Advances in Pure Mathematics, 2012, 2, 274-279
http://dx.doi.org/10.4236/apm.2012.24035 Published Online July 2012 (http://www.SciRP.org/journal/apm)
The Commutants of the Dunkl Operators on d
d

d
C
*
Mohamed Sifi1, Fethi Soltani2
1Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis, Tunis, Tunisia
2Higher College of Technology and Informati cs, Tunis, Tunisia
Email: mohamed.sifi@fst.rnu.tn, fethisoltani10@yahoo.com
Received January 17, 2012; revised Februa ry 28, 2012; accepted March 8, 2012
ABSTRACT
We consider the harmonic analysis associated with the Dunkl operators on . We study the Dunkl mean-periodic
functions on the space (the space of
-functions). We characterize also the continuous linear mappings-
from into itself which commu te with the Dunkl operators.

d
d
Cd
Keywords: Du nkl O perators on ; -Functions on ; Dunkl Intertwining Operator; Mean-Periodic Functions;
Continuous Linear Mappings
1. Introduction

 
,: ,;
and .
tkk
dd
VT gTVg
Tg


tV

d
The Dunkl operators
j
1, ,jd;
d
Pd
, on , are dif-
ferential-difference operators associated with a positive
root system and a non negative multiplicity func-
tion k, introduced by Dunkl in [1]. These operators ex-
tend the usual partial derivatives and lead to a generali-
zations of various analytic structure, like the exponential
function, the Fourier transform, the translation operators
and the convolution product [2-4]. Dunkl proved in [2]
that there exists a unique isomorphism k
V from the
space of homogeneous polynomials n on of de-
gree n onto itself satisfying the transmutation relations:

11, ;
kjkk
j
VVV
x1,2,,.jd 

d
Cd
 
d;
,,
kx
dd
Vfxfy y
This operator is called the Dunkl intertwining operator.
It has been extended to a topological automorphism of
(the space of -functions on ) (see [5]).
The operator Vk has the integral representation (see [6]):

d
fx

(1)
where
x
is a probability measur e on , such that
d


:.
d
x
s
uppy x 
tVk

y
The dual intertwining oper ator k of V defined on
(the dual space of
d
We use the Dunkl intertwining operator k
V and its
dual k to study the harmonic analysis associated with
the Dunkl operators (Dunkl translation operators, Dunkl
convolution, Dunkl transform, Paley-Wiener theorem,
etc.). As applications of this theory we study the mean-
periodic functions on the space in the Dunkl
setting. We characterize also the continuous linear map-
pings from
d
V V
into itself which commute with the
Dunkl operators.
The contents of this paper are as follows. In the second
section we recall some results about the Dunkl operators.
In particular, we give some properties of the operators
k and tk. Next, we define the Dunkl translation op-
erators
x
, and the Dunkl convolution product
d
x
k
by
 

1
:dd,
,,
dd
xkxy
dd
yVfztzt
fy





and

 
:, ,
,.
kyx
dd
TfxT fy
Tf
 



d
d
), by In Section 3, we study the mean-periodic functions as-
sociated to the Dunkl operators on . We prove
that every continuous linear mapping from
*Authors partially supported by DGRST project 04/UR/15-02 and
CMCU program 10G 1503.
d
C
opyright © 2012 SciRes. APM
M. SIFI, F. SOLTANI 275
into itself such that
j
j
 

k
T fx

, , has the
form 1, ,jd

,.
d
T

d

fx
d
In the one-dimensional case (d = 1), the Dunkl convo-
lution operators and the Dunkl mean-periodic functions
are studied in [7-9], on the space of entire functions on
.
2. The Dunkl Harmonic Analysis on
We consider with the Euclidean inner product .,.
and norm :,
y
yy

\0 .
For , let
d
be the reflection in the
hyperplane orthogonal to
d
H
:
2
2,
:.
y
yy


\0
d

A finite set is called a root system, if
.
,

  and
 for all
. We
assume that it is normalized by 22
for all
.
For a root system , the reflections
,
generate a finite group
dGO
\
d
, the reflection group
associated with . All reflections in G, correspond to
suitable pairs of roots. For a given
H

,
we fix the positive subsystem:

::,0.

