Continuous Gabor transform for a class of non-Abelian groups

0.21 MB·English

by  Arash Ghaani Farashahi

CONTINUOUS GABOR TRANSFORM FOR A CLASS OF NON-ABELIAN GROUPS
ARASHGHAANIFARASHAHI∗ ANDRAJABALIKAMYABI-GOL
2
1
Abstract. InthisarticlewedefinethecontinuousGabortransformforsecondcountable,non-abelian,unimodularand
0
typeIgroupsandalsoweinvestigateaPlancherelformulaandaninversionformulaforourdefinition. Asanexample
2
weshowthathowtheseformulasworkfortheHeisenberggroupandalsothematrixgroupSL(2,R).

n
a
J

4
1. Introduction

]
Many physical quantities including pressure, sound waves, electro fields, voltages, electronic currents and electro-
A
magnetic fields vary with time. These quantities are called signals or waveforms. Signals can be described in a time
F
domainorinafrequencydomainbythetraditionalmethodsofFouriertransform. Thefrequencydescriptionofsignals
.
h is known as the frequency analysis or the spectral analysis. It was recognized long times ago that a global Fourier
t
a transform of a long time signal has little practical value in analyzing the frequency spectrum of the signal. From
m the Fourier transform f(w) of a signal f(t), it is always possible to determine which frequencies were present in the
[ signal. However, there is absolutely no indication as to when these frequencies existed. So, the Fourier transform

