International Journal of Scientific & Engineering Research, Volume 5, Issue 5, May-2014 ISSN

Transcription

1 1601 Non Commutative Fourier Transform and Partial Di erential Equation on the Motion roup *ahar El-Hussein **Badahi Ould Mohamed *Department of Mathematics, Faculty of Science, Al Furat University, Dear El ore, Syria & Department of Mathematics, Faculty of Science and Arts Al Qurayat, Al-Jouf University, ingdom of Saudi Arabia, kumath@ju.edu.sa, kumath@hotmail.com * Badahi Ould Mohamed Department of Mathematics, Faculty of Arts and Science at Al Qurayat, Al Jouf University, ingdom of Saudi Arabia, badahi1977@yahoo.fr May 23, 2014 Abstract Let be the n dimensional real vectoriel group, a connected compact Lie group and o be the motion group, which is the semi-direct product of the group and : Let U be the enveloping algebra of, which is the algebra of all the invariant partial di erential equations on. In this paper, we will de ne the Fourier transform on and we demonsrate the Plancherel theorem. Besides we give necessary and su cient condition for the existence of a fundemantal solution for any invariant partial di erential equations P on. eywords: ey words : Motion roup, Semidirect Product of Two Lie roups, Fourier Transform, Plancherel Theorem, Invariant Partial Differential Operators. AMS 2000 Subject Classi cation: 43A30&35D

2 Introduction and Results. 1. Let be the n dimensional real vectoriel group, a compact Lie group and :! L( ) a continuous linear representation from in. Let o be the motion group, which is the semi-direct product of the group and : We supply by invariant scalar product which is denoted by h ; i: Let S( ) be the Schwartz space of. We denote by S() the complemented of the space S( ) C 1 () tensor product of S( ) and C 1 (): The topology of the space S() which is de ned by the family of l ;(f) sup jpj; sup (1 + jvj 2 ) Q v D l f(v; y) 2 (1) (v;y)2 turns S() a Frechet space wich can be called the Schwartz space of, where j j signi es the norm associated to (p); see [3]: De nition 1.1. Let P be an invariant di erential operator on a connected Lie group H. by de nition P is said to be semi-globally solvable if there exist a distribution T on H such that P T H ; where H ias the Dirac measure at the identity element of H: De nition 1.2. Let P be an invariant di erential operator on a connected Lie group H by de nition P is said to be globally solvable if for any function g 2 C 1 (H); there exist a function f 2 C 1 (H) such that P f g (2) For all the following notations and results, see [2] 2. Let k be the Lie algebra of and ( 1 ; 2 ; :::::; m ) a basis of k, such that the both operators m i 2 (3) D q 0lq i1! l m i 2 (4) are left and right invariant (bi-invariant) on ; this basis exist see [2; p:564). For l 2 N, let D l (1 ) l, then the family of semi-norms f l, l 2 Ng such that l (f) ( D l f(y) 2 dy) 1 2 ; f 2 C 1 () (5) 2 i1 2014

3 1603 de ne on C 1 () the same topology of the Frechet topology de ned by the semi-norms k fk 2 de ned as k fk 2 ( j f(y)j 2 dy) 1 2 ; f 2 C 1 () (6) where ( 1 ;..., m ) 2 N m ; see [2; p:565] 3. Let b be the set of all irreducible unitary representations of : If 2 b, we denote by E the space of representaion and d its demension then we get 1 hd q tr(r); tr(r)i d q () (7) If u() is a polynomial on R n valued in E ; we denote by eu() the matrix de ned by [eu ij ()] (8) where eu ij 2 uij () () 2 A (9) jjm ( 1 ; 2 ; :::::; m ); 1 j n; 1 :::::: m ; and ( 1 ; 2 ; ::::; n ) is basis of R n : In this case we get. keu()k 2 u() () 2 (10) jjm De nition 1.3. de ned as The Fourier transform of a function f 2 C 1 () is T f() f(x)(x 1 )dx (11) where T denotes the Fourier transform on Theorem (A. Cerezo) 1.1. Let f 2 C 1 (); then we have the inversion of the Fourier transform f(x) 2 b dtr[t f()(x)] (12)

