Advances in Pure Mathematics, 2012, 2, 203-210
http://dx.doi.org/10.4236/apm.2012.23029 Published Online May 2012 (http://www.SciRP.org/journal/apm)
Weinstein Gabor Transform and Applications
Hatem Mejjaoli1, Ahmedou Ould Ahmed Salem2
1Department of Mathematics, College of Sciences, King Faisal University, Ahsaa, KSA
2Department of Mathematics, College of Education, King Khalid University, Mohayil, KSA
Email: hmejjaoli@kfu.edu.sa, ahmdoo@kku.edu.sa
Received January 12, 2012; revised February 28, 2012; accepted March 7, 2012
ABSTRACT
In this paper we consider Weinstein operator. We define and study the continuous Gabor transform associated with this
operator. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. As applications,
we obtain analogous of Heisenberg’s inequality for the generalized continuous Gabor transform. At the end we give the
practical real inversion formula for the generalized continuous Gabor transform.
Keywords: Weinstein Gabor Transform; Practical Real Inversion Formula; Kernel Reproducing Theory
1. Introduction
We consider the Weinstein operator defined on
0,
d by :
2
121
:,
d
2
111
1
2d
idd
ixx
x



 
where d is the Laplacian for the d-first variables and
the Bessel operator for the last variable, given by
2
2
11
21 1
,.
2
dxx


 
>2d
1dd
x
For , the operator
is the Laplace-Beltrami
operator on the Riemanian space
0,
d

equipped
with the metric
1
1 1
22
1
1
dd
d
d
di
i
22
s
xx

(cf. [1]).
The Weinstein operator
2
L
has several applications
in pure and applied Mathematics especially in Fluid Me-
chanics (cf. [3]).
The harmonic analysis associated with the Weinstein
operator is studied by Ben Nahia and Ben Salem (cf.
[1,2]). In particular the authors have introduced and
studied the generalized Fourier transform associated with
the Weinstein operator. This transform is called the
Weinstein transform. In this work we are interested to the
Gabor transform associated with Weinstein operator.
Time-Frequency analysis plays a central role in signal
analysis. Since years ago, it has been recognized that the
global Fourier transform of a long time signal has a little
practical value to method is preferred to the classical
Fourier method, whenever the time dependence of the
analyzed signal is of the same importance as its fre-
quency dependence.
However, there exist strict limits to the maximal Time-
Frequency resolution of this transform, similar to Heisen-
berg’s uncertainty principles in the Fourier analysis.
In fact, Dennis Gabor [4] was the first to introduce the
Gabor transform, in which he uses translations and modu-
lations of a single Gaussian to represent one dimensional
signal. Other names for this transform used in literature,
are: short time Fourier transform, Weyl-Heisenberg trans-
form, windowed Fourier transform.
In this paper, we are interested a generalized Gabor
transform associated for the Weinstein transform. More
precisely, we give here general reconstruction formulas
and we give many applications. In the classical case the
Gabor transform is very fundamental and has many ap-
plications to Mathematical Sciences.
The paper is organized as follows. In Section 2, we re-
call the main results about the harmonic analysis related to
the Weinstein operator. In Section 3, we introduce the
analogue of the continuous Gabor transform associated
with the Weinstein operator and we give some harmonic
properties for it (Plancheral formula,
inverse formula,
weak uncertainty for it). The Section 4 is devoted to prove
the analogous of Heisenberg’s inequality for the general-
ized continuous Gabor transform. In Section 5 using the
kernel reproducing theory given by Saitoh [5] we study the
problem of approximative concentration. In the last section
we give a practical real inversion formulas and extremal
function for the Weinstein-Gabor transform.
2. Preliminaries
In order to confirm the basic and standard notations we
C
opyright © 2012 SciRes. APM
H. MEJJAOLI, A. O. A. SALEM
204
briefly overview the Weinstein operator and related har-
monic analysis. Main references are [1,2].
1
,1,forall, .
d
xyxy
 
In the following we denote by

0,
dd

1 1ddd



1
1
,d
d
xx


1
.
22
:.

