 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Cd Keywords: Du nkl O perators on ; -Functions on ; Dunkl Intertwining Operator; Mean-Periodic Functions; Continuous Linear Mappings 1. Introduction  ,: ,;and .tkkddVT gTVgTg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 . 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  that there exists a unique isomorphism kV from the space of homogeneous polynomials n on of de- gree n onto itself satisfying the transmutation relations: 11, ;kjkkjVVVx1,2,,.jd dCd d;,,kxddVfxfy y This operator is called the Dunkl intertwining operator. It has been extended to a topological automorphism of (the space of -functions on ) (see ). The operator Vk has the integral representation (see ): dfx (1) where x is a probability measur e on , such that d:.dxsuppy x tVky The dual intertwining oper ator k of V defined on (the dual space of d We use the Dunkl intertwining operator kV 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 dxk by  1:dd,,,ddxkxyddfyVfztztfy and  :, ,,.kyxddTfxT fyTf 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 Copyright © 2012 SciRes. APM M. SIFI, F. SOLTANI 275into itself such that jj kT fx, , has the form 1, ,jd,.dTdfxd 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 :,yyy\0 . For , let d be the reflection in the hyperplane orthogonal to dH: 22,:.yyy\0d 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: kw,,dy22:,kkwy 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1Cd  :.,jjjfyf yyydyfy fy k For , the initial problem ; .,uy ,jjx yuxy1, ,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 : d,kExy,,ddwhere 1i and ,: diiyz 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 , Theorem 6.3). The Dunkl inter- twining operator Vk defined by d;,,dkxddVfxfy yfxd is a topological isomorphism from onto itself, and satisfies: ;and1,, ,jkk jdVf Vfx;,,yzddzkxExzey x (2) fjd (3) 00.kVf fkVkV From Theorem 1, we deduce also the following re- sults. Theorem 2. The dual intertwining operator t of defined on d by  ,: ,;and ,tkkddVT gTVgTgd1tkV (4) is a topological isomorphism from onto itself. Its inverse operator  is given by   11,, ;and .tkkddVTgTVgTg (5) We denote by dHd the space of entire functions on which are rapidly increasing and of exponential type. We have 0,ddaaHH where dHda is the space of entire functions f on satisfying Im,1 ,supdNaNfe  where 2211,,,.ddd k d by We define the Dunkl transform on :, ,.;and .kkddTTEiT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 .iddTTed,.tdTT (7) Proposition 1. admits on  the following decomposition: k kkTV (8) Proof. In (4), we take ,.ige and applying rela- tion (2) we obtain  ,.; .tdTk,.,,ikkVTeTE i Then the result follows from (6) and (7). Theorem 3. (Paley-Wiener theorem). is a topo- logical isomorphism from d onto . dHProof. The result follows from (8), Theorem 2 and Paley-Wiener theorem for the Fourier transform (see ). Definition 1. The Dunkl translation operators (see ) are the operators x, , defined on , by dxd :,,dy y1,,xkxkykfyVVVf x (9) which can be written as:  dd.xyz tf,dxy1ddxkfyVf zt We next collect some properties of Dunkl translation operators (see ). Proposition 2. Let and . Then d1) 0ff, xyfyfx and xy yxff. 2) jxxj,1,,ffj d . 3) Product formula:  ,yd,.xk kEy ,kExE, . 4) The Dunkl translation operators x, , are continuous from onto itself. dxdTdf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kTfd:,, .dkyxTfxT fyx 0  (10) We notice that agrees with the convolution * that is given by TfxTf:, ;,.yddTfxyTdf 1tkkkkVTVfVTf (11) Theorem 4. Let and . Then d1) . 11tkk kkVT VfVTf. 