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 Total Downloads : 185
 Authors : Kahar ElHussein
 Paper ID : IJERTV2IS100367
 Volume & Issue : Volume 02, Issue 10 (October 2013)
 Published (First Online): 24102013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Non Commutative Fourier Transform on Some Lie Groups and Its Application to Harmonic Analysis
Non Commutative Fourier Transform on Some Lie Groups and Its Application to Harmonic Analysis.
Kahar ElHussein
Department of Mathematics, Faculty of Science, Al Furat University, Dear El Zore, Syria and Department of Mathematics, Faculty of Science, AlJouf University, KSA
October 23, 2013
Abstract
This paper will focus on Fourier transform on the semidirect prod uct of two Lie groups to obtain some results in abstract harmonic analysis. In fact the combining of the classical Fourier transform on R, and on a compact Lie group permits us to define the Fourier trans form, and then to obtain the Plancherel formula on these groups. In the end we will introduce some interesting new groups.
Keywords: Key words : Semidirect Product of Two Lie Groups, Fourier Transform, Plancherel Formula
AMS 2000 Subject Classification: 43A30&35D 05


Noncommutative Fourier analysis is a beautiful and powerful area of pure mathematics that has connections to, theoretical physics, chemistry
analysis, algebra, geometry, and the theory of algorithms. In mathematics, abstract harmonic analysis is the field in which results from Fourier anal ysis are extended to topological groups which are not commutative. 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. How ever, with the dual object not being a group, it is not possible to define the Fourier transform and the inverse Fourier transform between G and G. These difficulties of Fourier analysis on noncommutative groups makes the noncommutative version of the problem very challenging. It was necessary to find a subgroup or at least a subset of locally compact groups which were not pathological, or wild as Kirillov calls them [13]. Unfortunatly If the group G is no longer assumed to be abelian, it is not possible anymore to
consider the dual group G (i.e the set of all equivalence classes of unitary
irreducible representations). Abstract harmonic analysis on locally compact groups is generally a difficult task. Still now neither the theory of quantum groups nor the representations theory have done to reach this goal. So the important and interesting question is: One can do abstract harmonic analy sis on Lie groups i.e. the Fourier transform can be defined to solve the major problems of abstract harmonic analysis. Here are some interesting examples of these groupse.
( )
( )

The linear group GL(n, R) consisting of all matrices of the form
GL(n, R) = { aij : aij R, 1 i n, 1 j n} (1) The orthogonal group O(n, R)
O(n, R) = {A GL(n, R) : AA* = I and det A = 1 } (2)
Ã—
Ã—

The general Lorentz groupThe Lorentz group provides another in teresting example. Moreover, the Lorentz group O(3, 1) shows up in an interesting way in computer vision. Denote the p p identity matrix by Ip,p and define
Ip,q
= Ip,p 0
(
(
0 Iq,q
We denote by O(p, q) the group consisting of all matrices of the form
O(p, q) = {A GL(n, R), AtIp,q A = Ip,q } (3)
rv
rv
rv
rv
Ã—
Ã—
rv Ã—
rv Ã—

The n dimensional real Heisenberg group H. the Galelian group GA which is isomorphic onto the group HXi SO(3, R) semidirect product of H by SO(3, R), i.e GA HXi SO(3, R). Let SL(2, R) be the 2 2 real semisimple Lie group and let SL(2, C) be the 2 2 complex semisimple Lie group, then we get the Jacobi group GJ HXi SL(2, R) and the Poincare group(Space time) R4 Xi SL(2, C).
Recently, these problems found a satisfactory solution with the papers [6, 8, 10, 11] . The ways were introduced in those papers will be the business of the expertise in the theory of abstract harmonic analysis, and in theoretical physics, and that is what I am interested In this paper I will define the Fourier transform and establishing Plancherel formula for the semidirect of
two vector groups Rn Xi Rm (m n) and the motion group.