11
,,,
dd
xx xx
.


11
:
dd
xn



C1d
0, :Bnx
1d
.
the space of continuous functions on

1pd
C
,
even with respect to the last variable.
the space of functions of class
p
C on
, even with respect to the last variable.
1d
1d
the space of C
-functions on 1d

1d
1d
1d
D
C1d
, even
with respect to the last variable.
the Schwartz space of rapidly decreasing
functions on , even with respect to the last variable.
the space of -functions on
which
are of compact support, even with respect to the last
variable.
We consider the Weinstein operator
defined by

,0,
d
 

1
1, ,
x d
fxx


1d
xxx
 ,


1,
21
() ,
,
xd d
d
fx fxx
fC


 
(1)
where
x
1
,d
x
is the Laplace operator on , and
d
the Bessel operator on
0, given by
1
2
,2
1
21
d
d
x
d
dd
xx
x

11
1
:,.
d2
dd


(2)
The Weinstein kernel is given by


11
11
,
,
dd
dd
j xz




11dd
jxz

1
,d
zt
 
;,01and
,forall.
z
 
1x1d
z

,
,:
for all,
ixz
xz e
xz

 (3)
where is the normalized Bessel function.
The Weinstein kernel satisfies the following properties:
1) For all , we have
 

,,
,,
zt tz
ztz t


 (4)
2) For all , and
d
1d
, we
have


xpIm,x z,e
z
Dxz x
 (5)
where 1
11
d
1d
zz
z
D
 and 11
.
d
 
 In
particular
(6)
We denote by
1pd
L
1d
the space of measurable
functions on
such that

 


11
1
1
1
:d,if1,
:ess ,
sup
pdd
dd
pp
L
Lx
ffxxp
ffx

 

d
where
is the measure on given by
1d

 
21
1
1
2
d,: d.
2π21
d
dd
x
x
xx

The Weinstein transform is given for f in
11d
L
by
 
1
1
,d ,
for all.
d
W
d
f
yfxxyx
y

(7)
Some basic properties of this transform are the fol-
lowing:
11d
L
, 1) For f in
 
11
1.
d
d
WL
L
ff

(8)
1d
we have 2) For f in
 
21
,for all.
d
WW
fy yfyy
  (9)
11d
L

W
, if 3) For all f in
f
belongs to
11d
L
, then
 
1,d ,a.e
dW
fyf xxyx

. (10)
1d
f
, if we define 4) For

,
WW
fyf y
then
.
WW WW
I
d 

1d
(11)
Proposition 1. 1) The Weinstein transform W is a
topological isomorphism from onto itself and
for all f in
1d

,
 
11
22
dd
dd
W
fx xf

.





(12)
W
2) In particular, the Weinstein transform
f
f
can be uniquely extended to an isometric isomorphism
from
21d
L
The generalized translation operator
onto itself.
x
, 1d
x
, as-
sociated with the operator
is defined by
 

π2
122
11 11
0
1
,,2cossind
1
π
2
xd
ddd
y fxyxyxy
d
yf



 
 





Copyright © 2012 SciRes. APM
H. MEJJAOLI, A. O. A. SALEM 205
where .

1d
fC
d,d d
x
yxy


such that
By using the Weinstein kernel, we can also define a
generalized translation. For a function
21d
fL
1d
y
y
and the generalized translation

f
is de-
fined by the following relation:

 
Wy
fx x


,
W
y fx
0t
. (13)
For example, for , we see that

22
2tx y
t
yexe


2, .ityx (14)
By using the generalized translation, we define the
generalized convolution product
f
g

11d
L
 
d.
of functions
as follows:
,fg


11
,
dxd
f
gxfyy

gy y

11d
(15)
This convolution is commutative and associative and
satisfies the following propositions:
Proposition 2. 1) For all ,
,fg L
f
g
belongs to and

