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: [email protected], [email protected]
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
1 2 1
: d 2 ,
1 1 1
1 2 d
i xi xd xd
where d is the Laplacian for the d-first variables and
the Bessel operator for the last variable, given by
2
2
1 1
2 1 , 1.
2
d x x
> 2
d
1 d d
x
For , the operator is the Laplace-Beltrami operator on the Riemanian space d
0,
equipped with the metric
1 1 1
2 2
1
1
d d d d
d i
i
2 2
s x x
(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
briefly overview the Weinstein operator and related har-monic analysis. Main references are [1,2].
1
, 1, for all , d .
x y x y
In the following we denote by
0,d d
1 1
d d d
11
, d
d
x x
1
.
2 2: .
1, , ,d d 1
x x x x .
1 1:
d d
x n
C d 1
0, :
B n x
1 d
.
the space of continuous functions on
1
p d
C
, even with respect to the last variable.
the space of functions of class p
C on
, even with respect to the last variable.
1
d
d1
the space of C-functions on d1
d1
1
d
d1D C d 1
, 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
,
d
0,
1
1 , ,
x f x xd
1 d
x x x
,
1 ,
2 1
( ) ,
,
x d d
d
f x f x x
f C
(1)
where x is the Laplace operator on , and ,xd1
d
the Bessel operator on
0, given by1
2
, 2
1
2 1 d
d
x
d
d d
x x
x
1 1
1
: , .
d 2
d d
(2) The Weinstein kernel is given by
1 1
1 1
, ,
d d
d d
j x z
d 1 d 1
j x z 1
, d
z t
; ,0 1 and, for all .
z
1 x d 1
z
,
, : for all ,
i x z
x z e
x z
(3)
where is the normalized Bessel function. The Weinstein kernel satisfies the following properties:
1) For all , we have
, ,
, ,
z t t z
z t z t
(4)
2) For all d , d1 and
, we have
, exp
x Imz
, zD x z x (5)
where
1
11 dd1
z z
z D
and 1 d1. In
particular
(6)
We denote by p d 1
L 1d
the space of measurable functions on such that
1 1
1
1
1
: d , if 1 ,
: esssup ,
p d d d
d
p p
L
L
x
f f x x p
f f x
d
where is the measure on d1 given by
2 1
1 1
2
d , : d .
2π 2 1
d
d d
x
x x x
The Weinstein transform is given for f in 1 d 1
L by
1
1
, d ,
for all .
d
W
d
f y f x x y x
y
(7)
Some basic properties of this transform are the fol- lowing:
1 d 1
L ,
1) For f in
d1 1
d1 .W f L f L
(8)
d1
we have 2) For f in
2
, for all d 1.W f y y W f y y
(9)
1 d 1
L , if W
3) For all f in f belongs to
1 d 1
L , then
d1 W
, d
, a.ef y f x x y x
. (10)
d 1
f , if we define
4) For
,W f y W f y
then
.
W W W W Id
d1
(11) Proposition 1.1) The Weinstein transform W is a
topological isomorphism from onto itself and
for all f in
d1
,
1 1
2 2
d d
d f x x d W f .
(12)
W
2) In particular, the Weinstein transform f f
can be uniquely extended to an isometric isomorphism from 2
d1
L
The generalized translation operator onto itself.
x
, x d1
, as- sociated with the operator is defined by
π
21 2 2
1 1 1 1
0
1
, , 2 cos sin d
1
π
2
x y f x y xd yd xd yd
d
y f
where fC
d1
. d
x y, d
x d
y
such that
By using the Weinstein kernel, we can also define a generalized translation. For a function f L2
d 1
1
d
y y
and the generalized translation f is de-
fined by the following relation:
W yf x x
,y
W f x0
t
. (13) For example, for , we see that
t 2
t x
2 y2y e x e
2 , .
i ty x
(14) By using the generalized translation, we define the generalized convolution product f g
1 d 1
L
d
. of functions as follows:,
f g
d1 x
, d 1
f g x f y y
g y y
1 d1
(15) This convolution is commutative and associative and satisfies the following propositions:
Proposition 2. 1) For all f g, L , f g belongs to 1
d1
and
L
W .W f g W f g
, p q r
(16) 2) Let 1 , such that 1 1 1 1.
p q r
1
q d
gL
If and , then
and
1p fL
f gL
d
1
r d
1
1* Lp d g Lq d .