cannot provide any infbormation regarding either a time evolution of spectral characteristic or possible localization
1
with respect to the time variable. Some signals such as speech signals or ECG signals require the idea of frequency
v
4 analysis that is local in time.
3 In general,the frequency ofa signalvaries with time, so there is a need for a joint time-frequency representationof
9 a signalin orderto describe fully the characteristicsof the signal. This requires specific mathematical methods which
0
go beyond the classicalFourier analysis. The first who introduced the joint time-frequency representationof a signal,
.
1 was Gabor (1946). Gabor transform originates from the work of Dennis Gabor [8], in which he used translations and
0 modulationsofthe Gaussiansignalto representonedimensionalsignals. Also,thistransformiseventuallycalledsuch
2
as; the short time Fourier transform, the Weyl-Heisenberg transform or the windowed Fourier transform. Usually
1
Gabor analysis is investigated on R and recently other settings have been looked at (See [5]). Using discrete signals,
:
v GabortheoryisdoneonZ,andnumericalimplementationsrequiretoconsiderfiniteperiodicsignalsandconsequently
Xi Gabor theory on finite cyclic groups. In image processing Gabor theory on R2 or Rn and also its discrete version
r on Zn and finite abelian groups is necessary. Computer scientists might even argue that the right setting for Gabor
a analysis are the p-adic groups, because their group operation imitate the computer arithmetic most closely. Since
Gabor analysis resets mainly on the structure of translations and modulation, it is possible to extend it to other
abelian groups. For more explanation, we refer the readers to the monograph of K.Gr¨ochenig [9] or complete work
of H.G.Feichtinger and T.Strohmer [5]. One can find a complete extension of this theory to the set up of locally
compact abelain groups. But many groups in physics such as the Heisenberg group and also many applicable groups
in engineeringsuchas Motion groupsare non-abelianand so that the standardSTFT theory in abeliancase fails. On
theotherhandgeneralizationoftheFeichtingeralgebraandalsoGelfandtripletothesetupofnon-abeliangroupsvia
theshorttimeFouriertransformapproachwillbeuseful(see[3]and[4]). Thesefactspersistustofindageneralization
of the basic STFT theory for non-abelain groups. We recall that passing through the harmonic analysis of abelian
2000 Mathematics Subject Classification. Primary43A30,43A32, 43A65,22D10.
Keywordsandphrases. continuousGabortransform,Fouriertransform,Plancherelformula,Plancherelmeasure,unitaryrepresentation,
irreduciblerepresentation, primaryrepresentation, typeIgroup,unimodulargroup,measurablefieldofoperators.
∗Correspondingauthor.
E-mailaddresses: [email protected](A.GhaaniFarashahi),[email protected](R.Kamyabi-Gol).
1
2 ARASHGHAANIFARASHAHIANDRAJABALIKAMYABI-GOL
groups to the harmonic analysis of non-abelian groups, many useful results and basic concepts in abelian harmonic
analysiscollapse,whichplayimportantrolesinthe usualGabortheory. Thus,the extensionofGaboranalysis(STFT
theory) for non-abelian groups is not trivial and as we shall discuss, the shape of results in this theory change a lot.
But fundamental properties are still valid, with different proofs. In this article we would like to find an appropriate
notationofthecontinuousGabortransformonaclassofnon-abelaingroupsandalsoweinvestigatethegeneralization
of its fundamental properties.
Throughout this paper which contains 4 sections, we assume that G is a second countable, type I and unimodular
locally compact group. Section 2 is deduced to fix notations and a brief summery on non-abelian Fourier analysis.
In section 3, we define the continuous Gabor transform of a square integrable function f on G, with respect to the
window function ψ, as a measurable fields of operators defined on G×G by
Gψf(x,π):= f(y)ψ(x−1y)π(yb)∗dy.
ZG
Finally,insection4,westudyexamplesofcontinuousGabortransformfortheHeisenberggroupandthematrixgroup
SL(2,R).
2. Preliminaries and notations on non-abelian Fourier analysis
Let H be a separable Hilbert space. An operator T ∈ B(H) is called a Hilbert-Schmidt operator if for one, hence
foranyorthonormalbasis{e }ofH wehave kTe k2 <∞. The setofallHilbert-SchmidtoperatorsonH denoted
k k k
by HS(H) and for T ∈ HS(H) we define Hilbert-Schmidt norm of T as kTk2 := kTe k2. It can be checked that
P HS k k
HS(H) is a self adjoint and two sided ideal in B(H) and when H is finite-dimensional we have HS(H ) = B(H),
π
P
also we call an operator T ∈ B(H) of trace-class, whenever kTk := tr[|T|] < ∞, where tr[T] := hTe ,e i and
tr k k k
|T|=(TT∗)1/2. For more details about trace-class and Hilbert-Schmidt-operators, we refer the readers to [16].
P
Let (A,M) be a measurable space. A family {H } of non zero separable Hilbert spaces indexed by A will be
α α∈A
calleda fieldofHilbert spacesoverA. A mapΦ onAsuchthatΦ(α)∈H for eachα∈Awillbe calleda vectorfield
α
on A. We denote the inner product and norm on H by h.,.i and k.k , respectively. A measurable field of Hilbert
α α α
spaces over A is a field of Hilbert spaces {H } together with a countable set {e } of vector fields such that the
α α∈A j
functions α 7→ he (α),e (α)i are measurable for all j,k and also the linear span of {e (α)} is dense in H for each