11d
L


.
WWW
f
gfg
,pqr
 (16)
2) Let 1, such that  111
1.
pqr


1qd
gL
If
and , then
and
1p
fL
fgL


d

1rd
  
11
*.
pq
dd
LL
g





21d
L


1rd
L
fgf
(17)
Proposition 3. Let . Then
,fg

21d
L
 fg if and only if
WW
f
g

21d
L
 
.
W
 be-
longs to , and in this case we have

WW
f
gfg

21d
L

An immediate consequence of Proposition 3 and the
Plancherel formula that will be used in the next section is
the following.
Proposition 4 Let f and g be in . Then, we
have
 
 
2
d
1
1
2
2
d
d
dWW
fgx x
fg



(18)
where both sides are finite or infinite.
3. The Continuous Weinstein Gabor
Transform
Notations. We denote by:
,1,
p
Xp

11dd
the space of measurable functions f on
with respect to the measure


11
1
,:,d,,
1
dd
pp
p
ff xyxy
p






1
,
,
: esssup,.
d
xy
ffxy


21d
L
1d
Definition 1. For any function g in and
any
, we define the modulation of g by v as:

2
:: ,
W
gg g
 



 (19)
1d
y
where
y
,
, are the Weinstein translation
operators given by (13).
21d
L
, we have Remark 1. For g in
 
21
21 .
d
dL
L
gg
,
g
We consider the family
y
1
,d
y
, defined by

1
,,,
d
yy
gx gxx




21d
L
Definition 2. Let g be in . For a function f
in
21d
L
 
we define its continuous Weinstein Gabor
transform by

1,
,:d ,
d
gy
f
yfxgxx


(20)
which can also be written in the form

,:
g
f
yfgy


 
(21)
hxhx
where
2
L.
Theorem 1. (
inversion formula).


21 1
\0.
dd
LL



 Then, for Let g be in
21d
L
, we have any function f in


11
0, ,d,
dd
ngy
Bn
xfygxy

 



f
(22)
21d
L
, where in

d,:d dyy


and satisfies

21
lim 0.
d
nL
nff

For proof this theorem we need the following Lem-
mas.
Lemma 1. Let g be as above. For any positive integer
n define the two functions



  
11
2
0,
:,dd,
dd
n
W
Bn
Gx
xg






Copyright © 2012 SciRes. APM
H. MEJJAOLI, A. O. A. SALEM
206
for 1d
x
, and




2
:d,



21 111
,,
and .
ddd
nn
Wnn
GL HLL
GH





10,
d
nW
Bn
Hg

1d
for . Then
Proof. Using the Cauchy-Schwartz inequality we obtain






  



  
111
11
2
2
2
0, 0,
2
2
0,
d,dd
,dd.
ddd
dd
n W
Bn Bn
W
Bn
Gx Cxg
Cxg


 









3 Therefore by Fubini theorem, the inversion theorem, the Plancherel formula and Proposition
 



  


  

 

 




1111
11
111 1
11
2
2
2
0,
0,
2
2
0,
22
0,
2
0,
d,ddd
dd
d
d<.
dddd
dd
ddd
dd
nW
Bn
Bn
Bn
Bn LL
Bn L
Gxx Cxgx
Cg
Cgg
Cg



 

 

 















That

11
2
2
1dd
dd
WW
Cg
xx





11 1dd
n
HL L




is easily checked. Finally, using Fubinis theorem we obtain

 



 



 
1
11
1
2
0,
,d
,dd.
d
dd
Wn n
Wn
Bn
Hy Hy
yg Gy

 

11
2
0,
,d
d
dd
W
Bn
yg








 

Lemma 2. Let g be as above. For any positive integer
n the function






 
11
2
0,
:,dd,
dd W
Bn xg

nxG





can be written

d.



