=
lim (1/α)[Dkα f − f ],
α→0
α→0
f ∈ D(T ).
Our goal here is to study spectral properties of T. We indicate some necessary
notation and definitions, needed in the sequel. We denote by B(L2k (RN )), the set
of all linear bounded operator in L2k (RN ). The resolvent set of T is the set ρ(T )
consisting of all scalars λ for which the linear operator λI −T is a 1-1 mapping from
its domain D(λI − T ) = D(T ) on to the Hilbert space L2k (RN ) with (λI − T )−1 ∈
B(L2k (RN )). The spectrum of T is the set σ(T ) that is the complement of ρ(T ) in
C. The function R(λ, T ) = (λI − T )−1 from ρ(T ) into B(L2k (RN )) is the resolvent
of T.
As T is the generator of the C0 -group {Dkα }α∈R , some elementary properties of
T and Dkα are listed in the following proposition (see [9], [10]).
Proposition 4.1. Let α ∈ R. The following properties hold.
i) If f ∈ D(T ), then Dkα f ∈ D(T ) and
d α
D f = Dkα T f = T Dkα f.
dα k
ii) For every t ∈ R and f ∈ L2k (RN ), one has
Z t
Dkα f dα ∈ D(T ).
(4.1)
0
iii) For every α ∈ R, one has
Dkα f − f
α
Z
=
T
Z
=
Dks f ds,
if f ∈ L2k (RN )
(4.2)
if f ∈ D(T ).
(4.3)
0
α
Dks T f ds,
0
Remark 4.1. If we apply the Proposition 4.1, iii) to the rescaled semigroup
S(α) := e−λα Dkα ,
α∈R
whose generator is B := T − λI with domain D(B) = D(T ), we obtain for every
λ ∈ C and α ∈ R,
Z α
−e−λα Dkα f + f = (λI − T )
e−λs Dks f ds; f ∈ L2k (RN ),
(4.4)
0
Z α
=
e−λs Dks (λI − T )f ds; f ∈ D(T ).
(4.5)
0
Now we are interesting with the eigenvalues of T by giving an important formula
relating the semigroup {Dkα }α∈R , to the resolvent of its generator T.
Proposition 4.2. For the operator T, the following properties hold:
1) T is closed and densely defined.
2) The operator iT is self-adjoint.
3) σ(T ) = σp (T ) ⊂ iZ, and for each λ ∈ C\iZ and for all f ∈ L2k (RN ),
Z 2π
−2πλ −1
R(λ, T )f = (1 − e
)
e−λs Dks f ds.
0
(4.6)
10
S. GHAZOUANI, F. BOUZAFFOUR
Here σp (T ) = {λ ∈ C : λI − T is not injective} .
Proof 1) The fact that T is closed and densely defined follows from the HilleYosida Theorem (see[[10], p. 15]).
2) Since {Dkα }α∈R is unitary, it follows from Stone’s Theorem [[10], p. 32] that T
is skew-adjoint (T ∗ = −T ) and therefore iT is self-adjoint.
3) If we replace α by 2π in (4.4) and (4.5) and we use the fact that {Dkα }α∈R is a
periodic C0 -group with period 2π, we get
Z 2π
e−λs Dks f ds;
f ∈ L2k (RN ),
(4.7)
(1 − e−2πλ )f = (λI − T )
0
Z
=
2π
e−λs Dks (λI − T )f ds;
f ∈ D(T ).
(4.8)
0
Let λ 6∈ iZ. Then 1 − e−2πλ 6= 0. By the use of (4.7) and (4.8), λI − T is invertible
(λ ∈ ρ(T )) and
Z 2π
(λI − T )−1 f = R(λ, T )f = (1 − e−2πλ )−1
e−λs Dks f ds.
0
The previous Proposition indicates that every point in the spectrum of T is an
isolated point of the set iZ. Let in be an element of the spectrum of T and
Z
1
Pn =
R(λ, T ) dλ,
2iπ Γ
the associated spectral projection, where Γ is a Jordan path in the complement
of iZ\{in} and enclosing in. The function λ 7−→ R(λ, T ) can be expanded as a
Laurent series
+∞
X
R(λ, T ) =
(λ − in)k Bk
k=−∞
for 0 < |λ − in| < δ and some sufficiently small δ > 0. The coefficients Bk of this
series are bounded operators given by the formulas
Z
R(λ, T )
1
Bk =
dλ, k ∈ Z.
2iπ Γ (λ − in)k+1
The coefficient B−1 is exactly the spectral projection Pn corresponding to the decomposition σ(T ) = {in}∪{iZ\{in}} of the spectrum of T. From (4.6), one deduces
the identitie
Pn
=
=
B−1 = lim (λ − in)R(λ, T )
λ→in
Z 2π
1
e−ins Dks ds,
2π 0
(4.9)
which allows as to interpret Pn as the nth Fourier coefficient of the 2π-periodic
function s 7−→ Dks .
In the following Proposition we gather some properties of the operator Pn .
Proposition 4.3. Let n, m ∈ Z such that n 6= m and f, g ∈ L2k (RN ). Then
i) T Pn = inPn ,
ii) Dks Pn = eins Pn ,
iii) Pn Pm = 0,
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
11
iv) hPn f, gi = hf, Pn gi . In particular hPn f, Pm gi = 0.
v) The linear span
[
lin
Pn L2k (RN )
n∈Z
is dense in L2k (RN ).
Proof i) It follows directly from (4.7) applied to λ = in.
ii) Applying Dks to each member of (4.9) then according to Proposition 3.1, 3), we
obtain
Z 2π
1
s
e−int Dks Dkt f dt
Dk Pn f =
2π 0
Z 2π
1
=
e−int Dks+t f dt.
2π 0
The change of variables u = s + t gives the desired result.
iii) From (4.9) and ii), we have
Z 2π
1
e−ins Dks (Pm f ) ds
Pn Pm f =
2π 0
Z 2π
1
=
ei(m−n)s ds Pm f
2π 0
= 0.
iii) Obvious.
iv) Assume that the linear span
lin
[
Pn L2k (RN )
n∈Z
is not dense in
functional
L2k (RN ).