Let L = Rn Ã— Rm Ã— Rm be the group with law:
(x, t, r)(y, s, q) = (x + (r)y, t + s, r + q)
–
–
for all (x, t, r) L and (y, s, q) L. In this case the group G can be identified with the closed subgroup Rn Ã—{0} Ã— Rm of L and B with the subgroup RnÃ— Rm Ã— {0}of L.
–
–
Definition 2.1. For every f C(G), one can define a function f C(L) as follows:
f (x, t, r) = f ((t)x, r + t) (4)
–
–
–
–
for all (x, t, r) L. So every function (x, r) on G extends uniquely as an invariant function (x, t, r) on L.
Remark 2.1. The function f is invariant in the following sense:
– –
– –
f ((s)x, t s, r + s) = f (x, t, r) (5)
for any (x, t, r) L and s Rm.
Lemma 2.1. For every function F C(L) invariant in sense (5) and for every uU, we have
u F (x, t, r) = u c F (x, t, r) (6)
for every (x, t, r) L, where signifies the convolution product on G with re spect the variables (x, r) and csignifies the commutative convolution product on B with respect the variables (x, t).
r
r
Proof: In fact we have
PuF (x, t, r) = u F (x, t, r) =
G
F (y, s)1(x, t, r)u(y, s)dyds
= r F [((s)(y), s)(x, t, r)] u(y, s)dyds
G
G
r
r
= r F [(s)(x y), t, r s] u(y, s)dyds
G
G
= F [x y, t s, r] u(y, s)dyds = u c F (x, t, r) = QuF (x, t, r) (7)
G
S
S
S Ã— S
S Ã— S
where Pu and Qu are the invariant differential operators on G and B respec tivel. As in [9], we will define the Fourier transform on G. Therefor let (G) be the Schwartz space of G which can be considered as the Schwartz space of (Rn Rm), and let !(G) be the space of all tempered distributions on
G. The action of the group Rm on Rn defines a natural action of the dual group (Rm)of the group Rm ((Rm) rv Rm) on (Rn), which is given by :
((t)(), x) = (, (t)(x)) (8) for any = (1, 2, …, n) Rn , t = (t1, t2, …, tm) Rm and x = (x1, x2, …, xn)
Rn. Also we have, for every u S(G) and f S(G)
u f(x, t, r) = u c f(x, t, r) (9)
S F
S F
Definition 2.1. If f (G), one can define its Fourier transform f
r
r
by :
Ff (, ) =
G
f (x, t) e i ((,x)+(,t)) dxdt (10)
for any = (1, 2, …, n) Rn, x = (x1, x2, …, xn) Rn, = (1, 2, …, m)
Rm and t = (t1, t2, …, tm) Rm, where (,x) = 1×1 + 2×2 + … + nxn and
(,t) = 1t1 + 2t2 + … + mtm . It is clear that Ff S(Rn+m) and the mapping f Ff is isomorphism of the topological vector space S(G) onto S(Rn+m).
Definition 2.2. If f S(G), we define the Fourier transform of its
– r
– r
invariant f as follows
F(f )(, , 0) =
LÃ—Rm
f(x, t, s)e i ((,x)+(,t)) e i (Âµ,s) dxdtdsdÂµ (11)
where (Âµ,s) Rn+m and (Âµ, s) = Âµ1s1 + Âµ2s2 + … + Âµmsm
r
r
– r
– r
Corollary 2.1. For every u S(G), and f S(G), we have
F(f)(, , Âµ)F(u)(, )dÂµ
F(f)(, , Âµ)F(u)(, )dÂµ
F(u f )(, , Âµ)dÂµ =
Rm Rm
–
–
= F(f )(, , 0)F(u)(, ) (12)
for any = (1, 2, …, n) Rn, = (1, 2, …, m) Rm and Âµ = (Âµ1, Âµ2, …, Âµm)
Rm, where u(x, t) = u(x, t)1
Proof : By equation (9)we have
– –
– –
u f (x, t, r) == u c f (x, t, r) (13)
r
r
Applying the Fourier transom we get
– –
– –
F(u f )(, , Âµ)dÂµ = F(u c f )(, , 0)
Rm
F(f)(, , 0)F(u)(, ) (14)
Theorem 2.1.(Plancherels formula). For any f L1(G) L2(G),
r
r
f (x, t) dxdt = r
f (x, t) dxdt = r
2
2
we get
2
G Rn+m
F f (, ) dd (15)
f
f
–
–
Proof: First, let – be the function defined by
f (x, t, r) = f (((t)x, r + t)1
) (16)
then we have
f
f
f
f
f – (0, 0, 0) = r – r(x, t)1(0, 0, 0)1 f (x, t)dxdt
G
f
f
= r – [(t)((x) + (0)), 0, 0 t] f (x, t)dxdt
G
f
f
= r – [(t)(x), 0, t] f (x, t)dxdt
G
G
= r [(t)(x), t] f (x, t)dxdt = r f (x, t)f (x, t)dxdt
f
f
r
r
G G
2
2
= f (x, t) dxdt (17)
G
Second by (12), we obtain
f
f
r
r
F(f )(, , Âµ)dddÂµ = r
F(f )(, , Âµ)dddÂµ = r
f – (0, 0, 0)
f
f
=
r
r
F()(, , 0)F(f )(, )dd = r
F()(, , 0)F(f )(, )dd = r
Rn+2m
f
Rn+2m
F(f c – )(, , Âµ)dddÂµ
=
r
r
Rn+m
f
F(f )(, ) dd = r
F(f )(, ) dd = r
2
2
Rn+m
F(f ) (, )F(f )(, )dd
= 2
Rn+m G
f (x, t) dxdt (18)
which is the Plancherels formula on G. So the Fourier transform can be extended to an isometry of L2(G) onto L2(Rn+m).
Corollary 2.2. In equation (18), replace the second f by g, we obtain
f (x, t)g(x, t)dxdt =
r
r
r
r
G Rn+m
F(f )(, Fg(, )dd (19)
which is the Parseval formula on G.