10,
d
nBn
Gxg gx



roposition 3 we have

Proof. From P



 


 
1
2
2
1
0,
d
d
d
n
WW
Bn
xg
gx






10, d.
d
Bn
gg
x
 
11
0, ,d
dd W
Bn
Gx





Lemma 3. Let g be in








1
\0
dd



.
21
LL
Then, for any function f in
21d
L
,
.
nn
we have
f
Gf
 (23)
Proof. We have

 


 


 



11
1
11
1
0,
0,
,d,
d
d
d,
dd
d
.
dd
d
dd
d
n
gy
Bn
xBn
xn
n
fx
fy gxy
x
fg gx
fygg yy


 

1
0,
0, .,
d
Bn
g
Bn fg

11
0,
dd
x
Bn
f
yggy y
fyG yy
fGx

 
 

 

















On the follow we justify the use of Fubinis theorem in
the last sequence of equalities observe that

 





 
11
1
0,
0,
d,
d.
dd
d
x
Bn
Bn
f
ygg yy
fg gx
 
 







Now, using Proposition 3 and hypothesis on g we see
Copyright © 2012 SciRes. APM
H. MEJJAOLI, A. O. A. SALEM 207
that gg

ity and Parsev

21d
L
 
. Next using Young’s inequal-
al theorem we obtain

 
  
21
1 1
d
ddd
L
L
g
g




and
1
21
212
d
d
L
L
LL
fg g
fg
Cfg







 


  
21 1
.
d
L
gg

The proof is complete.
ma 1 that

n
fL
and

.
nW
1
21
1
0,
0,
d
(d
d
dd
d
Bn
LL
Bn
fg gx
Cf

 






Proof. of Theorem 1.
It follows from Proposition 3 and Lem
21d

Wn
f
Hf
By this, the Plancherel formula, thehat 1
n
H
em, it follows that
fact t
pointwise as n, and the dominated convergence
theor

  
 


2
2
d
1d0
d
d
nL
WnW
Wn
fH f
fH
21
1
1
2
d
ff
 




which achieves the proof.
ositi r f in
as n
Propon 5. Fo
21d
and g in
L
21d
L
we have
 
2121
.
dd
LL
g




()
erel formula).
The in
,
gff
24
Proposition 6. (Planch
Let g be in

21d
L
.
n, for all f
21d
L
,
we have
 
21 21
.
dd
LL
g f




(25)
Proof. Using relation (18), Fubini’s theorem and
Pl , we
2,
gf
ancherel’s formula for the Weinstein transform
have
  


 


  


  
 
21 21
.
dd
LL
11
2d
dd
fgy
11
11
11
2
2
22
22
2
d
dd
dd
dd
dd
dd
dd
WW
W
W
y
2
f
gv
f
gv
fgvv
 





 
















e classical case, the continuous Weinstein Ga-
bor transform preserves the orthogonality re
ever, we have the following result. Then, for all f, h
in
fg




As in th



lation. How-
Corollary 1. Let g be in

21d
L
.
21d
L
, we have
 


11
1
2
,,dd
() d.
dd
d
gg
fy hyy
gfxhxx


21d
L





e continuous Weinstein Gabor transform.
Proposition 7. Let g be in

21d
L
such that
In what follows, we show the weak uncertainty prince-
ple for th
(26)
21d
L
. Suppose that

21
1
d
L
g
and f in

21
1
d
L
f
Then, for 11dd
U


and 0
satisfying
 
2
,d,1,
g
Ufy y
 


we have
 
1,U

where
:U

d ,.
Uxy

Proof. From the relation (24) we deduce that
,1.
gf
Thus,
 

,

2.
2
1,d,
g
Ufy y
g
f
UU




This achieves
Proposition

21d
L
, g i
the proof.
8. Let f be inn

1d
2
L
such that

21
1
d
L
g
and
2, .p
Then,
 

21
11 ,d ,.
d
dd
pp
gL
fyy f


(27)
Proof. Using Proposition 5 and Proosition 6 the result
follows by applying the Riesz-Thorinterpolation theo-
4. Uncertainty Principles of Heisenberg Type
Inequality
for the generalized continuous Gaboansform.
Proposition 9. (Uncertainty principle of Heisenberg
type for ).
n

p
n i
rem.
this section we will to prove the Heisenberg in
r tr
W
Let f be i
21d
L
, the following inequality holds
Copyright © 2012 SciRes. APM
H. MEJJAOLI, A. O. A. SALEM
Copyright © 2012 SciRes. APM
208
 