4 1604 f(i) and the Plancherel formula kf(x)k 2 2 2SO(3) dtr[t f()] (13) jf(x)j 2 dx 2 b d kt f()k 2 H:S (14) where I is the identity element of ;where kt f()k 2 H:S Hilbert-Schmidt of the operator T f() is the norm of the 2 Fourier Transform on the Motion roup De nition 2.1. For every function f belongs to L 1 ( ); one can de ne the Fourier transform of f by the following manner T Ff(; ) f(v; x)e ih ; vi (x 1 )dvdx (15) for all 2 ' and for all 2 ; b where F is the partial Fourier transform on the real vector Lie group Then we get the the Fourier inversion f(v; x) d tr[t Ff(; )e ih ; vi (x 1 )]d (16) 2 b d tr[t Ff(; )e ih ; vi (x 1 )]d (17) get f(0; 1) 2 b 2 b d tr[t Ff(; )]d (18) When SO(n);our result is Theorem 2.1. (Plancheral s formula) For any f 2 L 1 ()\ L 2 ();we f f(0; _ 1) jf(v; x)j 2 dvdx 2 b d kt Ff(; )k 2 2 d (19)

5 1605 where _ f is the function de ned by Proof: _ f(v; x) f(v; x) 1 (20) _ f f(0; 1) d tr 2 b T F( _ f f )(; )d d tr[ f _ f(v; x))e ih ; vi (x 1 )]dvdxd 2 b d tr[ f((u; y) 1 (v; x)) f(u; _ y)e ih ; vi (x 1 )]dvdxdudyd 2 b d tr[ f((u; y)(v; x))f(u; y) 1 e ih ; vi (x 1 )]dvdxdudyd 2 b d tr[ f(u + yv; yx)f(u; y)e ih ; vi (x 1 )]dvdxdudyd 2 b Chinging variables u + yv w, yx z; we have _ f f(0; 1) d 2 b d 2 b tr[ tr[ f(w; z)f(u; y)e ih ; y 1 (w u)i (z 1 y)]dwdzdudyd f(w; z) f(u; y)e ihy ; (w u)i (z 1 )(y)]dwdzd Using the fact that the lebesgue d is invariant by the group rotation SO(n); we get

6 1606 _ f f(0; 1) d tr[ f(w; z) f(u; y)e ihy ; (w u)i (z 1 )(y)]dwdzd 2 b d tr[ f(w; z) f(u; y)e ih; (w u)i (z 1 )(y 1 ) 1 ]dwdzdudyd 2 b d tr[ f(w; z)e ih; wi f(u; y)e ih; ui (z 1 ) (y 1 )]dwdzdudyd 2 b d tr[f(w; z)e ih; wi (z 1 ) f(u; y)e ih; ui (y 1 )]dwdzdudyd 2 b d tr[t Ff(; ) T Ff(; )]d d kt Ff(; )k 2 H:S d 2 b 2 b Hence our theorem Let L be the group with law: (v; x; y)(w; s; t) (v + (y)w; xs; yt) (21) Let D( ); S( ), and C 1 ( ) be C 1 with compact support, the Schwartz space and the space of C 1 - functions of the group L: De nition For any f 2 S(); we can de ne a function f e 2 S( ) as follows ef(v; x; y) f((xv); xy) f(xv:xy) (22) for all 2 ' and for all 2 b ;where xv signi es (xv): Note here that the function e f is invariant in the following sense ef(tv; xt 1 ; ty) e f(v; x; y) (23) We denote by D ( ); S ( ), and C 1 ( ) the spaces C 1 with compact, the Schwartz space and the space of C 1 - functions of the group L, which are invariant in sense (21)

7 1607 De nition for any two function f 2 S() and F 2 S ( ), we can de ne a convolution product of f and F on as f F (v; x; y) F ((w; z) 1 (v; x; y)f(w; z)dwdz F (z 1 (v w); x; z 1 y)f(w; z)dwdz (24) This leads to obtain Lemma 2.1. If F is invarant in sense (21), then we get f F (v; x; y) f c F (v; x; y) (25) for every f 2 C 1 ( ); (v; x; y) 2 L, where signi es the convolution product on fi g with respect the variables (v; y) and c signi es the convolution product on the subgroup B fi g of L;which is the direct product of and with respect the variables (v; x) Proof: Let f 2 S() and F 2 S ( ); then we have f F (v; x; y) F ((w; z) 1 (v; x; y)f(w; z)dwdz F (z 1 (v w); x; z 1 y)f(w; z)dwdz F ((v w); xz 1 ; y)f(w; z)dwdz F c f(v; x; y) (26) So the lemma is proved. De nition 2.4. If f 2 S(); one can de ne the Fourier transform of its invariant f e as T Ff(; e ; 1) d tr[ T Ff(v; e x; y)(y 1 )dy](x 1 )dxe ih ; vi dv 2 b 7 (27) 2014