S
S
S S
S S
S S
S S

Let V be the n dimensional vectoriel group, K a compact Lie group and : K GL(V ) a continuous linear representation from K in V . Let G = V Xi K be the motion group, which is the semidirect product of the group V and K. We supply V byK invariant scalar product which is denoted by (1). Let (V ) be the Schwartz space of V . We denote (G) the complemented of the space (V ) C(K) tensor product of (V ) and C(K). The topology of the space (G) which is defined by the family of seminormas
l ,
(f ) = sup
p,
sup
(v,y)V Ã—K
(1 + v ) Qv D f (v, y) 2
(20)
2 l
2 l
turns S(G) a Frechet space wich can be called the Schwartz space of G, where  signifies the norm associated to (1), see [5], lemma 2 and proposition
1. and 11, chap 45]. Let L = V Ã— K Ã— K be the group with law:
(v, x, y)(w, s, t) = (v + (y)w, xs, yt) (21)
S Ã— Ã—
S Ã— Ã—
D Ã— Ã— Ã— Ã—
D Ã— Ã— Ã— Ã—
Let (V K K ) and C(V K K) be C with compact support and the space of C functions of the group L.In the same manner we define the Schwartz space (V K K)
Ã— Ã—
Ã— Ã—
Definition 3.1.. For every function f belongs to L1(V K K), one can define the Fourier transform of f by the following manner
r
r
Ff (, ) =
V
f (v, x)e i( , v) (x1)dvdx (22)
r
r
K
rv
rv
–
–
for all V V and for all K. In the following we will use the Lie group L to prove by another way the plancherel formula.
Definition 3.2. For any f S(G), we can define an function f S(V Ã— K Ã— K) as follows
f(v, x, y) = f (x.v.xy) (23)
–
–
note here that the function f is invariant in the following sense
– –
– –
f (tv, xt1, ty) = f (v, x, y) (24)
S Ã— Ã— S Ã— Ã— S Ã— Ã—
S Ã— Ã— S Ã— Ã— S Ã— Ã—
We will denote by K (V K K), K (V K K), K (V K K)
S S Ã— Ã—
S S Ã— Ã—
Definition 3.3.. for any two function f (G) and F (V K K),
we can define a convolution product of f and F on G
r
r
f F (v, x, y) = r
G
G
=
G
F ((w, z)1(v, x, y)f (w, z)dwdz
F (z1(v w), x, z1y)f (w, z)dwdz (25)
This leads to obtain
Lemma 3.1. If F is invarant in sense (24), then we get
f F (v, x, y) = f c F (v, x, y) (26)
for every (v, x, y) L, where signifies the convolution product on G with respect the variables (v, y) and csignifies the convolution product on B with respect the variables (v, x)
Proof: Let f S(G) and F SK (V Ã— K Ã— K), then we have
f F (v, x, y) = r
G
G
r
r
= r
G
G
=
B
F ((w, z)1(v, x, y)f (w, z)dwdz
F (z1(v w), x, z1y)f (w, z)dwdz
F ((v w), xz1, y)f (w, z)dwdz (27)
So the lemma is proved.
Definition 3.4. If f S(G), one defines the Fourier transform of its
invariant f as follows
'
'
–
–
K
K
K
K
Ff(, , 1) = r
V
V
r [ d r
K
tr(<;f (v, x, y)(y1)dy)](x1)dxe i( , v)dv
(28)
<;
<;
where is the partial Fourier transform on the compact Lie group.
Theorem 3.1. For any two functions g and f belong to G , the we have
– –
– –
F(g f )(, , 1) = F(g)(, )F(f )(, , 1) (29)
Proof: By lemma 3.1. we have if f and g two functions from S(G)
r
r
r
r
' dtr[(g f)(v, x, )](x1)dxe i( , v)dv (30)
' dtr[(g f)(v, x, )](x1)dxe i( , v)dv (30)
F(g f)(, , 1)
=
=
V
K
V
K
=
=
r
r
r
r
'
'
r
r
K
= dtr[
K
K
V
V
K K
r
r
'
'
r
r
K
K
K
K
= dtr[r
V K
((g f)(v, x, y))(y1)dy](x1)dxe i( , v)dv (31)
((g c f)(v, x, y))(y1)dy](x1)dxe i( , v)dv(32)
Chinging variables v u = w, xt1 = z, this implies
r
r
r
r
r
r
r
r
F(g f)(, , 1)
–
–
= f (v u, xt1, 1)]g(u, t)dudt(x1)dxe i( , v)dv K K
V V
r
r
r
r
r
r
= r r r
V
K
V
V
K
V
=
K
V V
r f(w, z, 1)]g(u, t)(t1z1)dxe i( , w+u)dudtdwdzvdx
K
K
r f(w, z, 1)]g(u, t)(z1)(t1)dzdte i( , w)e i( , u)dudw
K
K
get
= Ff(, , 1)Fg(, ) (33)
Theorem 3.2.(Plancherals formula) For any f L1(G) L2(G),we
r
r
f f (0, 1) =
G
G
f (v, x) dvdx
2
2
= ' d r
2
2
IFf (, )I2 d (34)
K V
f
f
–
–
Proof: First, let – be the function defined by
then we have
f (v, x, y) = f ((xv, xy)1
) (35)
f
f
f
f
f – (0, 1, 1) = r ' ' ddtr(F(f – )(, , ))d
' dtr{[' dtr[<;F – (, , )]Ff (, ))}d f
' dtr{[' dtr[<;F – (, , )]Ff (, ))}d f
r V K K
=
=
=
=
= ' d
= ' d
tr[F – (, , 1)Ff (, )]d = ' d f
tr[F – (, , 1)Ff (, )]d = ' d f
V K r K r
tr[Ff (, t)Ff (, )]d
V
tr[Ff (, t)Ff (, )]d
V
V
V
= ' d
= ' d
r
r
K
V
V
'
'
K r
V
V
tr[(Ff (, ))Ff (, )]d
r
2
2
K
2
2
= d
K
IFf (, )I2 d =
G
f (v, x) dvdx (36)