2) dTdf and . Proof. Let 1) From (10) and (5), we have    111,,,.tkkktkxkyykyx kVTVfxVTVf yTVV fy But from (9), we obtain 1,,.ky xkkxVVfyVfxy  Thus  1,,,,.tkkkykxkx ykVTVfxTVfxyVT fxyVT fx 2) From (11) and (4), we have   111,,,.tkktkkyyky kVTVfxVTV fxyTVVfx y But from (9), we obtain 11,,.kykkx xVVfxyVfy  Thus  111,,1,,.tkkykx xkxyxkkVTV fxTVfyV TfyVT fxdT Which completes the proof of the theorem. Proposition 3. Let . The mapping kTfd is continuous from onto itself. fnf is a sequence in Proof. Assume that d such that nffg and knTfn, as , where f, g being in ddxxn x. According to Proposition 2 4), for every , ffn as , in d. Hence Tfx Tfx ndxkknk , as , for every . By using the closed graph theorem we conclude that the mapping fTf is continuous from d,dTS kTS 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,: ,,kxyxTSfTS fy , ,kTSfSd (12) where is the distribution in given by , ,dSf f,.dx x0,,Sf with  fx f We notice that agrees with the convolution * that is given by , .dS,: ,,;xyTSfTSfxyT  Proposition 4. Let ,dTS k. Then 1) and TTkkTSST. 2) kT S tT VSkSkk kTS. 3) . ttkkVT Sk kProof. 1) follows from (12 ) . V2) From Proposition 3, the distribution T be- longs to , and by (6), we have d ,,Ei,dkk kkTS TSx . Thus, by (7) and Proposition 2 3), we obtain  ,,kk xykkTS TS,..xkEi yTS  3) From 2) and (8) we obtain .ttk kT VStkkVT 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, fkTfxor all.dx For example, let 0\0xd00,xfx0,xkd. The function f in satisfying  is mean-periodic, because we have 0xfxfx 0x being the Dirac measure at 0x. We now characterize the Dunkl mean-periodic func- tions on d1kf. 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 dT0T and , such that 0.kTf 1kVApplying  to this equation, then Theorem 4 2) implies that 10.tkkVT Vf From Theorem 2, 0tVT1kVfk, thus is a clas- sical mean-periodic function. Conversely, if 1fdT010.kTV fkV is a classical mean-periodic function, there exists and T, such that kV10.tkkVTf10tkVT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  the functions ,,,,ix dFxixex are classical mean-periodic functions. Then from Theo- rem 5, the functions  ,, ,,, ,dkk kExiVFxDEixx  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 jjxx1, ,jd, d, then has the form , on 00,.dfxT fxTx Proof. For a fixed f, the map fx is a continuous form o n d. So there exists dxT, such that ,, .dxfx Tfx Copyright © 2012 SciRes. APM M. SIFI, F. SOLTANI 278 Using the fact jjxxjd, , on 1, ,d, we deduce xjxjTiTx,1,,.jd Then 00.TT,ixxxTe0,xxTT Thus,  and   0,,.Tftyfxd000,,xxfx TfTfxyT  Lemma 2. Every continuous linear mapping from into itself, such that jjx 1, ,jd, , has the form  kkfxTVfx,.dT1k Proof. Applying V to the relation jjx , , and using the fact that 1,jd,11kjkjVxV1,jd, , we obtain the deduce ,11kkjjVVxx,1,,.jd1k By applying Lemma 1, we deduce that V0,kkkkk, and Theorem 4 1) yields  10tkkfxVfxVTTVf1tVTfxVfxxTVTd where . 0kWe now establish the main result of this paragraph. Theorem 6. Every the continuous linear mapping from into itself, such that jj 1, ,jd,.dT, , has the form  kfx Tfx Proof. Using the relation jk kjVVx1, ,jd, , and the fact that jj , , we obtain 1,,jd,1,,.jkjk kjVVV jdx  kV By applying Lemma 2, we deduce that , and hence 1.kkfxVfxTfx  Remark 2. Let be continuous linear mapping from d into itself, such that jj 1, ,jd , . By virtue of Theorem 6, we can find dT such that ,, .dkyxfx TfxTfyf dx In particular (by Proposition 2 3)), for every , we have  .,, .,,, ,.kyxkkykEzxTEzyExzTEyz :, ,ykzTEyzWe put , we obtain .,, ,.dkkEzxExzz zdz Hence, for every .,Ez, k is an eigenfunc- tion of associated with the eigenvalue z. REFERENCES  C. F. Dunkl, “Differential-Difference Operators Associ-ated with Reflections Groups,” Transactions of the American Mathematical Society, Vol. 311, 1989, pp. 167- 183. doi:10.1090/S0002-9947-1989-0951883-8  C. F. Dunkl, “Integral Kernels with Reflection Group Invariance,” Canadian Journal of Mathematics, Vol. 43, No. 6, 1991, pp. 1213-1227. doi:10.4153/CJM-1991-069-8  M. F. E. de Jeu, “The Dunkl Transform,” Inventiones Mathematicae, Vol. 113, No. 1, 1993, pp. 147-162. doi:10.1007/BF01244305  K. 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