8 1608 where e f is the function de ned by ef(v; x; y) f(xv; xy) (28) Theorem 2.2. For any two functions g and f belong to C 1 ( ) C 1 (), then we have T F(g f)(; e ; 1) T F( f e c g)(; ; 1) F( f)(; e ; 1)T F(g)(; ) (29) Proof: By lemma 2.1. we have if f and g two functions from S() T F(g f)(; e ; 1) d tr[(g f)(v; e x; )](x 1 )dxe ih ; vi dv 2 b d tr[ ((g f)(v; e x; y))(y 1 )dy](x 1 )dxe ih ; vi dv 2 b d tr[ 2 b u w; xt 1 z; this implies Changing variables v (( e f c g)(v; x; y))(y 1 )dy](x 1 )dxe ih ; vi dv(30) T F(g f)(; e ; 1) ef(v u; xt 1 ; 1)]g(u; t)dudt(x 1 )dxe ih ; vi dv ef(w; z; 1)]g(u; t)(t 1 z 1 )dxe ih ; w+ui dudtdwdzvdx ef(w; z; 1)]g(u; t)(z 1 )(t 1 )dzdte ih ; wi e ih ; ui dudw ef(zw; z)]g(u; t)(z 1 )(t 1 )dzdte ih ; wi e ih ; ui dudw(31) T F e f(; ; 1)T Fg(; ) (32)

9 1609 Theorem 2.3. (Plancheral s formula) For any f 2 L 1 ()\ L 2 ();we get f f(0; e_ 1) jf(v; x)j 2 dvdx d kt Ff(; )k 2 2 d (33) 2 b where e_ f(v; x; y) f((xv; xy) 1 ) (34) The proof of this theorem results immediatly from Fourier Transform and Di erential Operators on the Motion roups f(v; x) 3. We denote by L 1 () the Banach algebra that consists of all complex valued functions on the group, which are integrable with respect to the Haar measure of and multiplication is de ned by convolution on, and we denote by L 2 () the Hilbert space of. So we have for any f 2 L 1 () and 2 L 1 () f((w; y) 1 (v; x))(w; y)dwdy (35) where dwdy is the Haar measure on, and denotes the convolution product on : Let U be the complexi ed universal enveloping algebra of the real Lie algebra of ; which is canonically isomorphic onto the algebra of all distributions on supported by fi g; where fi g is the identity element of : For any u 2 U one can de ne an left invariant di erential operator P on as follows: P u f() u f(v; x) f((w; y) 1 (v; x))u(w; y)dwdy (36) for any f 2 C 1 (): The mapping u! P u is an algebra isomorphism of U onto the algebra of all invariant di erential operators on : where c means the convolution product on the group A; and dwdy is the Haare measure on A: We denote by U B the complexi ed universal enveloping

10 1610 algebra of the real Lie algebra of B;which is canonically isomorphic onto the algebra of all distributions on B supported by the fi B g; where fi B g is the identity element of A: For any u 2 U one can de ne a di erential operator with constant coe cients Q on B as f c Q u (v; x) f c u(v; x) f(v w; xy 1 )u(w; y)dwdy (37) for any f 2 C 1 (A); where c means the convolution product on the group B:The mapping u! Q u is an algebra isomorphism of U B onto the algebra of all invariant di erential operators on B: For more details see [4; 11]: Theorem 3.1. Let P be a right invariant di erential on the motion group, then P has a fundamental solution if and only if there is a constant C and a number q 2 N, such that co T Fu(0; det T Fu(0; ) 6 0; and e ) det T Fu(0; e ) d (q) (38) Proof: Let u be the distribution associted to the right invariant di erential operator P: Let End(E ) be the space of all enomorphisms of E and let P be the polynomial valued in End(E ); de ned by (; ) 7! T F( u)((; _ ) + (; )) T F( u)( _ + ; ) (39) For any f 2 D(), we put B hs; fi co T F( u)( _ + ; ) d tr[ det T F( u)( _ + ; ) T F(f)( + ; )](P ; )ddd!ddd (40) 2 b where is a ball in C n with center 0; is Hormander,s function[12]: Then by [2; ]; S de nes distribution on ; so we can de ne a new distribution es assiciated to S as