In this section I will introduce new group whose names are not known for Mathematicians and Physicians. But they will be very useful.

The first new group is: R*
= {x R; x (0 } , with law
x Â· y = x .y (37)
for all x R* and y R* , where Â· signifies the product in R*
and . signifies
pro
pro
tw
tw
the ordinary
duct of
o real numbers.
Theorem 4.1. (R* , Â·) with the law Â· becomes a commutative group
isomorphic with the ative group (R* , .) and with the real vector
group (R, +).
multiplic +
Proof: the identity element is 1 because
(1) Â· x = (1) Â· x = x, and
x Â· (1) = x .(1) (38)
If (x ,y) R* Ã— R* , such that x Â· y = 1, then we get
x Â· y = x .y = 1 (39)
From this equation we obtain y = x1 = 1 R* , which is the inverse of
x. Now let x, y, and z be three elements b x R* , then we have
elong to
and
(x Â· y) Â· z = (x Â· y).z = (x .y).z = x .y.z (40)
x Â· (y Â· z) = x Â· (y.z) = x.(y.z) = x.y.z (41)
So the law is associative and clearly commutative.
+
+
Let : R*
R*
the mapping defined by
(x) = x (42)
then we get
(x Â· y) = (x Â· y)
= x.y = (x).(y) = (x).(y) (43)
That means is homomorphism from R* to R* and evidently is onetoone
+ * *
and surjective, and so is a group isomorphism from R onto R+. Then
R* = R*
R*
= R*
R*
+
+
*
+
+
+
+
D
D
Definition 4.1. Let f belongs to (R), on can define the Fourier
transform of by
Ff () = r
R
R
dy
i
i
f (y)( y)
y
(44)
Corollary 4.1. For any f L1(R* ) L2(R* ), we get
r
r
r
r
2 dx
f (x) x =
R
R
R
2
2
F f () d (45)

The second group is: GL(n, R): Let GL(n, R) be the general linear group consisting of all matrices of the form
GL(n, R) = {X = ( aij ) , aij R, 1(i, j (n, and det A =
0} (46)
and let be the subset GL(n, R) of GL(n, R), which is defined as
GL(n, R) = {A GL(n, R), det A (0 } (47)
Â·
Â·
Definition 4.2. We supply GL(n, R) by the law noted following struc ture
A B = IA.B (48)
for any A GL(n, R) and B GL(n, R), where Â· signifies the multiplica tion in GL(n, R) and . signifies the usual multiplication of two matrix and I is the the following matrix defined as
(aij ) (49)
where a11 = 1 and aii = 1 for any 1(i n, and aij = 0, i /= j.
Theorem: (i) (GL(n, R), ) is group, and is isomorphic onto the sub group GL+(n, R)
Proof: It sufficies to consider the mapping
(A) = IA, A GL(n, R) (50)

The third new group is: O(n, R): Let O(n, R) be the orthogonal Lie group consisting of all matrices of the form

O(n, R) = {A GL(n, R), AA = I } (51)
where I is the identity matrix. Let O(n, R) be the subset of O(n, R) that is defined by
O(n, R) = {A O(n, R), det A (0 } (52)
It is easy to show that (O(n, R), ) becomes group isomorphic onto SO(n, R.

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