The Commutants of the Dunkl Operators on d

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 mappingsfrom into itself which commute with the Dunkl operators.   d   d  C d 


Introduction
and .
, are differential-difference operators associated with a positive root system  and a non negative multiplicity function k, introduced by Dunkl in [1].These operators extend the usual partial derivatives and lead to a generalizations of various analytic structure, like the exponential function, the Fourier transform, the translation operators and the convolution product [2][3][4].Dunkl proved in [2] that there exists a unique isomorphism k V from the space of homogeneous polynomials n on of degree n onto itself satisfying the transmutation relations: 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 V k has the integral representation (see [6]): where x  is a probability measure on , such that ).As applications of this theory we study the meanperiodic functions on the space  in the Dunkl setting.We characterize also the continuous linear map- 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 t k .Next, we define the Dunkl translation operators x  , and the Dunkl convolution product ), by In Section 3, we study the mean-periodic functions associated to the Dunkl operators on .We prove that every continuous linear mapping from In the one-dimensional case (d = 1), the Dunkl convolution operators and the Dunkl mean-periodic functions are studied in [7][8][9], on the space of entire functions on .

The Dunkl Harmonic Analysis on
We consider with the Euclidean inner product a function which is constant on the orbits under the action of G).For abbreviation, we introduce the index: Moreover, let denotes the weight function: which is G-invariant and homogeneous of degree  .
The Dunkl operators j  ; , on associated with the finite reflection group G and multiplicity function k are given for a function f of class on , by 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 where 1 i and , : is the measure on given by (1). We denote by the space of C -functions on , and by the space of distributions on of compact support.
Theorem 2. The dual intertwining operator t of defined on and , is a topological isomorphism from onto itself.Its inverse operator  is given by , , ; and .


We denote by the space of entire functions on which are rapidly increasing and of exponential type.We have is the space of entire functions f on satisfying , , , .
and .
We notice that agrees with the Fourier transform that is given by  : , ; and .
admits on  the following decomposition: Proof.In (4), we take and applying relation (2) we obtain .
Definition 1.The Dunkl translation operators (see [4]) are the operators x  , , defined on , by which can be written as: We next collect some properties of Dunkl translation operators (see [4]).
We notice that agrees with the convolution * that is given by , .
, where . According to Proposition 2 4), for every , , as , for every .By using the closed graph theorem we conclude that the mapping into itself. The Proposition 3 used to investigate the following definition.
Definition 3. Let .The Dunkl convolution product of T and S, is the distribution  in where is the distribution in given by   , , We notice that agrees with the convolution * that is given by , , ; and T T Proof. 1) follows from (12). V 2) From Proposition 3, the distribution T  belongs to , and by ( 6), we have Thus, by (7) and Proposition 2 3), we obtain .
3) From 2) and ( 8) we Then we deduce the result from the injectivity of the Fourier transform on

Commutators and Mean-Periodic Functions
In this section, we use Theorem 4 to study the Dunkl mean-periodic functions on   , and to give a characterization of the continuous linear mappings from into itself which commute with the Dunkl operators j  being the Dirac measure at 0 x .We now characterize the Dunkl mean-periodic func-

Theorem 5. A function f is mean-periodic function if and only if the function V
is a classical meanperiodic function.
Proof.Let f be a mean-periodic function, then there exists is a classical mean-periodic function, there exists and T , such that Applying to this equation, then Theorem 4 1) implies that From Theorem 2, , thus f is a meanperiodic function.
Remark 1.Let and .From [11] the functions , , , , are classical mean-periodic functions.Then from Theorem 5, the functions are mean-periodic functions.

Commutator of Dunkl Operators
In this section, we give a characterization of the contenuous linear mappings from into itself which commute with the Dunkl operators j  1, , , and using the fact that , we obtain the deduce ,  where .

k
We now establish the main result of this paragraph.

y
The dual intertwining operator 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.