International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 11, November 2013
ISSN 2319 - 4847
Fourier Transform and Plancherel Formula for
Galilean Group (Spacetime).
Kahar El-Hussein
Department of Mathematics, Faculty of Science, Al Furat University, Dear El Zore, Syria &
Department of Mathematics, Faculty of Science and Arts Al Qurayat, Al-Jouf University, KSA
Abstract
The Galilean group  is one of the most important group in non-relativistic physics, which is used to transform between the
coordinates of two reference frames. The well known Galileo relativity principal is equivalent to the invariance of the motion
equation and in the special relativity the Galilean group are replaced by Lorentz group. In this paper, we will de.ne the Fourier
transform on  in order to prove the Plancherel theorem.
Keywords: Galilean Group, Semi direct Product of Two Lie Groups, Fourier Transform and Plancherel Formula
AMS 2000 Subject Classification: 4330&3505
1 INTRODUCTION
The Galilean group is the group of transformations which connect inertial frams of reference. such farms of reference can
be displaced from one another, shifted in time, rotated in space, and can move constant relative velocity. The Galilean
group is non commutative Lie group of the 10-parameters of transformations in the four dimensional (), and is
one of the most important group in non-relativistic physics. The mathematical essence of the special relativity is the
Galilean group is replaced by the Lorentz group. One of the interesting branch in mathematical physics is the Fourier
transform on the non commutative Lie groups and which is the most widely used mathematical tools in engineering for
solving problems in robotics, image analysis, computer vision , mechanics. For a long time, people have tried to construct
objects in order to generalize Fourier transform and Pontryagin,s theorem to the non abelian case. However, with the dual
object not being a group, it is not possible to define the Fourier transform and the inverse of the Fourier transform
between and b. The goal of this paper is to define the Fourier transform on by combining the classical Foureir
transform on Rand on a connected compact Lie group, and then we demonstrate the Plancherel Theorem.
2 Notations and Results.
2.1. In the following we will introduce some notations and results on the Galilean group.
Definition 2.1. The special Galilean group is a 10- dimensional Lie
group and defined by
1 0 s 


GA={X€GL(n,5); X= v r x }


0 0 1 
(1)
(1)
where 2 R2 R 2 R , and 2 (3)The multiplication of two
elements and is given by
3
1 0
v r
1

0 0
3
s
x 
1 
1 0
w r
2

 0 0
t  1
0


y  = v  r1 w r1 r2
1   0
0
st

x  tv  r1 y 

1
(2)
and the inverse of any element 


 X
1
0
 1
1

1
  r v r1
0
 0
s

r (vs  x)
1

1
Volume 2, Issue 11, November 2013

(3)
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 11, November 2013
ISSN 2319 - 4847