2 d 1
L
1
r d
L
f g f
(17)
Proposition 3.Let f g, . Then
2 d 1
L
f g if and only if W f W
g
2 d 1
L
W . be-
longs to , and in this case we have
W f g W f g
2 d 1
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
d W W
f g x x
f g
(18)
where both sides are finite or infinite.
3. The Continuous Weinstein Gabor
Transform
Notations. We denote by:
, 1,
p
X p
1 1
d d
the space of measurable functions f on
with respect to the measure
1 1
1
, : , d , ,
1
d d
p p
p
f f x y x y
p
1
, ,
: esssup , .
d
x y
f f x y
2 d 1
L 1
d
Definition 1. For any function g in and any , we define the modulation of g by v as:
2: : W ,
g g g
(19)
1
d
y
where y , , are the Weinstein translation
operators given by (13).
2 d 1
L , we have
Remark 1. For g in
2
1 2 d1 L d .L
g g
, g
We consider the family y , d 1 y
, defined by
1, , ,
d
y y
g x g x x
2 d 1
L
Definition 2.Let g be in . For a function f in 2
d 1
L
we define its continuous Weinstein Gabor transform by
1 ,
, : d d ,
gf y f x gy x x
(20)
which can also be written in the form
,
:
gf y f g y
(21)
h x h x
where
2 L
.
Theorem 1. ( inversion formula).
2 d1 d 1
\ 0 .
L L Then, for
Let g be in
2 d 1
L , we have
any function f in
d1 0, d1
,
d
,
n x B n g f y yg x y
f
(22)
2 d 1
L , where
in
d ,y : d y d
and satisfies
2 1
lim n L d 0.
n f f
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
1 1
2
0,
: d d , d d ,
n
W
B n
G x
x g
for d 1 x , and
2
: d ,
2 1 , 1 1 1 ,
and .
d d d
n n
W n n
G L H L L
G H
10,
d
n B n W
H g
1
d
for . Then Proof. Using the Cauchy-Schwartz inequality we obtain
1 1 1
1 1
2 2
2
0, 0,
2 2
0,
d , d d
, d d .
d d d
d d
n B n B n W
W
B n
G x C x g
C x g
3 Therefore by Fubini theorem, the inversion theorem, the Plancherel formula and Proposition
1 1 1 1
1 1
1 1 1 1
1 1
2 2
2
0,
0,
2 2 0,
2 2
0,
2
0,
d , d d d
d d
d
d < .
d d d d
d d
d d d
d d
n B n W
B n
B n
B n L L
B n L
G x x C x g x
C g
C g g
C g
That
1 1
2 2
1 d d
d d W W
C g x x
1 d 1 d1
n
H L L is easily checked. Finally, using Fubinis theorem we obtain
1
1 1
1
2
0,
, d
, d d .
d
d d
W n n
W n
B n
H y H y
y g G y
1 1
2
0,
, d d
d y Bd n W g
Lemma 2. Let g be as above. For any positive integer n the function
1 1
2
0,
: Bd n d ,x W g d d ,
n x G
can be written
d .
d1 0,
n B n
G x g g x
roposition 3 we have
Proof. From P
12
2 1
0,
d d
d
n
W W
B n
x g
g x
10, d .
d
B n g g x
10, 1 , d
d d W
B n
G x
Lemma 3. Let g be in
d d1
\ 0
.
2 1
L L
Then, for any function f in 2
d1
L ,.
n n
we have
f G f (23)
Proof. We have
1 1
1
1 1
1
0,
0,
, d ,
d
d
d ,
d d
d
.
d d
d
d d
d
n
g y
B n
x B n
x n
n f x
f y g x y
x
f g g x
f y g g y y
1
0,
0, .,
d
B n
g
B n f g
10, 1
d d x
B n
f y g g y y
f y G y y
f G x
On the follow we justify the use of Fubinis theorem in the last sequence of equalities observe that
1 1
1
0,
0,
d ,
d .
d d d
x
B n
B n
f y g g y y
f g g x
that g g
ity and Parsev
2 d 1
L
. Next using Young’s inequal- al theorem we obtain
2 1
1 1
d
d d d
L
L g
g
and
1 2 1
2 1 2
d d
L
L
L L
f g g
f g
C f g
g 2
1 g L
d1 .
The proof is complete.
ma 1 that
n
f L and
. n W1
2 1 1
0,
0,
d
( d
d
d d d
B n
L L
B n
f g g x
C f
Proof. of Theorem 1.