 

21
1
22
2
d1.
2d
L
d
y yf




1
1
2
22
2
d
dd
W
xf
x yf



lt by combining the Heisen-
besical Fourier tsform and
Fo m 2. (Uncertrinciples berg
x
Proof. We obtain the resu
rg inequalities for the cl
Type for
g
).
as ran
urier-Bessel transform. Let g be in
21d
L
. Then, for all f in
21d
L
,
the following inequality holds
Theore ainty pof Heisen

 

 
21 21
111
1
2
2
1
2
2
g
fy
2
2
d,1
2
dd
d
xf
x xy





Proof. Let us assume the non-trivial case that both in-
grals on the left hand side of (28) are finite. Fixing
d, .
dd
LL
d
yg f





(28)
W


arbitrary, Heisenberg’s inequality for einstein trans-
form gives that
W
te





 
11
1
22
22
., d
dWg
yf
yyyf


 1
22
1
2
, 1.
2
dgd
y y
 
 


d|, d
dg
y fy
 
Integrating over
and using Cauchy Schwartz inequality we obtain




 
 
11
1,
d,
2ddgfy y



11 11
11
22
2
22 2
2
.,dd, d,
.
dd dd
Wg g
yfyyyfy y
d
 
 
 
 


 
 
 





Thus, using the fact that




 
2
,y
we obtn
21
11 1
22
22
.,d dd
d
dd d
Wg W
L
yfyygyfx




 

 
ai

 
 
 

1
2
2
22
2
2
222
d,d,
1,d,1.
22
dddd
dd
Wg
L
gLL
gxfxxyfyy
dd
fyyg f








 
 
 
 






2. (Reproducing Kernel). Let g be in
21 111
22
11
1

\0LL



This proves the result. 21 1dd . Then,

21d
gL
is
t space in a reproducing kernel Hilber2
5. Reproducing Kernel
X
with kernel
function
Corolla ry





.
y
ggy


(29)
Th
1
21 21
22
11
,;,:d:
d
d d
gyy
LL
yygxgx x
gg
 

 
 
 

 


e kernel is pointwise bounded:
 
11
1;forall,,
.
dd
y

,
y
,;,
gyy



(30)
Proof. We have
 
1
,d.
d
g
f
,y
yfxx

gx
Using the relation (26), we obtain


 

 
11
21
,
2
,,
,dd
dd
d
gggy
L
fyfxgxx
g

1








H. MEJJAOLI, A. O. A. SALEM 209
On the other hand using Proposition 3, one can easily
1
,, d

The essential result of this section is
Theorem 3. (Conce ntration of g
see that for every y
, the function





21
,
2
2
1,
1
d
gy
y
L
xgx
21d
L
g
g
gx
g



o

21d
L
. Therefore, the result is obtained.
llowing theorem, we will show that the por-
tion of the continuous Weinstein Gabor transform lying
utside some sufficiently small set of finite measure
a
f the cont
the following
22
:PX X
be ngs t
In the fo
lo
o
c nnot be arbitrarily too small. Then, in order to prove a
concentration result oinuous Weinstein Gabor
transform, we need notations:
g
the orthogonal projecton from 2
i
X


21d
gL
.
22
:
U
PXX
onto
the orthogonal prn from 2
ojectio
X
o the subspace of function supported in the subset

<.U
ont
U11dd
 with
We put

2
2,
2,
sup,;1.
Ug Ug
PPPPvv Xv
 (31)
the following.
g be in
in small sets) Let

21 1
\0
dd
LL



 and
U
<1.U
Then, for all f in
11dd

with
21d
L
we have


 
21 21
1.
dd
LL
Ug f



 
(32)
Proof. From the definition of U
P and
2,
gUg
ff

g
P we have

2, 2,
gUgUg

, sing the Propo ge
.ff IPPf

g
usition 6 we Thent

2, 1
g Ug
2,
gU g
f
ffPP

(33)
 