The following matrix of the form
1 0 0 
G1  { X  GA; X  0 r 0  SO(3)}
0 0 1 
(4)
1 0 s 
G2  { X  GA; X  v I x  7 }
0 0 1 
1 0 s 
G3  { X  GA; X  0 I x  5 }
0 0 1 
(5)
(6)
1 0 0
G4  { X  GA; X  v I 0 3 }
0 0 1
(7)
are all subgroups of 
2.2. If we refer to [8910]then it is easy to show that 2 =3 o4 is the semidirect product of the two groups 4
with 3 and G=2o1 is the semidirect product of the two groups 2 with 1
So we get G=(3o4 )o1 ' ((4o 3 ) o(3)where is the group homomorphism from (3) into
(4 o 3 ) of all automorphisms of (4o3)which is defined by
1 ( x)( x, s, v)  (rx, s, rv)
(8)
and is the group homomorphism from 3 into (4 )of all automorphisms of 4which is defined by
 2 (v)( y, s)  ( y  vs, s)
(9)
where =(123)=(123). Then
X .Y  ( x, s, v, a)( y, t , w, b)  ( x, s, v)( 1 (a)( y, t , w), ab)  (( x, s, v)(ay, t , w), ab)
(( x, s)( 2 (v)(ay, t )), v  aw, ab)  (( x, s)  (ay  tv, t ), v  aw, ab)
(10)
 ( x  ay  tv, s  t , v  aw, ab)
2.3. We denote by 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
2() the Hilbert space of . So for any 2 ()and 2 (), we have
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Volume 2, Issue 11, November 2013
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  f (X ) 
 f (Y
1
X ) (Y )dY
GA
(11)
where = dy7 dy6dy5dy4dy3dy2dy1dr is the Haar measure on 
=(y7, y6, y5, y4, y3, y2, y1, r ), X=(x7, x6, x5, x4, x3, x2, x1, r ), and  denotes the convolution product on For
more details see [313]. Let B=4x4x(3) be the group of the direct product of 4 , 3 and SO(3), then for 2
() and 2 (), we get
 c f ( X , r )  f c  ( X , r )   f ( X  Z , q 1r ) ( Z )dZdr
(12)
B
where  signify the convolution product on the abelian group B, r  SO(3), , q  SO(3), dz7dz6dz5dz4dz3dz2dz1 and
Z=(z7, z6, z5, z4, z3, z2, z1). In this paper we will prove the Plancherel theorem
3 Fourier transform and Plancherel Theorem
4
3
3
3.1. Consider the set K        SO(3)  SO(3) with law
XY  (( x1 , s1 , u1 , v1 , q1 , r1 )( x 2 , s 2 , u 2 , v 2 , q 2 , r1 )
 (( x1 , s1 , u1 , v1 , q1 )( 1 (r1 )(( x 2 , s 2 ), u 2 , v 2 , q 2 )), r1r2 )
 (( x1 , s1 , u1 , v1 )(1 (r1 )((x 2 , s 2 ), u 2 , v2 )), q1q 2 , r1r2 )
 (( x1 , s1 , u1 , v1 )(r1 x 2 , s 2 ), u 2 , r1v 2 , q1 q 2 , r1r2 )
 (( x1 , s1 , u1 )(  2 (v1 )((r1 x2 , s 2 ), u 2 )), v1  r1v2 , q1 q 2 , r1r2 )
 (( x1 , s1 )(  2 (v1 )(r1 x 2 , s 2 )), u1  u 2 , v1  r1v 2 , q1q 2 , r1r2 )
 (( x1 , s1 )  (r1 x 2  v1 s 2 .s 2 ), u1  u 2 , v1  r1v 2 , q1q 2 , r1 r2 )
 ( x1  r1 x 2  v1 x 2 , s1  s 2 , u1  u 2 , v1  r1v2 , q1 q 2 , r1r2 )
(13)
for all X  K and Y  K. In this case the group can be identified with the closed subgroup R4 x{0}xR3 x{0}xSO(3) of
and with the closed subgroup R4 xR3 x{0}x SO(3)x fg of K, where {0}=(0,0,0) is the identity element of R3 and fg
is the identity element of SO(3).
1
Definition 3.1. For every f  L (GA) one can define function
 ( f ) on as
f (( x, s ), u, v, q, r )  f (( 1 (q)(  2 (u )( x, s )), v  u ), qr )
Remark 3.1. The function f is invariant in the following sense
f (( 1 (h)(  2 (k )( x, s )), u  k , v  k ), qh 1 , hr )  f (( x, s ), u, v, q, r )
(14)
So any function  ( x, s, v, r ) on GA can be extended as a unique function  ( x, s, u , v, q, r ) on K
1
1
Theorem 3.1. For every function   L ( K ) invariant in sense (14) and for every   L ( K ) , we have
   ( x, s , u , v, q , r )    c  ( x , s , u , v, q , r )
(15)
for every ( x, s, u , v, q , r )  K , where  signifies the convolution product on GA with respect the variable ( x, s, v, r )
and  is the convolution product on B with respect the variables ( x, s, u , q ) .
Proof: Since is invariant in sense (14), we get
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   (( x, s ), u , v, q, r )
  [((x , s ), v , r ) 1 (( x, s, )u , v, q, r )] (( x, s ), v , r )dx ds dvdr 
GA
  [( 1 (r  1 )( x , s )), v , r  1 ](( x, s ), u , v, q, r )] (( x , s ), v , r )dx ds dv dr 
GA
  [( 1 (r  1 )(( x , s ), v ) 1 (( x, s), u , v)), q , r  1r )] ( x , s , v , r )dx ds dvdr 
GA
  [((1 (r  1 )(  2 (v)(( x , s ) 1 ( x, s )), u , v  v )), q, r  1 r ] ( x , s , v , r )dx ds dv dr 
GA
   [((1 (r  1 )( x  x , s  s ), u  v , v)), q , r  1 r ] ( x, s , v , r )dx ds dv dr 
   [( x  x , s  s ),u  v , v, qr  1 , r ] ( x , s , v, r )dxds dv dr     c  (( x, s ), u , v, q, r )
B
(16)


Let SO (3) be the set of all irreducible unitary representations of SO(3) . If   SO (3), we denote by E the space of
the representation of  and d its dimension.