It follows from Proposition 3 and Lem
2 d1
W fn H f
By this, the Plancherel formula, the hat Hn1 em, it follows that
fact t
pointwise as n , and the dominated convergence
theor
2
2
d
1 d 0
d
d
n L
W n W
W n
f H f
f H
2 1 1 1
2
d
f f
which achieves the proof. ositi r f in
as n
Prop on 5. Fo 2
d1
and g in
L L2
d 1
we have
2 d1 2 d1 .
L g L ( )
erel formula).
The in
,
gf f
24
Proposition 6. (Planch
Let g be in L2
d1
. n, for all f L2
d1
,we have
2 d1 2 d1 .
L L
g f
(25)
Proof. Using relation (18), Fubini’s theorem and
Pl , we
2, gf
ancherel’s formula for the Weinstein transform have
2 d1 2 d1 .
L L
1 1
2
d
d d f g y
1 1 1 1 1 1
2 2
2 2
2 2
2
d
d d
d d
d d
d d d d
d d
W W
W
W
y
2
f g v
f g v
f g v v
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
f g
As in th
lation. How- Corollary 1. Let g be in L2
d1
.
2 d 1
L , we have
1 1 1
2
, , d d
( ) d .
d d d
gf y gh y y
g f x h x x
2 d1
L
e continuous Weinstein Gabor transform. Proposition 7. Let g be in L2
d1
such that
In what follows, we show the weak uncertainty prince- ple for th
(26)
2 d 1
L . Suppose that
2 d1 1
L g
and f in
2 d1 1
L f
Then, for U d1d1 and 0
satisfying
2
, d , 1 ,
g
U f y y
we have
U 1
,
where
U :
d
, .U x y
Proof. From the relation (24) we deduce that
, 1.
gf
Thus,
,
2
.
2
1 U gf y, d y,
gf
U U
This achieves
Proposition L2
d1
, g i the proof.
8. Let f be in n
d1
2 L such that
2 d1 1
L g
and p
2,
. Then,
2
11 1 , d , d .
d d
p p
gf y y f L
(27)Proof. Using Proposition 5 and Pro osition 6 the result
follows by applying the Riesz-Thori nterpolation theo-
4. Uncertainty Principles of Heisenberg Type
In equality
for the generalized continuous Gabo ansform.
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 L2
d1
, the following inequality holds
2
11
2d 2 1 2 .
2 L d
d
y y f
1
1 2
2 d 2 2
d x W f x d y f
lt by combining the Heisen- be sical Fourier t sform and Fo
m 2. (Uncert rinciples berg
x
Proof. We obtain the resu
rg inequalities for the cl Type for g
).
as ran
urier-Bessel transform. Let g be in L2
d1
. Then, for all f in L2
d1
,the following inequality holds
Theore ainty p of Heisen
1
1 1
2
1 2
11 2
2
1 2
2 gf y
2
2
d , 1
2
d x f x x d d y
Proof. Let us assume the non-trivial case that both in-
grals on the left hand side of (28) are finite. Fixing
d , L d L d .
d
y g f
(28)
W
arbitrary, Heisenberg’s inequality for einstein trans- form gives that
W
te
1
1
1
2 2
2 2
., d
d y W gf y y y f
12 12 2
, 1 .
2
d g
d
y y
d y d | gf y, d
Integrating over and using Cauchy Schwartz inequality we obtain
1 1
1 , d ,
2 d d gf y y
1 1 1 1
1 1
2 2 2
2 2 2
2
., d d , d ,
.
d d y W gf y y d d y gf y y d
Thus, using the fact that
2
,
y
we obt n
2 1
1 1 1
2 2
2 2
., d d d d
d d y W gf y y g L d y W f x
ai
1 2
2
2 2 2 2
2 2 2
d , d ,
1 , d , 1 .
2 2
d d d d
d d
W g
L
g L L
g x f x x y f y y
d d
f y y g f
2. (Reproducing Kernel). Let g be in
2 1 1 1 1
2 2
1 1
1
L L
\ 0
This proves the result. 2 d1 d1
. Then, g
L2 d1
is
t space in
a reproducing kernel Hilber 2
5. Reproducing Kernel
X with kernelfunction
Corollary
.
yg g y
(29)
Th
1
2 1 2 1
2 2
1 1
, ; , : d d :
d d
g y y
L L
y y g x g x x
g g
e kernel is pointwise bounded:
1 1
1; for all , , .
d d
y
y, , ; ,g y y
(30)
Proof. We have
, d1
d
.gf y f x ,y x
g x
Using the relation (26), we obtain
1 1
2 1
, 2
, d d , , d d
d
g g g y
L
f y f x g x x
g
1
On the other hand using Proposition 3, one can easily
1
, , d
The essential result of this section is Theorem 3. (Concentration of g
see that for every y , the function
2 1
, 2
2
1
,
1
d
g y
y L
x g x
2 d1
L g
g g x
g
o 2
d 1
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
2 2
:
P X X
be ngs t In the fo lo
o
c nnot be arbitrarily too small. Then, in order to prove a concentration result o inuous Weinstein Gabor transform, we need notations:
g the orthogonal project on from 2
i X
2 d1
g L
.