By the Hahn-Banach theorem there exists a nonzero linear
ϕ : L2k (RN ) −→ C
2
N
vanishing on each Pn Lk (R ), n ∈ Z. By the Riesz representation theorem, there
exists a unique vector g ∈ L2k (RN )\{0} such that
ϕ(f ) = hf, gi for all f ∈ L2k (RN ).
Hence for all n ∈ Z and f ∈ L2k (RN ),
Z 2π
1
−ins s
e
Dk f ds, g
0 = hPn f, gi =
2π 0
Z 2π
1
=
e−ins hDks f, gi ds.
2π 0
For each f ∈ L2k (RN ), the function s 7−→ hDks f, gi has all its Fourier coefficients
equal to zero, then it vanishes. This cannot be true, since if we take f = g and
s = 0,
0
Dk g, g = kgk22 > 0.
Proposition 4.4. Let f ∈ D(T ). Then
f
=
+∞
X
n=−∞
Pn f,
(4.10)
12
S. GHAZOUANI, F. BOUZAFFOUR
and therefore, if f ∈ D(T 2 )
Tf
=
+∞
X
inPn f.
(4.11)
X
Pn f is summable for all f ∈
n=−∞
Proof We are going to show that the series
n∈Z
D(T ). For this, let f ∈ D(T ) and put g = T f. The commutativity of T and Pn
together with Proposition 4.3 gives:
Pn g = Pn T f = T Pn f = inPn f.
By the Cauchy-Schwartz inequality, it follows that
X
X
−1
(in) hPn g, hi
hPn f, hi = n∈H
n∈H
!1/2
!1/2
X
X
2
−2
,
≤
n
|hPn g, hi|
n∈H
n∈H
L2k (RN )
2
where h ∈
and H be a finite subset of Z\{0}. The function s 7−→ hDks g, hi
belongs to L ([0, 2π]), then we obtain from Bessel’s inequality
Z 2π
X
1
2
|hPn g, hi| ≤
|hDks g, hi|2 ds
2π 0
n∈H
Z
khk22 2π
≤
kDks gk22 ds = khk22 kgk22 .
2π 0
Therefore, for any h ∈ L2k (RN ),
*
+ X
X
Pn f, h = hPn f, hi
n∈H
!1/2
≤
khk2 kgk2
n∈H
X
n−2
.
n∈H
Taking supremum over h ∈ L2k (RN ) with khk2 ≤ 1, we get
!1/2
X
X
−2
Pn f ≤ kgk2
n
,
n∈H
n∈H
2
X
which means that the series
Pn f converges converges in L2k (RN ).
n∈Z
Set
f1 =
+∞
X
Pn f
n=−∞
and let g ∈ L2k (RN ). As the Fourier coefficients of the continuous, 2π-periodic
functions
s 7−→ hDks f1 , gi and s 7−→ hDks f, gi
coincide. Then, for all s ∈ R,
hDks f1 , gi = hDks f, gi .
In particular, for s = 0, hf1 , gi = hf, gi and therefore f1 = f.
Replacing f in (4.10) by T f, then we get (4.11).
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
13
At the end of the section 4, we will show that Pn = 0, for any negative integer
n 6= 0.
5. Integral representation.
In this section, we shall derive an integral representation for the fractional Dunkl
transform Dkα defined by (3.2), for suitable function f.
α
We define the operator Dk,r
on L2k (RN ) by
X
α
r|ν| ei|ν|α hf, hν ihν ,
Dk,r
f :=
(5.1)
ν∈ZN
+
α
where 0 < r ≤ 1 and so Dkα = Dk,1
.
α
In the next proposition, we collect some properties of Dk,r
.
Proposition 5.1. Let α ∈ R and r ∈]0, 1]. Then
α
α
1) Dk,r
is a bounded operator on L2k (RN ) satisfying kDk,r
f k2 ≤ kf k2 .
2
2
N
α
α
N
2) For all f ∈ Lk (R ), Dk,r f → Dk f in Lk (R ) as r → 1− .
Proof Let f ∈ L2k (RN ). 1) According to Parseval’s formula, we have
X
2
α
kDk,r
f k22 =
r2|ν| |hf, hν i|
ν∈ZN
+
≤
X
2
|hf, hν i| = kf k22 .
ν∈ZN
+
2) It is easy to see that
α
Dk,r
f − Dkα f =
X
(r|ν| − 1)ei|ν|α hf, hν ihν .
ν∈ZN
+
Then
X
α
2
Dk,r f − Dkα f 2 =
|r|ν| − 1|2 |hf, hν i| .
2
ν∈ZN
+
α
2
By the dominated convergence theorem it follows that lim Dk,r
f − Dkα f 2 = 0.
r→1−
−
Corollary 5.1. For each fixed f ∈ L2k (RN ), there exists {rj }∞
j=1 , with rj → 1 as
j → ∞, such that
α
Dkα f (x) = lim Dk,r
f (x)
j
j→∞
for almost all x ∈ RN .
Proof This is a consequence of a standard result that if a sequence {fn } converges in L2k (RN ) to f, then there exists a subsequence {fnk } that converges pointwise almost everywhere to f.
α
The operator Dk,r
defined above have the integral representation given in the next
lemma.
Lemma 5.1. For f ∈ L2k (RN ) and 0 < r < 1, we have
Z
α
Dk,r f (x) =
Kα (r, x, y)f (y)ωk (y) dy,
RN
(5.2)
14
S. GHAZOUANI, F. BOUZAFFOUR
where
X
Kα (r, x, y) =
r|ν| ei|ν|α hν (x)hν (y).
(5.3)
ν∈ZN
+
Proof Let x ∈ RN and H be a finite subset of ZN
+ . Then
2
X
X
|hν (x)|2 |r|2|ν| .
hν (x)hν (y)(reiα )|ν| =
ν∈H
2
(5.4)
ν∈H
Since the series (see Theorem 3.12 in [17])
X
hν (x)hν (y)(reiα )|ν|
ν∈ZN
+
converges absolutely for all x, y ∈ RN , then according to (5.4), the series
X
hν (x)hν (y)(reiα )|ν|
ν∈H
L2k (RN )
to a function denoted by Kα (r, x, .).
converges in
By the use of Cauchy-Schwartz inequalities, we obtain
Z
X
α
Dk,r
f (x) =
(reiα )|ν| hν (x)
f (y)hν (y)ωk (y) dy
RN
ν∈ZN
+
Z
=
f (y)
RN
X
hν (x)hν (y)(reiα )|ν| ωk (y) dy
ν∈ZN
+
Z
=
Kα (r, x, y)f (y)ωk (y) dy.