 
Then for each
 either
 or


.
Let be a multiplicity function on
:k
(i.e.
a function which is constant on the orbits under the
action of G). For abbreviation, we introduce the index:
 
:.kk



Moreover, let denotes the weight function:
k
w


,,
d
y

2
2
:,
k
k
wy y

which is G-invariant and homogeneous of degree
.
The Dunkl operators
j
; , on asso-
ciated with the finite reflection group G and multiplicity
function k are given for a function f of class on ,
by
1,,jd d
1
Cd
  
:.
,
jj
j
f
yf y
yy
d
y
fy fy k



For , the initial problem
;

.,uy
 
,
jj
x yuxy1, ,jd
, with
0, 1uy
admits a unique analytic solution on , which will be
denoted by and called Dunkl kernel [2,3].
This kernel has the Laplace-type representation [6]:
d

,
k
Exy

,
,d
d
where 1
i and
,: d
ii
yz yz
x
d
is the measure on
given by (1).
We denote by
d
d
the space of C-functions on
, and by
d
d
the space of distributions on
of compact support.
Theorem 1. (See [5], Theorem 6.3). The Dunkl inter-
twining operator Vk defined by


d;
,,
d
kx
dd
Vfxfy y
fx




d


is a topological isomorphism from onto itself,
and satisfies:

;
and1,, ,
jkk j
d
Vf Vf
x
;,,
y
zdd
z
kx
Exzey x


(2)
f
jd





(3)

00.
k
Vf f
k
V
k
V
From Theorem 1, we deduce also the following re-
sults.
Theorem 2. The dual intertwining operator t of
defined on
d
by

 
,: ,;
and ,
tkk
dd
VT gTVg
Tg


d

1
tk
V
(4)
is a topological isomorphism from onto itself.
Its inverse operator
is given by

 
 
11
,, ;
and .
tkk
dd
VTgTVg
Tg

(5)
We denote by
d
H
d
the space of entire functions
on which are rapidly increasing and of exponential
type. We have

0
,
dd
a
a
HH

where
d
H
d
a is the space of entire functions f on
satisfying


Im
,1 ,
sup
d
Na
Nfe

 
where

22
11
,,,.
d
dd
 

k
d
by We define the Dunkl transform on


:, ,.;
and .
kk
dd
TTEi
T



0
(6)
We notice that agrees with the Fourier transform
Copyright © 2012 SciRes. APM
M. SIFI, F. SOLTANI
276
that is given by


T
,.
:, ;
and .
i
dd
TTe


d

,.
td
TT

(7)
Proposition 1. admits on the following
decomposition: k
 
kk
TV (8)
Proof. In (4), we take ,.i
ge
and applying rela-
tion (2) we obtain
 

,.; .
td
T
k
,.
,,
i
kk
VTeTE i

Then the result follows from (6) and (7).
Theorem 3. (Paley-Wiener theorem). is a topo-
logical isomorphism from
d
onto .

d
H
Proof. The result follows from (8), Theorem 2 and
Paley-Wiener theorem for the Fourier transform
(see [10]).
Definition 1. The Dunkl translation operators (see [4])
are the operators
x
, , defined on , by
d
x

d
 
:,,
d
y y

1
,,xkxkyk
fyVVVf x
(9)
which can be written as:
 

dd.
xy
z t
f,d
xy
1
dd
xk
fyVf zt



We next collect some properties of Dunkl translation
operators (see [4]).
Proposition 2. Let and . Then

d
1) 0ff
,

xy
f
yfx

and
xy yx
f
f

.
2)


jxxj,1,,
f
fj d

 .
3) Product formula:
 