Definition 3.2. The Fourier transform of a function f  C (SO(3)) is defined as

f ( ) 
 f ( x) ( x
1
) dx
(17)
SO ( 3)
1
2
Theorem 3. 2. Let f  L ( SO(3))  L ( SO(3) , then we have the Fourier inversion

f ( x)  
d

tr
(
f
( ) ( x ))

(18)
 SO ( 3)
and the Plancherel formula

SO ( 3)

where f ( )
2
2
f ( x ) dx 



SO ( 3 )

d f ( )
2
(19)

is the norm of Hilbert-Schmidt of the operator f ( ) , see [2].
1
Definition 3. 3. If f  L (GA) , we can define the Fourier transform of f as follows
f (( ,  ), ,  )    f (( x, s ),v, r ) e  i  (( , ), ),( x , s,v )   (r 1 ) dxdsdvdr
(20)
where
( x, s )  ( x1 , x 2 , x3 , s)   3     4 , v  (v1 , v 2 , v3 ),   (1 ,  2 ,  3 ),   ,  (1 , 2 , 3 )   3 ,
dx  dx1dx2 dx3 , dv1dv2 dv3
1
Definition 3. 4. If f  L (GA) , we can define the Fourier transform of its invariant f by
(f )( ,  , ,0,  , I )

  
7
 

3 SO ( 3)

SO ( 3)
d  tr[
 f (( x, s), u, v, q, r ) (r
)] (q 1 )dqe i  ( , , ),( x ,s ,u ) e i   ,v  dxdsdudvd Lemma 3. 1.
SO ( 3 )
     d  tr[ f (( x, s), u, v, q,  )] (q

 7 3 SO ( 3) SO ( 3)
For every
1
1
)dqe i  ( , , ),( x ,s ,v ) e i   ,v  dxdsdudvd
  L1 (GA), and f  L1 (GA), we have
(  f )(( ,  , ,0,  , I )  (f )( ,  , ,0, I ) ( ,  , ,  )
(21)
Proof: By (15) we have
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 (  f )( ,  , ,  ,  , I )d   ( 
3
c
f )( ,  , ,  ,  , I )d
3
(22)
 ( )( ,  , ,  )(f )( ,  , ,  ,  , I )d   (f )( ,  , ,0,  , I )( )( ,  , ,  )
3
1
2
Theorem 3.3 (Plancherel formula). For any f  L (GA)  L (GA) , we get

2
f ( x, s, v, r ) dxdsdvdr 
GA
Proof: Let
   d

 7 SO ( 3 )
2
f ( ,  , ,  ddd
(23)
~
 ( f ) be the function defined by
~
~
 ( f )(( x, s ), u , v, q, r )  f ( 1 (q )(  2 (u )(( x, s )), u  v), qr )
(24)
 f ( 1 ( q )(  2 (u )(( x, s )), u  v, qr ) 1
then first we get
~
f   ( f )(( x, s ), u , v, q, r )
~
   ( f )[((x, s), v, r ) 1 ((0,0, ),0,0, I , I )] f (( x, s ), v, r )dxdsdvdr
GA
~
   ( f )[( 1 ( r 1 )(( x, s )), v ) 1 , r 1 )((0,0),0,0, I , I )] f (( x, s ), v, r ) dxdsdvdr
GA
~
   ( f )[( 1 ( r 1 )(( x, s ), v) 1 ((0,0),0,0)), I , r 1 )] f ( x, s, v, r ) dxdsdvdr
(25)
GA
~
   ( f )[(( 1 ( r 1 )(  2 ( v)(( x, s) 1  (0,0)),0,0  v )), I , r 1 ] f ( x, s, v, r ) dxdsdvdr
GA

~
 f [(( (r
1
1
)(  2 ( v)((0  x,0  s ),v )), r 1 ] f ( x, s, v, r ) dxdsdvdr
GA

 f ( x, s, v, r ), f ( x, s, v, r )dxdsdvdr  
GA
2
f ( x, s, v, r ) dxdsdvdr
GA
Secondly
~
f   ( f )(0,0,0,0, I , I ) 

   

 7  3  7 SO ( 3)
d
~
 d  [   ( f  ( f ))(( x, s), u, v, q, r) (r


SO ( 3)
1
)dr (q 1 )dq]
SO ( 3 ) SO ( 3)
e i  ( ,  , ),( x , s ,u ) e i   ,v  dxdsdudvdddd

     d  tr[  ( f   ( f ))(( x, s ), u , v, q , I ) (q 1 ) dq]e i  ( , , ),( x , s ,u ) e i   ,v  dxdsdudvdddd

 7  3  7  SO ( 3)


SO ( 3)
~
      d  tr[( f  ( f ))( ,  , ,  ,  , I )]dddd

 3   3  3 SO ( 3)
~
     d  tr[( f  ( f ))( , , ,0,  , I )]ddd
   SO ( 3)
3

3
     d  tr[( f 
c
~
 ( f ))( ,  , ,0,  , I )]ddd
   SO ( 3)
3

3
~
     d  tr[( ( f ))( , , ,0,  , I )( f )( , , ,  )]ddd
   SO ( 3)
3


3
   d


 3   3  SO ( 3)
tr[( f )( ,  , ,   )( f )( ,  , ,  )]ddd
   ( f )( ,  , ,  )
2
ddd
3  3
Whence our theorem
Volume 2, Issue 11, November 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 2, Issue 11, November 2013
ISSN 2319 - 4847
4 Acknowledgment
This work is not support by any university or any research center
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