2 2
:
U
P X X
onto
the orthogonal projection from X2
o the subspace of function supported in the subset
U < .
ont
U d1d1 with
We put
2
2, 2,
sup , ; 1 .
U g U g
P P P P v v X v
(31)
the following.
f
g be in
in small sets) Let
2
d 1
d 1
\ 0
L L and
Ud1d1 with
U < 1. Then, for all f in
2 d 1
L we have
1 U
g L2
d1 f L2
d1 .
(32)
Proof. From the definition of PU and
2,
gf U gf
g
P we have
2, 2,
g U g U g
, sing the Propo ge
.
f f IP Pg f
u sition 6 we
Then t
2, 1
g 2, U g
gf U f gf P P
(33)
2
1
1 U g
.L f L P P (34) As
2 d1 d
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
, d1 d1
,
g y, ; ,y
d
y,
,g
P F y F y
with g defined by (29). Hence, for
2
FX arbitrary, we have
, d1 d1
,
,
y, ; ,y
d
y,
U g U g
P P F y y F y
and its Hilbert-Schmidt norm
2 2 2 21 2
2 2
, , ; , d , d , .
d d U y g y y y y
lity we see tha
U g HS
P P
By the Cauchy-Schwartz inequa t
.
U g HS U g
P P P P (35)
On the other hand, from (29) and Fubini’s theorem, it is easy to see that
. U gP P U (36) Let s HS
5) sult.
6. Practical Real Inversion Formulas for
gThus, from the relations (34), (3 and (36) we obtain the re
. We define the space s
d 1
H by
2
2
1 : 2 d 1 : 1 s 2 d 1 .
W
f L f L
The space H
provided with the inner products d
H
1 s d
2
f g d
, sW W
1 1
, Hs d d 1
f g
(37) and the norm
1
1s d s d
H H
2
,
f f f
, is a Hilbert space.
Proposition 10. Let g be a function in
2 d1 d1 ,
form g
., , is a bounded linear operator from
1s d
H , s in , into L2
d1 , and we have
., 2
d1
d1 s
d1 .gf g L f H
L
Proof. We proceed as [6] we obtain the result.
Definition 3. Let g be a function in
L L > 0, d1
and
s efine the
f Hs
d1
product:
2 1
1
. Let r
d d
. We d ilbert space Hr s,
d 1
as the
H
with the inner subspace o
., 2
1 ,.
d
L
rm associated to the inner product is defined by:
, 1
1
, = , ., ,
,
r s s d g g
H H
s d
f h r f h f h
f h H
The no
1
, 2
1 .s d L d
, : .
r s g
H H
f r f
proceed
Proposi . L a function
L L
2
2 2
f
We as [6] we prove the following results. tion 11 et g be in
2 d 1
d1
. For 1
2
s , the Hilbert
r s
H admits the following reproducing
d
space ke
, d1 rnel:
1
2 2
d
, d .
g
P x y
,x ,y
1
r g
s
4. Let g be
L L
Theorem a function in
2 d 1
d 1
. Let 1 2
d
s .
1) Fo 2
1
L for any r0, the
infin
r any h in d and
itum
2
12
infs d Hs d g ., L d
f H
r f h f
(38)
is
1 1
2
attained by a unique function fr h,
given by
1
, d , d
r h r
,f x h y Q x y y
(39)e
wher
1
,
2
2 2
, ,
, ,
d .
1
d
r r s
s
Q x y Q x y
g x y
r g
(40)
2) Let r, 0 2
d 1
L such that
and h h, in
2 d1 .
L h h
Then
1, , .
2
s d
r h r h H
f f
r
3) Let r> 0. If f is in Hs
d1
and h gf
.,Then
.
12
, s d s 0.
r h
f f r
0 a
H
REFERENCES
[1] Z. B. Nahia and N. B. Salem, “Spherical Harmonics and Applications Associated with the Weinstein Operator,” Po- tential Theory—ICPT 94, 1996, pp. 235-241.
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