 

21
1.
Ug
LL
f PP


 (34)
As
21dd
g

g
P is a projection onto a reproducing k
bert space, then, from Saitoh [5],
ernel Hil-
g
P can be re
by
presented

,;,d,,
g
y yy
11
,,
dd
g
PF yFy






with
g
defined by (29). Hence, for 2
F
X
arbitr, warye have

,;,d,yyy
11
,,,
dd
UgU g
PPF yyF y


 

 

and its Hilbert-Schmidt norm
 
22 22
1
2
22
,,;,d,d,.
dd
U g
yy yyy

 


 

lity we see tha
Ug
HS
PP
By the Cauchy-Schwartz inequat
.
Ug Ug
HS
PPPP (35)
On the other hand, from (29) and Fubini’s theorem, it
is easy to see that

.
Ug
PP U
(36) Let s
HS
5)
sult.
6. Practical Real Inversion Formulas for g
Thus, from the relations (34), (3 and (36) we obtain
the re
. We define the space

1sd
H
by
 



2
2
121 21
::1 .
s
d d
W
fL fL
 
 
 
  
The space

H
provided with the inner product
sd
H
1sd

 
dfg

2,
s
WW
11
,1
sd d
H
fg


(37)
and the norm

 
11sd sd
HH


2,fff

, is a Hilbert
space.
Proposition 10. Let g be a function in

21 1
,
dd
LL



  and 1d
. The integral trans-
Copyright © 2012 SciRes. APM
H. MEJJAOLI, A. O. A. SALEM
210
form

.,
g
, is a bounded linear operator from

1sd
H
,o s in
, int
21d
L
, and we have

 
11
21
., .
dsd
d
gLH
fgf
 
L



Proof. We proceed as [6] we obtain the result.
Definition 3. Let g be a function in
LL >0, 1d
and
sefine the
f

1sd
H
product:

21

1
. Let r
dd



. We dilbert space

,1rs d
H
as the H
with the inner subspace o

 

21
., ,
.
d
L

rm associated to the inner product is defined by:

,1
1
,=, .,,
,
rss dgg
HH
sd
fh rfhfh
fh H

The no


121
,.
sd d
L
,:.
rs g
HH
frf
proceed
Proposi . L a function
LL



2
22 f
We as [6] we prove the following results.
tion 11et g be in

21

1dd

. For 12
s
, the Hilbert

rs
H
admits the following reproducing

d
space
ke
,1d
rnel:
 


122
d
,.
d
g
Px
y
,,xy
1rg
s


4. Let g be
LL


Theorem a function in

21

1dd



. Let 12
d
s
.
1) Fo

21
L
for any 0r, the
infin
r any h ind and
itum
 


21
2
inf .,
sd d
sd g
HL
fH rfh f



(38)
is
1
1
2

attained by a unique function ,rh
f
given by
 

1
,,d
d
rh r
,
f
xhyQxyy
(39)
e
wher



 
1
,
2
22
,,
,,
d.
1
d
rrs
s
QxyQ xy
gxy
rg

 


(40)
2) Let ,0r

21d
L
such that and ,hh in

21 .
d
L
hh
Then

1
,, .
2
sd
rhrhH
ff r


3) Let >0r. If f is in

1sd
H
and
.,
gfh
Then
.

1
2
,s 0.
sd
rh
ff r
 0a
H
REFERENCES
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Associated with the Weinstein Operator,” Potential The-
ory—ICPT 94, 1996, pp. 243-253.
[3] relot, “E et Pote
cel Riesz,” Lecture Notes in Mathematics 681, Séminaire
de Théorie de Potentiel Paris, No. 3,, pp. 18-38.
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the Institute of Electrical Engineers, Vol. 93, No. 26,
1946, pp. 429-457.
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ejjaoli and N. Sraieb, “Gabor Transform in Quantum
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M. Bquation de Weinsteinntiels de Mar-
1978
D. Ga
plications,” Longman Schnical,
[6] H. M
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