,y
d
,.
xk k
Ey


 ,k
ExE
, .
4) The Dunkl translation operators
x
, , are
continuous from onto itself.
d
x
d
T

d
f
The 4) of Proposition 2 used to investigate the
following definition.
Definition 2. Let and . The
Dunkl convolution product of T and f, is the function
in defined by

d
k
Tf

d

:,, .
d
kyx
TfxT fyx
 
0
 
(10)
We notice that agrees with the convolution * that
is given by


Tfx
Tf


:, ;
,.
y
dd
Tfxy


T

d
f

 
1
tkkkk
VTVfVTf
(11)
Theorem 4. Let and . Then

d
1) .

11tkk kk
VT VfVTf

. 2)
d
T

d
f and . Proof. Let
1) From (10) and (5), we have
 

 
 
1
1
1
,
,
,.
tkkk
tkxk
y
ykyx k
VTVfx
VTVf y
TVV fy




But from (9), we obtain

1
,,
.
ky xkkx
VVfyVfxy
 


Thus
 


1
,,
,,
.
tkkk
ykxkx y
k
VTVfx
TVfxyVT fxy
VT fx


2) From (11) and (4), we have


 
 
1
1
1
,
,
,.
tkk
tkk
y
yky k
VTVfx
VTV fxy
TVVfx y




But from (9), we obtain


11
,,
.
kykkx x
VVfxyVfy


 

Thus

 


1
11
,,
1
,,
.
tkk
ykx xkxyx
kk
VTV fx
TVfyV Tfy
VT fx





d
T
Which completes the proof of the theorem.
Proposition 3. Let . The mapping
k
Tf

d
is continuous from onto itself.
f
n
f is a sequence in Proof. Assume that
d
such that n
f
fg and kn
Tf
n, as , where
f, g being in
d
d
xxn x
. According to Proposition 2 4), for
every ,
f
f
n as , in
d
.
Hence

Tfx Tfx n
d
x
k
knk , as , for every
. By using the closed graph theorem we conclude
that the mapping
f
Tf is continuous from
d

,d
TS
k
TS
into itself.
The Proposition 3 used to investigate the following
definition.
Definition 3. Let . The Dunkl convolu-
tion product of T and S, is the distribution
in
d
defined by
Copyright © 2012 SciRes. APM
M. SIFI, F. SOLTANI 277

,: ,,
kxyx
TSfTS fy
 , ,
k
TSf
S

d
(12)
where is the distribution in given by

, ,
d
Sf f
,.
d
x x
0
,,Sf
with
 
fx f
We notice that agrees with the convolution * that
is given by

, .
d
S
,: ,,;
xy
TSfTSfxyT 
Proposition 4. Let
,d
TS
k
. Then
1) and TT
kk
TSST

.
2)

k
T S
 
t
T VS
kS
kk k
TS.
3) .

tt
kk
VT S
k k
Proof. 1) follows from (12 ) .
V
2) From Proposition 3, the distribution T
be-
longs to , and by (6), we have

d

 
,,Ei

,d
kk kk
TS TSx
 .
Thus, by (7) and Proposition 2 3), we obtain

 

,,
kk xy
kk
TS TS




,.
.
xk
Ei y
TS

 
3) From 2) and (8) we obtain


.
tt
k k
T VS
t
kk
VT SV
Then we deduce the result from the injectivity of the
Fourier transform on
d
.
3. Commutators and Mean-Periodic
Functions
In this section, we use Theorem 4 to study the Dunkl
mean-periodic functions on
d

d
, and to give a char-
acterization of the continuous linear mappings from
into itself which commute with the Dunkl op-
erators
j
; . 1,,jd
3.1. Mean-Periodic Functions
Definition 4. A function f in is said mean-

d
d
0Tperiodic, if there exists T and
, such
that

0, f
k
Tfxor all.
d
x
For example, let
0\0x
d

00,
xfx

0,
xk
d. The function f in
satisfying
is mean-periodic, because we have

0x
f
xfx


0
x
being the Dirac measure at 0
x
.
We now characterize the Dunkl mean-periodic func-
tions on
d

