CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION

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1 Avilble online hp://scik.org Eng. Mh. Le. 15, 15:4 ISSN: CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION PANDEY, C. P. 1, RAKESH MOHAN AND BHAIRAW NATH TRIPATHI 3 1 Deprmen o Mhemics, Ajy Kumr Grg Engineering College, Ghzibd 19, Indi Deprmen o Mhemics, Dehrdun Insiue o Technology, Dehrdun 489, Indi 3 Deprmen o Mhemics, Urkhnd Technicl Universiy, Dehrdun 487, Indi Copyrigh 15 Pndey, Mohn nd Triphi. This is n open ccess ricle disribued under he Creive Commons Aribuion License, which permis unresriced use, disribuion, nd reproducion in ny medium, provided he originl work is properly cied. Absrc. Clderon-ype reproducing ormul or Dunkl convoluion is esblished using he heory o Dunkl rnsorm. Keywords: wvele rnsorm; Dunkl convoluion; Dunkl rnsorm. 1 Mhemics Subjec Clssiicion: 4C4, 44A35, 65T6, 65R1. 1. Inroducion Clderon ormul [8] involving convoluion reled o he Fourier rnsorm is useul in obining reconsrucion ormul or wvele rnsorm besides mny oher pplicions in decomposiion o cerin uncion spces. I is expressed s ollows: * Corresponding uhor d ( ) ( )( ), (1.1) x x n n where : C nd ( x) ( x / ),. For condiions o vlidiy o ideniy (1.1), we my reer o [8]. On he rel line, he Dunkl operor re dierenil-dierence operor inroduced by Dunkl [1] nd re denoed by, where is rel prmeer 1/.These operor ssocied wih he relecion group on. The Dunkl kernel E is used o deine he Dunkl rnsorm which ws inroduced by Dunkl in []. Rosler in [3] show h he Dunkl kernels veriy produc ormul. This llows o deine he Dunkl rnslion. As resul, we hve he Dunkl convoluion. Received December 11, 14 1

2 PANDEY, MOHAN AND TRIPATHI Dunkl Operor hs unique soluion E x, clled Dunkl kernel nd given by x E x j i x j i x 1 1, x R, (1.) where j is he normlized Bessel uncion o he irs kind nd order. Le 1/ be ixed number nd be he weighed Lebesgue mesure on R, given by : 1 d x x dx. (1.3) We deine L p, (, ), 1 p, which s he spces o hose rel mesurble uncion on(, ) or 1 p p x d x i p 1, p, (1.4) R nd ess sup ( x) i p =. xr The Dunkl kernel gives rise o n inegrl rnsorm, clled Dunkl rnsorm on R, which ws inroduced nd sudied in [7]. The Dunkl rnsorm F o uncion L1, ( R), is given by F E ix x d x ; R (1.5) An inversion ormul or his rnsorm is given by R F 1 x E ix d (1.6) An Prsevl ormul or his rnsorm is given by R x g x dx g (1.7) To deine Dunkl convoluion, we deine where W x, y, z E ( x) E ( y) E ( z) d ( ) (1.8) x, y, z x y z xy 1 x, y, z z, x, y z, y, x x, y, z, i x, y R \ oherwise

3 CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION 3 nd is he Bessel kernel. Clerly W x, y, z is symmeric in x, y, z. Apply inversion ormul (1.6) o (1.8), we ge E ( z) W x, y, z d ( z) E ( x) E ( y). (1.9) Now seing, we obin W x, y, z d ( z) 1. (1.1) Le p, q, r [1, ) nd 1. Then Dunkl convoluion o Lp, ( R) nd g Lq, ( R) r p q is deined by [7] ( g)( x) z g y W ( x, y, z) d y d z (1.11) RR p nd 1 r p q Le, q, r 1, * g x sisies he ollowing norm inequliy, L R nd g L R p, q,. Then convoluion * g 4 g (1.1) (i) r, p, q, Moreover or ll L R nd g L R 1, (ii) * g g,, we hve (1.13). Clderon s ormul In his secion, we obin Clderon s reproducing ideniy using he properies o Dunkl rnsorm nd Dunkl convoluions. Theorem.1 Le nd 1, [, ) be such h ollowing dmissibiliy condiion holds: L d ( ) ( ) 1 (.1) or ll [, ). Then he ollowing Clderon s reproducing ideniy holds: 1 d ( x) * * ( x), L ( R). (.) Proo: Tking Dunkl rnsorm o he righ hnd side o (.), we ge