RN
Now, we summarize some properties of the kernel Kα (r, x, y).
Proposition 5.2. Let x, y ∈ RN , α, r ∈ R such that 0 < |α| < π and 0 < r < 1,
then we have
1)
Kα (r, x, y)
= ck
e
−
1+r 2 e2iα
(|x|2 +|y|2 )
2(1−r 2 e2iα )
(1 − r2 e2iα )γ+N/2
Ek
2reiα x
,y ,
1 − r2 e2iα
(5.5)
2)
lim Kα (r, x, y) = Aα Kα (x, y),
(5.6)
r→1−
where
Kα (x, y)
Aα =
=
e
− 2i cot(α)(|x|2 +|y|2 )
Ek
ix
,y ,
sin α
ck ei(γ+N/2)(α̂π/2−α)
and α̂ = sgn(sin α).
(2| sin α|)γ+N/2
(5.7)
(5.8)
3)
)|y|2
2r 2 (1−r 2 ) cos2 (α)|x|2
− (1+r2 e2iα
2reiα x
e 2(1−r2 e2iα ) Ek
≤ e (r4 −2r2 cos 2α+1)(r2 +1) .
,
y
1 − r2 e2iα
(5.9)
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
15
Proof 1) According to (2.9), we have
Kα (r, x, y)
2
= ck e−(|x|
+|y|2 )/2
X
(reiα )|ν|
ν∈ZN
+
Hν (x)Hν (y)
.
2|ν|
Using Mehler’s formula for the generalized Hermite polynomials (see Theorem 2.5,
4)) and setting z = reiα with |z| = r < 1, we obtain the desired result.
2) Clearly
1 + r2 e2iα
r→1− 1 − r 2 e2iα
reiα
lim
r→1− 1 − r 2 e2iα
lim
N
lim (1 − r2 e2iα )−(γ+ 2 )
r→1−
= i cot α,
=
i
2 sin α
=
(1 − e2iα )−(γ+ 2 )
=
ei(γ+N/2)(α̂π/2−α)
, where α̂ = sgn(sin α).
(2| sin α|)γ+N/2
N
Then, for 0 < |α| < π,
lim Kα (r, x, y) = Aα Kα (x, y),
r→1−
where Kα (x, y) and Aα are defined respectively in (5.7) and (5.8).
3) It is straightforward to show that
1 + r2 e2iα
(1 − r4 )
> 0,
=
ar = <
1 − r2 e2iα
(1 + r4 ) − 2r2 cos 2α
2reiα
2(r − r3 ) cos α
br = <
.
=
1 − r2 e2iα
1 + r4 − 2r2 cos 2α
(5.10)
From (2.5) and (5.10), we deduce the following majorization:
2reiα x
E k
, y ≤ e|br | |x||y| .
2
2iα
1−r e
Hence,
)|y|2
− (1+r2 e2iα
2
2reiα x
e 2(1−r2 e2iα ) Ek
, y ≤ e−ar |y| +|br |
2
2iα
1−r e
|x||y|
.
(5.11)
As ar > 0, we deduce that
sup(−ar s2 + |br | |x|s) = −
s≥0
b2r |x|2
.
4ar
(5.12)
Combining (5.11) and (5.12), we see that
)|y|2
2r 2 (1−r 2 ) cos2 (α)|x|2
− (1+r2 e2iα
2reiα x
e 2(1−r2 e2iα ) Ek
≤ e (r4 −2r2 cos 2α+1)(r2 +1) .
,
y
1 − r2 e2iα
Proposition 5.3. Let α ∈ R \πZ and f ∈ L1k (RN ) ∩ L2k (RN ). Then the fractional
Dunkl transform Dkα have the following integral representation
Z
α
Dk f (x) = Aα
f (y)Kα (x, y)ωk (y)dy, a.e,
(5.13)
RN
16
S. GHAZOUANI, F. BOUZAFFOUR
where
Kα (x, y)
=
i
2
e− 2 cot(α)(|x|
+|y|2 )
Ek
ix
,y
sin α
and
Aα =
ck ei(γ+N/2)(α̂π/2−α)
.
(2| sin α|)γ+N/2
Proof Dkα is periodic in α with period 2π, we can assume that 0 < |α| < π. Let
f ∈ L1k (RN ) ∩ L2k (RN ). From Corollary 5.1,
Z
Dkα f (x) = lim
Kα (rj , x, y)f (y)ωk (y) dy, a.e.
j→∞
RN
From Proposition 5.2, 2) we see that
lim Kα (rj , x, y)f (y) = Aα Kα (x, y)f (y).
j→∞
Using again Proposition 5.2, 3), we obtain
!
(1+rj2 e2iα )|y|2
iα
− 2(1−r2 e2iα )
2r
e
x
j
e
j
Ek
, y f (y) ≤ Mx |f (y)|,
2
2iα
1 − rj e
2r 2 (1−r 2 ) cos2 (α)|x|2
where Mx = sup e (r4 −2r2 cos 2α+1)(r2 +1) .
0≤r<1
Hence, the dominated convergence theorem gives
Z
Dkα f (x) = Aα
f (y)Kα (x, y)ωk (y)dy, a.e.
RN
Definition 5.1. We define the fractional Dunkl transform Dkα for f ∈ L1k (RN ) by
Z
α
Dk f (x) = Aα
f (y)Kα (x, y)ωk (y)dy.
RN
Remark 5.1.
• For α = − π2 , the fractional Dunkl transform Dkα is reduces to the Dunkl transform
Dk and when the multiplicity function k ≡ 0, Dkα coincides with the fractional
Fourier transform F α [1]
Z
2
2
i
i
e(iN/2)(α̂π/2−α)
F α f (x) =
e− 2 (|x| +|y| ) cot α+ sin α hx,yi f (y) dy.