11 1611 h e S; f i hs; f e i co T F( u)( _ + ; ) d tr[ det T F( u)( _ + ; ) T F( f)( e + ; ; I)](P ; )dd 2 b where (P ; ) is the Hormander function [12]: Then we have by [2; ] and by Hormander construction, we have h ]u S; f i _ hu S; f e i hs; u f e i co T F( u)( _ + ; ) d tr[ 2 SO(3) \ det T F( u)( _ + ; ) T F(_ u f)( e + ; ; I))](P ; )dd co T F( u)( _ + ; ) d tr[ 2 b det T F( u)( _ + ; ) T F(_ u c f)( e + ; ; I)](P ; )dd co T F( u)( _ + ; )F( u)( _ + ; ) d tr[ T F( f)( e + ; ; I)](P 2 b det T F( u)( _ ; )dd + ; ) d tr[t F( f)( e + ; ; I)](P ; )dd 2 b d tr[t F( f)(; e ; I)]d f(0; e I; I) h ; fi e (41) 2 b Consequently, we have Hence the proof of our theorem. References u S(v; q) (v; q) (42) [1] Chirikjian,. S., and A. yatkin, A., (2000), Engineering Applications in Non-commutative Harmonic Analysis, Johns Hopkins University, Baltimore, Maryland, CRC Press

12 1612 [2] A. Cerezo and F. Rouviere, "Solution elemetaire d un operator di erentielle lineare invariant agauch sur un group de Lie reel compact" Annales Scientiques de E.N.S. 4 serie, tome 2, n o 4;p , [3]. El- Hussein, "Eigendistributions for the Invariant Di erential operators on the A ne roup". Int. Journal of Math. Analysis, ol. 3, no. 9, , [4]. El- Hussein, "Fourier transform and invariant di erential operators on the solvable Lie group 4", in Int. J. Contemp. Maths Sci. 5. No. 5-8, , [5]. El- Hussein, "On the left ideals of group algebra on the a ne group", in Int. Math Forum, Int, Math. Forum 6, No. 1-4, , [6]. El- Hussein, "Non Commutative Fourier Transform on Some Lie roups and Its Application to Harmonic Analysis", International Journal of Engineering Research & Technology (IJERT), ol. 2, Issue 10, , [7]. El- Hussein, "Abstract Harmonic Analysis and Ideals of Banach Algebra on 3-Step Nilpotent Lie roups", International Journal of Engineering Research & Technology IJERT, ol. 2 Issue 11, November [8]. El- Hussein, "Fourier Transform and Plancherel Formula for alilean roup (Spacetime)", International Journal of Application or Innovation in Engineering & Management, olume 2, Issue 11, November 2013, [9]. El- Hussein, "Fourier Transform and Plancherel Formula for the Lorentz roup", International Journal of Advanced Scienti c and Technical Research, Issue 3 volume 6, Nov.-Dec. 2013, [10] Harish-Chandra, "Plancherel formula for 2 2 real unimodular group", Proc. nat. Acad. Sci. U.S.A., vol. 38, pp , [11] S. Helgason, roups and eometric Analysis, Academic Press, [12] L. Hormander, "The analysis of Linear Partial Di erential Operator I", Springer-erlag,

13 1613 [13] M. Mursaleen and. El-Hussein, A Convolution Equation on Heisenberg roup and its ector roup. Southeast Asian Bulletin of Mathematics 31(2007), [14] W. Rudin, Fourier Analysis on roups, Interscience Publishers, New York, NY, [15]. Warner, Harmonic Analysis on Semi-Simple Lie roups, Springerverlag Berlin heidelberg, New york,