4 4 PANDEY, MOHAN AND TRIPATHI d F * * ( x) = Now, by puing Hence he resul ollows. 垐 d ( ) ( ) ( ) = 垐 d ( ) ( ) ( ) (.3) = 垐 ( ) ( ) ( ) d = ( ) ( ) ( ) d ( ) ( ) d (.4) 1. Theorem. Suppose 1, [, ) is rel vlued nd sisies For 1,, L d ( ) 1. (.5) L [, ) L [, ), suppose h Then, s &., d, ( x) * * ( x) (.6) Proo: Tking Dunkl rnsorm o boh sides o (.6) nd using Fubini s heorem, we ge By [4], we hve, ( ) ( ) ( ) * * * d, 1,,. 1,, (.7) (.8) Now using bove inequliy nd Minkowski s inequliy [6, pge 41], we ge

5 CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION 5 d d x * * ( ), x Hence by Prsevl ormul, we ge Since lim * * ( x) d x d d * * ( x) (.9), 1,, d log 1,,. lim,,,, lim ( ) 1 d ( ) d x. (.1) ( ) 1 d ( ) ( ), hereore by he domined convergence heorem, he resul ollows. The reproducing ideniy (.) holds in he poin wise sense under dieren se o nice condiions. Theorem.3 Suppose, L1, [, ). Le L1, [, ) be rel vlued nd sisies Then Proo: Le d ( ) 1, R. (.11) By [4, pge 311], we hve d lim * * ( x) ( x). (.1) d, ( x) * * ( x). (.13)

6 6 PANDEY, MOHAN AND TRIPATHI * * * 1, 1, 1, 1, 1, (.14) Now Thereore, 1, L(, ) d, d x * * ( ) 1, x * * ( x) d x d d * * ( x) (.15) 1, 1, 1, d log 1, 1,.. Also using Fubini s, we ge heorem nd king Dunkl rnsorm o (.13), we ge ( ) ( ) ( * * )( ) d, E x x d d E ( x )( * * )( x) d x (.16) ( ) ( ) d ( ) ( ) [ ( )] Thereore, by (.11),, ( ) ( ). d I ollows h, 1, [, ).By inversion, we hve L. Puing ( x) ( x) E ( x )[ ( ) ( )] d, x [, ) (.17),, h, ( : x) E ( x ) ( ), ( )

7 CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION 7 we ge Now using (.11) in (.18), we ge ( ) E ( ) 1 x [ ( )] d (.18) ( x), ( x) ( ) ( ), ( ) E x d (.19),, h ( : x) d. lim h ( : x), R. (.) Since h, ( : x) ( ), he Lebsegue domined convergence heorem yields lim ( x), ( x), x. (.1) Conlic o Ineress The uhors declre h here is no conlic o ineress. REFERENCES [1] C.F.Dunkl, Dierenil-dierence operors ssocied wih relecions groups, Trns. Amer. Mh. Soc. 311(1989), [] C.F.Dunkl, Hnkel rnsorms ssocied o inie relecion groups, Amer. Mh. Soc. 138 (199), [3] M.Rosler, Bessel-ype signed hypergroups on, in Probbily mesures on groups nd reled srucure, XI (Oberwolch, 1994), H.Heyer nd A.Mukherje, Eds., 9-34, World Scieniic, River edge, NJ, USA, [4] E.Gorlich nd C.Mrke, A convoluion srucure or Lguerre series, Indg.Mh.44 (198), [5] F.M.cholewinski nd D.T.Him, The dul Poisson-Lguerre rnsorm, Trns.Am.Mh.Soc.144 (1969), [6] H.L.Ellio nd M.Loss, Anlysis, Nros Publishing House, New Delhi, [7] Vgi S.GULIYEV nd Ygub Y.MAMMADOV, Funcion Spces nd Inegrl Operors or he Dunkl Operors on he Rel Line, Khjr Journl o Mhemics 4 (6), [8] M.Frzier, B.Jwerh, nd G.Weiss, Lilewood-Pley heory nd he sudy o uncion spces, CBMS Regionl Conerence Series in Mhemics, Vol.79, Americn Mhemicl Sociey, Rhode Islnd, 1991.