N/2
(2π| sin α|)
RN
• In the one-dimensional case (N = 1), the corresponding reflection group W is Z2
and the multiplicity function k is equal to ν + 1/2 > 0. The kernel Kα (x, y) defined
by (5.7) becomes
ix
− 2i cot α(x2 +y 2 )
,y ,
(5.14)
Kα (x, y) = e
Eν
sin α
where Eν (x, y) is the Dunkl kernel of type A2 given by (see [19])
K(ix, y) = jν (xy) +
ixy
jν+1 (xy),
2(ν + 1)
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
17
and jν denotes the normalized spherical Bessel function
jν (x) := 2ν Γ(ν + 1)
+∞
X
Jν (x)
(−1)n (x/2)2n
=
Γ(ν
+
1)
.
xν
n!Γ(n + ν + 1)
n=0
Here Jν is the classical Bessel function (see, Watson [21]). The related fractional
Dunkl transform Dkα in rank-one case takes the form
Z +∞
α
Dν f (x) = Bν
Kα (x, y)f (y)|y|2ν+1 dy,
(5.15)
−∞
where
Bν =
ei(ν+1)(α̂π/2−α)
.
Γ(ν + 1)(2| sin(α)|)ν+1
(5.16)
Note that if f is an even function then, the fractional Dunkl transform (5.15)
coincides with the fractional Hankel transform [13]
Z +∞
xy 2
2
i
Hνα f (x) = 2Bν
e− 2 (x +y ) cot α jν
f (y)y 2ν+1 dy.
sin
α
0
• More generally, for W = Z2 × · · · × Z2 and the multiplicity function k =
(ν1 , . . . , νN ), the kernel Kα (x, y) defined by (5.7) is given explicitly by
N
2
2 Y
i
ixj
, yj ,
Eνj
Kα (x, y) = e− 2 cot α(|x| +|y| )
sin α
j=1
where x = (x1 , . . . , xN ), y = (y1 , . . . , yN ) ∈ RN and Eνj (xj , yj ) is the function
defined by (5.14). In this case the fractional Dunkl transform will be
Z
Dkα f (x) = Aα
f (y)Kα (x, y)ωk (y) dy,
RN
where
Aα =
ei(γ+N/2)(α̂π/2−α)
Γ(ν1 + 1) . . . Γ(νN + 1)(2| sin α|)γ+N/2
and
ωk (y) =
N
Y
|xj |2νj .
j=1
5.1. Bochner type identity for the fractional Dunkl transform. In this
section, we start with a brief summary on the theory of k-spherical harmonics.
An introduction to this subject can be found in the monograph [8]. The space of
k-spherical harmonics of degree n ≥ 0 is defined by
Hnk = Ker∆k ∩ Pn .
Let S N −1 = x ∈ RN ; |x| = 1 be the unit sphere in RN with normalized Lebesgue
surface mesure dσ and L2 (S N −1 , ωk (x) dσ(x)) be the Hilbert space with the following inner product given by
Z
hf, gik =
f (ω)g(ω)ωk (ω) dσ(ω).
S N −1
18
S. GHAZOUANI, F. BOUZAFFOUR
As in the theory of ordinary spherical harmonics, the space L2 (S N −1 , ωk (x) dσ(x))
decomposes as an orthogonal Hilbert space sum
L2 (S N −1 , ωk (x) dσ(x)) =
∞
M
Hnk .
n=0
In [23], Y. Xu gives an analogue of the Funk-Hecke formula for k-spherical harmonics. The well-known special case of the Dunkl-type Funk-Hecke formula is the
following (see [20]):
Proposition 5.4. Let N ≥ 2 and put λ = γ + (N/2) − 1. Then for all Y ∈ Hnk and
x ∈ RN ,
Z
Γ(λ + 1)
1
K(ix, y)Y (y)ωk (y) dσ(y) = n
jn+λ (|x|)Y (ix), (5.17)
dk S N −1
2 Γ(n + λ + 1)
where
Z
dk =
ωk (y) dσ(y).
S N −1
In particular
1
dk
Z
K(ix, y)ωk (y) dσ(y) = jλ (|x|).
(5.18)
S N −1
An application of the Dunkl-type Funk-Hecke formula is the following:
Theorem 5.1. (Bochner type identity) If f ∈ L1k (RN ) ∩ L2k (RN ) is of the form
f (x) = p(x)ψ(|x|) for some p ∈ Hnk and a one-variable ψ on R+ , then
α
Dkα f (x) = einα p(x)Hn+γ+(N/2)−1
ψ(|x|).
(5.19)
In particular, if f is radial, then
α
Dkα f (x) = Hn+γ+(N/2)−1
ψ(|x|).
Proof Since Dkα is periodic in α with period 2π, we can assume that −π < α ≤ π.
We see immediately that
Dk0 f (x)
= f (x),
Dkπ f (x)
= f (−x)
= p(−x)ψ(−x)
(−1)n p(x)ψ(x).
=
Now, let 0 < |α| < π. By spherical polar coordinates, we have
Z
Dkα f (x) = Aα
f (y)Kα (x, y)ωk (y)dy
RN
Z +∞
= Aα
rN −1 F (r, x) dr,
0
where
F (r, x)
=
2π N/2
Γ(N/2)
Z
Kα (x, ry)p(ry)ψ(r|y|)ωk (ry) dσ(y).
S N −1
(5.20)
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
19
From (5.7) and the homogeneity of ωk and p, we obtain
F (r, x)
=
2π N/2 − i (|x|2 +r2 ) cot(α)
e 2
ψ(r)r2γ+n
Γ(N/2)
Z
p(y)Ek
S N −1
irx
, y ωk (y) dσ(y).
sin(α)
Using (5.17), we get
F (r, x)
=
2π N/2 dk
Γ(λ + 1)
Γ(N/2) 2n Γ(λ + n + 1)
2
i
× e− 2 (|x|
+r 2 ) cot(α)
ψ(r)r2γ+n p
irx
sin(α)
jλ+n
r|x|
sin(α)
,
where
λ = γ + (N/2) − 1.
Using again the homogeneity of p, we get
F (r, x)
=
Γ(λ + 1)
2π N/2 dk
Γ(N/2) 2n Γ(λ + n + 1)
× e
− 2i (|x|2 +r 2 ) cot(α)
ψ(r)r
i
2 sin α
2γ+2n
n
p(x)jλ+n
r|x|
sin(α)
.