1
kf
.
Theorem 5. A function f is mean-periodic function if
and only if the function V is a classical mean-
periodic function.
Proof. Let f be a mean-periodic function, then there
exists
d
T
0T and
, such that
0.
k
Tf
1
k
V
Applying
to this equation, then Theorem 4 2)
implies that

10.
tkk
VT Vf
From Theorem 2,
0
tVT

1
k
Vf
k, thus is a clas-
sical mean-periodic function.
Conversely, if
1f

d
T
0

10.
k
TV f
k
V is a classical mean-periodic
function, there exists and T, such that
k
V


10.
tkk
VTf



10
tk
VT
d
d

Applying to this equation, then Theorem 4 1)
implies that
From Theorem 2, , thus f is a mean-
periodic fu n ction.
Remark 1. Let and . From [11] the
functions
,
,,,
ix d
Fxixex




are classical mean-periodic functions. Then from Theo-
rem 5, the functions
 
,, ,,, ,
d
kk k
ExiVFxDEixx
 
 



d
are mean-periodic functions.
3.2. Commutator of Dunkl Operators
In this section, we give a characterization of the con-
tenuous linear mappings from into itself
which commute with the Dunkl operators
j
1, ,jd
;
.
Lemma 1. Let be a continuous linear mapping
d
into itself, such that from
j
j
x
x



1, ,jd
,
d
, then has the form , on

00
,.
d
fxT fxT

x
Proof. For a fixed
f
, the map
fx is a
continuous form o n
d
. So there exists
d
x
T
,
such that
,, .
d
x
fx Tfx
Copyright © 2012 SciRes. APM
M. SIFI, F. SOLTANI
278
Using the fact
j
j
x
x



jd, , on 1, ,
d
,
we deduce




xjx
j
TiT
x


,1,,.jd
Then







0
0.
T
T


,ix
x
x
Te


0,
xx
TT

Thus,

and
 
 
0
,
,.
Tfty
fx


d
0
00
,,
xx
fx Tf
TfxyT

 

Lemma 2. Every continuous linear mapping from
into itself, such that j
j
x
 1, ,
jd
,
,
has the form
 
kk
fxTVfx

,.
d
T
1
k
Proof. Applying V to the relation j
j
x
 ,
, and using the fact that
1,
jd,11
kjk
j
V
x

V
1,jd
,
, we obtain the deduce
,
11
kk
jj
VV
xx



,1,,.jd
1
k
By applying Lemma 1, we deduce that V





0
,
kk
kkk
,
and Theorem 4 1) yields
 



1
0
t
kk
f
xVfx
VT
TVf




1
t
VTfx
Vfx
x


TVT

d
where .
0
k
We now establish the main result of this paragraph.
Theorem 6. Every the continuous linear mapping
from into itself, such that
j
j
 
1, ,jd

,.
d
T
,
, has the form
 
k
fx Tfx
Proof. Using the relation jk k
j
VV
x
1, ,jd
,
,
and the fact that
j
j
 , , we obtain 1,,jd
,1,,.
jkjk kj
VVV jd
x
 
 
k
V
By applying Lemma 2, we deduce that
, and
hence

1.
kk
f
xVfxTfx

 
Remark 2. Let be continuous linear mapping
from
d
into itself, such that
j
j
 
1, ,jd ,
.
By virtue of Theorem 6, we can find
d
T
such that

,, .
d
kyx
fx TfxTfyf
 
d
x
In particular (by Proposition 2 3)), for every ,
we have

 
.,, .,
,, ,.
kyxk
kyk
EzxTEzy
ExzTEyz


:, ,
yk
zTEyz
We put , we obtain

.,, ,.
d
kk
EzxExzz z
d
z
Hence, for every

.,Ez
, k is an eigenfunc-
tion of associated with the eigenvalue
z.
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