Now we can express a relationship between dk and ck . In fact
Z
2
−1
ck
=
e−|y| ωk (y) dy
RN
=
=
=
2π N/2
Γ(N/2)
Z
2π N/2
Γ(N/2)
Z
+∞
rN −1 e−r
2
Z
ωk (ry) dσ(y) dr
S N −1
0
+∞
r2γ+N −1 e−r
2
Z
ωk (y) dσ(y) dr
S N −1
0
π N/2 Γ(λ + 1)dk
.
Γ(N/2)
(5.21)
Recall that
Aα = ck
ie−iα
2 sin α
γ+(N/2)
,
then by the use of (5.21), we obtain
N/2
Aα
Γ(λ + 1)
2π
dk
Γ(N/2) 2n Γ(λ + n + 1)
i
2 sin α
2
n
=
=
ie−iα
2 sin α
λ+n+1
Γ(λ + n + 1)
einα
2Bν einα .
Hence
F (r, x) = 2Bν e
inα − 2i (|x|2 +r 2 ) cot(α)
e
ψ(r)r
2γ+2n
p(x)jλ+n
r|x|
sin(α)
.
(5.22)
20
S. GHAZOUANI, F. BOUZAFFOUR
Substituting (5.22) in (5.20) to get
Dkα f (x)
=
×
2Bν einα p(x)
Z +∞
r|x|
2(λ+n)+1
− 2i (|x|2 +r 2 ) cot(α)
ψ(r)r
jλ+n
e
dr
sin(α)
0
=
α
einα p(x)Hn+λ
ψ(|x|)
=
α
ψ(|x|).
einα p(x)Hn+γ+(N/2)−1
Application
Now, we give the material needed for an application of Bochner type identity. Let
{pn,j }j∈Jn be an orthonormal basis of Hnk . Let m, n be non-negative integers and
j ∈ Jn . Define
1/2
m! Γ(N/2)
cm,n =
π N/2 Γ((N/2) + γ + n + m)
and
2
(n+γ+N/2−1)
ψm,n,j (x) = cm,n pn,j (x) Lm
(|x|2 ) e−|x|
/2
,
(5.23)
(a)
Lm
denote the Laguerre polynomial defined by
x−a ex dn
L(a)
e−x xn+a .
m (x) =
n
n! dx
It follows from Proposition 2.4 and Theorem 2.5 of Dunkl [6] that
where
{ψm,n,j : m, n = 0, 1, 2, . . . , j ∈ Jn }
forms an orthonormal basis of L2k (RN ).
Theorem 5.2. The family {ψm,n,j : m, n = 0, 1, 2, . . . , j ∈ Jn } are a basis of
eigenfunctions of the fractional Dunkl transform Dkα on L2 RN , ωk (x) dx , satisfying
Dkα ψm,n,j = eiα(n+2m) ψm,n,j .
(5.24)
Proof We need only to prove (5.24). Applying Theorem 5.3 with p replaced by
2
(n+γ+N/2−1) 2
(r ) e−r /2 , we obtain
pn,j and with ψ(r) = Lm
Dkα ψm,n,j (x) = cm,n einα pn,j (x)Hνα ψ(|x|),
where
ν = n + γ + (N/2) − 1,
and
Z
Hνα ψ(|x|)
=
2Bν
+∞
e
− 2i cot(α)(|x|2 +r 2 )
jν
0
r|x|
sin α
2 −
L(ν)
m (r )e
r2
2
r2ν+1 dr.
Observe that
Hνα ψ(|x|)
=
i
2
2Bν e− 2 cot(α)|x| Iν ,
where
=
=
+∞
r|x|
2 −( 12 + 2i cot(α))r 2
r2ν+1 L(ν)
(r
)e
j
dr
ν
m
sin α
0
ν Z +∞
sin α
r|x|
ν
ν+1 (ν) 2 −( 21 + 2i cot(α))r 2
2 Γ(ν + 1)
r
Lm (r )e
Jν
dr.
|x|
sin α
0
Z
Iν
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
21
To compute Iν , we need the following formulas (see 7.4.21 (4) in [11])
Z +∞
2
2
az 2
y ν+1 e−βy Lνm (ay 2 )Jν (zy) dy = dm z ν e−z /(4β) Lνm
4β(a − β)
0
where dm = ((β − a)m /(2ν+1 β ν+m+1 )), a, <β > 0, <ν > −1.
−iα
|x|
Let us take β = 21 + 2i cot(α) = 2iesin α , a = 1 and z = sin
α , then
dm
=
az 2
4β(a − β)
z2
−
4β
e2iαm
2ν+1 Aα Γ(ν
+ 1)
,
= |x|2 ,
= −
|x|2
i
+ cot(α)|x|2 .
2
2
Hence
ν
|x|2
Z +∞
2 i
e2iαm e− 2 + 2 cot(α)|x|
|x|
r|x|
(ν)
ν+1 (ν) 2 −( 21 + 2i cot(α))r 2
dr =
Lm
(|x|2 ),
Jν
r
Lm (r )e
sin α
2ν+1 Aα Γ(ν + 1)
sin α
0
and therefore
Hνα ψ(|x|)
2
−|x|
= e2iαm L(ν)
m (|x| ) e
2
/2
,
which finishes the proof.
5.2. Master Formula for the fractional Dunkl transform. In this section, we
are interesting with a master formula for the fractional Dunkl transform. For this
we need the following lemma
Lemma 5.2. Let p ∈ Pn and x = (x1 , . . . , xN ) ∈ CN . Then for ω ∈ C and
<(ω) > 0,
l(x)
Z
ω
e ω
−ω|y|2
ck
p(y)Ek (x, 2y)e
ωk (y) dy = γ+n+(N/2) e 4 ∆k p(x),
(5.25)
ω
N
R
where l(x) =
N
X
xj .
j=1
Proof First compute the above integral when ω > 0.
Z
Z
√
2
2
ck
p(y)Ek (x, 2y)e−ω|y| ωk (y) dy = ck
p(y)Ek (x, 2y)e−| ωy| ωk (y) dy.
RN
√
RN
By the change of variables u = ωy and the homogeneity of ωk and p, we obtain
Z
2
ck
p(y)Ek (x, 2y)e−ω|y| ωk (y) dy
RN
Z
2
ck
x
=
p(y)Ek √ , 2y e−|y| ωk (y) dy.
(5.26)
γ+(n+N
)/2
ω
ω
RN
Using Theorem 2.4,1), we deduce the following identity:
Z
∆k
2
ck
p(y)Ek (x, 2y)e−|y| ωk (y) dy = el(x) e 4 p(x).
RN
(5.27)
22
S. GHAZOUANI, F. BOUZAFFOUR
Combining (5.26) and (5.27) to get
Z
2
ck
p(y)Ek (x, 2y)e−ω|y| ωk (y) dy =
e
ω
RN
l(x)
ω
e
γ+(n+N )/2
∆k
4
p
x
√
ω
.
Now use Lemma 2.1 from [17] to obtain
∆k
x
1 ω
= n/2 e 4 ∆k p(x).
e 4 p √
ω
ω
Hence, we find the equality (5.25) for ω > 0. By analytic continuation, this holds
for {ω ∈ C : <(ω) > 0}.
We are now in a position to give the master formula.
Theorem 5.3. Let p ∈ Pn and x ∈ RN . Then
∆
∆k
|y|2
|x|2
α
− 4k
− 2
Dk e
e
p(y) (x) = einα e− 2 e− 4 p(x).
Proof It follows easily from (5.13) that
Z
∆
|y|2
− 4k
− 2i cot(α)|x|2
α
− 2
e
p(y) (x) = Aα e
Dk e
e
−
∆k
4
p(y)Ek
RN
(5.28)
2
ix
, y e−ω|y| ωk (y) dy,
sin α
where
ω=
1
i
ie−iα
+ cot(α) =
.
2 2
2 sin α
Since
(5.29)
n
e
−
∆k
4
p(y) =
[2]
X
(−1)s
s=0
s!4s
∆sk p(y),
we conclude that
[n
Z
Z
2]
X
∆
2
(−1)s
ix
ix
−ω|y|2
s
− 4k
,y e
ωk (y) dy =
∆
p(y)E
,
y
e−ω|y| ωk (y) dy.
p(y)Ek
e
k
k
s
sin
α
s!4
sin
α
N
N
R
R
s=0
(5.30)
For s ∈ Z+ with 2s ≤ n, the polynomial ∆sk p is homogeneous of degree n − 2s.
Hence by the previous Lemma, we obtain
l(Xα )
Z
ω
ix
e ω
−ω|y|2
s
ck
∆k p(y)Ek
,y e
ωk (y) dy = γ+n+(N/2) e 4 ∆k ω 2s ∆sk p (Xα ),
(5.31)
sin
α
ω
N
R
where
ix
Xα =
.
(5.32)
2 sin α
Substituting (5.31) in (5.30) to get
[n
l(Xα )
Z
2]
X
∆
ω
ix
e ω
(−1)s ω 2s s
− 4k
−ω|y|2
∆
k
4
p(y)Ek
ck
e
,y e
ωk (y) dy =
e
∆k p(Xα )
sin α
s!4s
ω γ+n+(N/2)
RN
s=0
=
=
e
l(Xα )
ω
ω γ+n+(N/2)
e
ω
e 4 ∆k e −
l(Xα )
ω
ω γ+n+(N/2)
e
ω−ω 2
4
ω2
4
∆k
∆k
p(Xα )
p(Xα ).
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
23
Replacing ω and Xα by their values given in (5.29) and (5.32) and use Lemma 2.1
in [17], we obtain
ω−ω 2
2
2
in
e 4 ∆k p(Xα ) =
e− sin (α)(ω−ω )∆k p(x)
n
2 sinn α
∆
in
− 4k
p(x).
=
n e
n
2 sin α
Also,
−iα n+γ+(N/2)
ie
in e−inα ei(γ+N/2)(α̂π/2−α)
ω n+γ+(N/2) =
= n n
2 sin α
2 sin α (2| sin α|)γ+(N/2)
e
Then
Z
e
RN
−
∆k
4
l(Xα )
ω
=
p(y)Ek
ieiα
2
e 2 sin α |x| .
iα
∆k
2
2
ix
inα 2ie
, y e−ω|y| ωk (y) dy = A−1
e sin α |x| e− 4 p(x). (5.33)
α e
sin α
2
i
Finally, if we multiply equation (5.33) by Aα e− 2 cot(α)|x| , we obtain the desired
result.
A consequence of the Master formula (5.28) is
Corollary 5.2. (Hecke type identity) If in addition to the assumption in Theorem
5.3, the polynomial p ∈ Hnk , then (5.28) becomes
|.|2
|x|2
Dkα e− 2 p (x) = einα e− 2 p(x).
(5.34)
Now, we are interesting to complete the spectral study of T started in proposition
4.4 by means of the Master formula. In fact we have the following
Corollary 5.3. L2 RN , ωk (x) dx decomposes as an orthogonal Hilbert space sum according to
M
L2 RN , ωk (x) dx =
Vn ,
n∈Z+
where
Vn =
e−
|x|2
2
e−
∆k
4
p(x);
p ∈ Pn
is the eigenspace of T corresponding to the eigenvalue in. In particular, T is essentially self-adjoint. The spectrum of its closure is purely discrete and given by
σ(T ) = iZ+.
Proof Let f be an element of the subspace Vn defined by
f (x) = e−
|x|2
2
e−
∆k
4
p(x),
where p ∈ Pn . From (5.28), the limits
Dkα f − f
=
α→0
α
exists in L2k (RN ) and equals inf. Then
lim
f ∈ D(T )
einα − 1
f
α→0
α
lim
and T (f ) = inf.
Hence, Vn is the eigenspace of T corresponding to the eigenvalue in.
(5.35)
24
S. GHAZOUANI, F. BOUZAFFOUR
6. Realization of the operator T.
The aim of the following is to find a subspace W ⊂ D(T ) of L2k (RN ) in which
we define T explicitly.
Lemma 6.1. For z ∈ CN set l(z) =
N
X
zi2 . Then for all z, ω ∈ CN ,
i=1
l(z)+l(ω)
Z
ck
Ek (2z, x)Ek (2ω, x)e
−A|x|2
RN
e A
ωk (x) dx = γ+N/2 Ek (2z/A, ω),
A
(6.1)
where A is a complex number such that <(A) > 0.
Proof The result is obtained by means of a similar technic used in the proof of
Lemma 5.2 and the following formula (see [6])
Z
2
ck
Ek (2z, x)Ek (2ω, x)e−|x| ωk (x) dx = el(z)+l(ω) Ek (2z, ω).
RN
Theorem 6.1. Let f ∈ L1k (RN ) ∩ L2k (RN ) such that Dk f ∈ L1k (RN ) and α 6∈
π
2 + kπ, k ∈ Z . Then
−iα γ+ N2 Z
2
2
i
ix
e
α
e 2 tan(α)(|x| +|y| ) Ek (
, y)Dk f (y)ωk (y)dy.(6.2)
Dk f (x) = ck
2 cos α
cos α
RN
Proof Let f ∈ L1k (RN )∩L2k (RN ) such that Dk f ∈ L1k (RN ). Let be an arbitrary
positive number and put
Z
F (x) =
f (y)g (y)ωk (y) dy,
RN
2
−(+ 2i
cot α)|y|
where g (y) = e
Ek
From (2.6), we deduce that
ix
sin α , y
.
|f (y)g (y)| ≤ |f (y)|,
so the dominated convergence theorem can be invoked again to give
i
lim F (x)
→0
=
2
e 2 |x| cot α α
Dk f (x).
Aα
(6.3)
Using Lemma 6.1, we can show
−
Dk g (ξ)
=
|x|2
|ξ|2
e 4 sin2 α+i sin 2α
− 4+2i cot α
e
Ek
(2 + i cot α)γ+N/2
x
, ξ .(6.4)
2 sin α + i cos α
Now applying the Parseval formula for the Dunkl transform (see Lemma 4.25, [2])
and using (6.4), we obtain
−
F (x)
=
|x|2
e 4 sin2 α+i sin 2α
(2 + i cot α)γ+N/2
Z
e
RN
|ξ|2
− 4+2i cot α
Ek
x
, ξ Dk f (−ξ)ωk (ξ) dξ.
2 sin α + i cos α
(2.5) gives again the following majorization:
2 sin α
x
|x||ξ|
Ek
, ξ ≤ e 42 sin2 (α)+cos2 (α)
.
2 sin α + i cos α
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
25
Hence,
− |ξ|2
2
x
e 4+2i cot α Ek
, ξ ≤ e−p |ξ| +qξ |ξ| ,
2 sin α + i cos α
(6.5)
where
p =
42 + cot2 α
and
q =
2 sin(α)|x|
.
42 sin2 (α) + cos2 (α)
As p > 0, we deduce that
sup(−p s2 + q s) = −
s≥0
q2
.
4p
Applying formula (6.5) and (6.6), we obtain
− |ξ|2
x
e 4+2i cot α Ek
, ξ Dk f (−ξ) ≤
2 sin α + i cos α
≤
where Bx = sup e
4|x|2
− 42 sin2 (α)+cos2 (α)
∈]0,1]
(6.6)
4|x|2
e
− 42 sin2 (α)+cos2 (α)
|Dk f (−ξ)|
Bx |Dk f (−ξ)|,
. The function ξ 7→ Dk f (−ξ) is in L1k (RN ),
then the dominated convergence theorem implies
i|x|2
Z
i|ξ|2 tan α
e sin 2α
ix
2
lim F (x) =
,
ξ
Dk f (−ξ)ωk (ξ)(6.7)
dξ.
e
E
−
k
→0
cos α
(i cot α)γ+N/2 RN
Hence, (6.3) and (6.7) gives after simplification
−iα γ+ N2
Z
i
i
e
ix
α
|x|2 tan α
|ξ|2 tan α
2
2
Dk f (x) = ck
e
e
Ek −
, ξ Dk f (−ξ)ωk (ξ)(6.8)
dξ.
2 cos α
cos α
RN
Finally, if we make the change of variables u = −y in (6.8), then we find (6.2).
Remark 6.1. Using (6.2) together with the dominated convergence theorem, we
get
lim Dkα f (x)
=
lim Dkα f (x)
=
α→0+
α→π −
lim Dkα f (x) = Dk2 f (−x) = f (x), a.e,
α→0−
lim Dkα f (x) = Dk2 f (x) = f (−x) a.e.
α→−π +
Corollary 6.1. Under the assumptions of Theorem 6.1, we have
Z
i
Dkα f (x) − f (x)
ck
ix
tan(α)(|x|2 +|y|2 )
2
= r1 (α) γ+(N/2)
e
Ek
, y Dk f (y)ωk (y)dy
α
cos α
2
RN
Z
ck
+
r2 (α, x, y)Dk f (y)ωk (y)dy, a.e,
(6.9)
N
2γ+ 2 RN
where
−iα γ+ N2
e
2
2
i
ix
−1
e 2 tan(α)(|x| +|y| ) Ek ( cos
cos α
α , y) − Ek (ix, y)
r1 (α) =
and r2 (α, x, y) =
.
α
α
Proof The result is consequence of (6.2) and (2.8).
26
S. GHAZOUANI, F. BOUZAFFOUR
Lemma 6.2. Let α0 ∈]0, π2 [ and x, y ∈ RN . Then
|r2 (α, x, y)| ≤
1
| sin(α0 )| √
(1 + tan2 α0 )(|x|2 + |y|2 ) +
N |x||y|,
2
cos2 (α0 )
(6.10)
where α ∈]0, α0 ].
Proof By the mean value theorem, we have
∂
|r2 (α, x, y)| ≤ sup r3 (α, x, y) ,
α∈[0,α0 ] ∂α
where
2
i
r3 (α, x, y) = e 2 tan(α)(|x|
+|y|2 )
Ek
ix
,y .
cos α
From (2.4), we get
Ek
ix
,y
cos α
iy
= Ek x,
.
cos α
Therefore,
r3 (α, x, y) = e
i
2
tan(α)(|x|2 +|y|2 )
iy
Ek x,
.
cos α
A simple calculations shows that
∂
r3 (α, x, y)
∂α
=
+
i
(1 + tan2 α)(|x|2 + |y|2 )r3 (α, x, y)
2
N
i sin(α) i tan(α)(|x|2 +|y|2 ) X
∂
iy
2
y
e
E
.(6.11)
x,
j
k
cos2 (α)
∂yj
cos α
j=1
From (2.5), the inequality
∂
iy
≤ |x|
E
x,
∂yj k
cos α holds and hence
∂
r3 (α, x, y) ≤
∂α
≤
≤
N
X
1
| sin(α)|
(1 + tan2 α)(|x|2 + |y|2 ) +
|x|
|yj |
2
2
cos (α)
j=1
1
| sin(α)| √
(1 + tan2 α)(|x|2 + |y|2 ) +
N |x||y|
2
cos2 (α)
1
| sin(α0 )| √
(1 + tan2 α0 )(|x|2 + |y|2 ) +
N |x||y|.
2
cos2 (α0 )
Which finishes the proof.
Theorem 6.2. Let
W = f ∈ L1k (RN ) ∩ L2k (RN ) ; |y|2 f ∈ L2k (RN )
and
|y|2 Dk f ∈ L1k (RN ) ∩ L2k (RN ) .
Then for all f ∈ W,
i
i
T f (x) = −i(γ + (N/2))f (x) + |x|2 f (x) + Dk |y|2 Dk (y) (−x) a.e.
2
2
(6.12)
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
27
Proof It is clear that
lim r1 (α) = −i(γ + (N/2)).
α→0
In view of (2.6), we deduce
ix
Ek
, y ≤ 1.
cos α
Then
i tan(α)(|x|2 +|y|2 )
ix
e 2
, y Dk f (y) ≤ |Dk f (y)|.
Ek
cos α
Let y ∈ RN such that |y| > 1. Then
|Dk f (y)| ≤ |y|2 |Dk f (y)|.
Since y 7−→ |y|2 Dk f ∈ L1k (RN ), it follows that Dk f ∈ L1k (RN ) and the dominated
convergence theorem implies
2
2
i
ix
2 tan(α)(|x| +|y| ) E
,
y
Dk f (y)ωk (y)dy
e
k
α→0
cos α
2γ+(N/2) RN
Z
ck
= −i(γ + (N/2)) γ+(N/2)
Ek (ix, y)Dk f (y)ωk (y)dy
2
RN
= −i(γ + (N/2))Dk2 f (−x)
lim r1 (α)
ck
Z
= −i(γ + (N/2))f (x), a.e.
From (6.11), we deduce
lim r2 (α, x, y) =
α→0
i
(|x|2 + |y|2 )K(ix, y).
2
By the previous Lemma, we have the following majorization:
|r2 (α, x, y)Dk f (y)| ≤ f1 (y) + f2 (y) + f3 (y),
where
f1 (y)
=
f2 (y)
=
f3 (y)
=
1
(1 + tan2 α0 )|x|2 |Dk f (y)|,
2
1
(1 + tan2 α0 )|y|2 |Dk f (y)|,
2
| sin(α0 )| √
N |x||y||Dk f (y)|.
cos2 (α0 )
28
S. GHAZOUANI, F. BOUZAFFOUR
Since y 7−→ |y|2 Dk f ∈ L1k (RN ), it follows that f1 , f2 , f3 ∈ L1k (RN ) and therefore
f1 + f2 + f3 ∈ L1k (RN ). By virtue of the dominated convergence theorem, we have
Z
ck
lim
r2 (α, x, y)Dk f (y)ωk (y)dy
N
α→0 2γ+ 2
N
Z R
i ck
(|x|2 + |y|2 )Ek (ix, y)Dk f (y)ωk (y)dy
=
2 2γ+ N2 RN
Z
i|x|2 ck
=
Ek (ix, y)Dk f (y)ωk (y)dy
2 2γ+ N2 RN
Z
i ck
|y|2 Ek (ix, y)Dk f (y)ωk (y)dy
+
2 2γ+ N2 RN
Z
i 2
i ck
=
|y|2 Ek (ix, y)Dk f (y)ωk (y)dy, a.e,
|x| f (x) +
2
2 2γ+ N2 RN
i
i 2
|x| f (x) + Dk |y|2 Dk (y) (−x) a.e.
=
2
2
Corollary 6.2.
1) S(RN ) ⊂ W ⊂ D(T ).
2) For all f ∈ S(RN ),
1
−iT f = −(γ + (N/2))f + (|x|2 − ∆k )f
2
Proof 1) Obvious.
2) Let f ∈ S(RN ). From Corollary 2.11 in [6], we deduce
−yj2 Dk f (y) = Dk [Tj2 f ](y),
where j ∈ {1, 2, . . . , N } . Then
−|y|2 Dk f (y) = Dk [∆k f ](y).
Therefore
−Dk |y|2 Dk (y) (−x)
=
Dk2 [∆k f (y)](−x)
=
∆k f (x).
(6.13)
Finally, from (6.12) and (6.13) we obtain the desired result.
Remark 6.2. It is clear that the operator 2iT − (2γ + N ) is an extension on W of
the Hermite operator Hk = ∆k − |x|2 studied by Rösler [17] where it used another
approach based on the notion of Lie algebra.
In the same context, we give a new proof of the following result established in [17]
Corollary 6.3. For n ∈ N and p ∈ Pn , the function f = e−
|x|2
2
e−
∆k
4
p(x) satisfies
(∆k − |x|2 )f = −(2n + 2γ + N )f.
(6.14)
(∆k − |x|2 )hν = −(2|ν| + 2γ + N )hν .
(6.15)
In particular
Proof Since f ∈ S(RN ), (6.14) is obtained by the use of the previous Corollary
and (5.35)
A FRACTIONAL POWER FOR DUNKL TRANSFORMS
29
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Sami Ghazouani
Institut Préparatoire aux Etudes d’Ingénieur de Bizerte, Université de Carthage, 7021
Jarzouna, Tunisie.
E-mail address: Ghazouanis@yahoo.fr
30
S. GHAZOUANI, F. BOUZAFFOUR
Fethi Bouzaffour
Department of mathematics, College of Sciences, King Saud University, P. O Box 2455
Riyadh 11451, Saudi Arabia.
E-mail address: fbouzaffour@ksu.edu.sa