j k j α
α ∈ A. Given a measurable field of Hilbert spaces ({H } ,{e (α)}) on A, a vector field Φ on A will be called
α α∈A j
measurable if hΦ(α),e (α)i is measurable function on A for each j. The direct integral of the spaces {H } with
j α α α∈A
L
respect to a measure dα on A is denoted by H dα. This is the space of measurable vectors fields Φ on A such
α
ZA
L
that we have kΦk2 = kΦ(α)k2dα < ∞. Then it is easily follows that H dα is a Hilbert space with the inner
α α
ZA ZA
product hΦ,Ψi= hΦ(α),Ψ(α)i dα.
α
ZA
Henceforth, when G is a locally compact group and dx is a Haar measure on G, C (G) consists of all continuous
c
complex-valuedfunctions on G with compactsupports andfor each1≤p<∞, let Lp(G) standfor the Banachspace
ofequivalence classesofmeasurablecomplex valuedfunctions onGwhose p-thpowersareintegrable. Now,let π be a
continuous unitary representation of G on the Hilbert space H , for more details and elementary descriptions about
π
the topologicalgrouprepresentations see [6] or [12]. The representationπ is called primary,if only scaler multiples of
the identity belongs to center of C(π). Note that, primary representations are also known as factor representations.
According to the Schur’s lemma, Theorem 3.5 of [6], every irreducible representation is primary. More generally, if
π is a direct sum of irreducible representations, π is primary if and only if all its irreducible subrepresentations are
unitarily equivalent. The group G is said to be type I, if every primary representation of G is a direct sum of copies
of some irreducible representation. Also, assume that the dual space G be the set of all equivalence classes [π] of
irreducibleunitaryrepresentationsπ ofGandwestilluseπ todenoteitsequivalenceclass[π]. Notethat,weequipped
G with the Fell topology. See [6], for a discussion of this topology on G.b
b b
CONTINUOUS GABOR TRANSFORM FOR A CLASS OF NON-ABELIAN GROUPS 3
There is a measure dπ on G, called the Plancherel measure, uniquely determined once the Haar measure on G is
fixed. Thefamily{HS(H )} ofHilbertspacesindexedbyGisafieldofHilbertspacesoverG. Recallthat,HS(H )
π π∈G π
is a Hilbert space with the inbnber product hT,Si = tr(S∗T). The direct integral of the spaces {HS(H )}
HS(Hπ) π π∈Gb
L b b
with respectto dπ, is denoted by HS(H )dπ andfor convenience we use the notation H2(G) for it. If f ∈L1(G),
π
ZGb
the Fourier transform of f is a measurable field of operators over G given by
b
(2.1) Ff(π)=f(π)= f(x)bπ(x)∗dx.
ZG
Let J1(G) := L1(G)∩L2(G) and J2(G) be the finbite linear combinations of convolutions of elements of J1(G). In
[18], Segal proved that, when G is a second countable, non-abelian, unimodular and type I group, there is a measure
dπ on G, uniquely determine once the Haar measure dx on G is fixed, which is called the Plancherel measure and
satisfies the following properties;
b
(1) The Fourier transform f 7→ f maps J1(G) into H2(G) and it extends to a unitary map from L2(G) onto
H2(G).
b b
(2) Also, each h∈J2(G) satisfies the Fourier inversion formula h(x)= tr[π(x)h(π)]dπ.
b
ZGb
Commonly class of type I and unimodular groups are compact groups. When G is a cbompact group due to Theorem
5.2 of [6], each irreducible representation (π,H ) of G is finite dimensional which implies that B(H )=HS(H ) and
π π π
alsoeveryunitary representationofG is adirectsumofirreducible representations. Ifπ is anyunitary representation
of G, for each u,v∈H the functions π (x)=hπ(x)u,vi are called matrix elements of π. If {e } is an orthonormal
π u,v j
basis for H , we put π (x) = hπ(x)e ,e i. Notation E stands for the linear span of the matrix elements of π and E
π ij j i π
for the linear span of E . The Peter-Weyl Theorem, Theorem 3.12 of [6], guarantee that E is uniformly dense
[π]∈G π
b
in C(G), L2(G) = [Sπ]∈GEπ, and also {dπ−1/2πij : i,j = 1...dπ,[π] ∈ G} is an orthonormal basis for L2(G). Thus,
b
according to the Peter-Weyl Theorem, if f ∈L2(G) we have
L
b
dπ
(2.2) f = cπ(f)π ,
ij ij
[πX]∈GbiX,j=1
where cπ (f) = d hf,π i . If we choose an orthonormal basis for H so that π(x) is represented by the matrix
i,j π ij L2(G) π
(π (x)), then f(π) is given by the matrix f(π) =d−1cπ(f) and satisfies
ij ij π ji
b dπ b dπ
cπ(f)π (x)=d f(π) π (x)=d tr[f(π)π(x)].
ij ij π ji ij π
i,j=1 i,j=1
X X
b b
So, for f ∈L2(G), (2.2) becomes a Fourier inversion formula, f(x)= d tr[f(π)π(x)]. The Parsevalformula
[π]∈G π
b
P
dπ b
(2.3) kfk2 = d−1|c (f)|2
L2(G) π ij
[πX]∈GbiX,j=1
becomes
kfk2 = d tr[f(π)∗f(π)].
L2(G) π
[πX]∈Gb
b b
4 ARASHGHAANIFARASHAHIANDRAJABALIKAMYABI-GOL
3. Continuous Gabor transform
Throughout this paper, let G be a second countable, non-abelian, unimodular and type I group. Let dσ be
the product of the Haar measure dx on G and the Plancherel measure dπ on G. For each (x,π) ∈ G × G, let
H =π(x)HS(H ), where π(x)HS(H )={π(x)T : T ∈HS(H )}. It can be checked that H is a Hilbert space
(x,π) π π π (x,π)
with respect to the inner product hπ(x)T,π(x)Si = tr(S∗T). The family {Hb } of Hilbert sbpaces
H(x,π) (x,π) (x,π)∈G×Gb
indexed by G×G is a field of Hilbert spaces over G×G. The direct integral of the spaces {H } with
(x,π) (x,π)∈G×G
b
respect to σ, is denoted by H2(G×G), that is the space of all measurable vector fields F on G×G such that
b b
kFbk2 = kF(x,π)k2 dσ(x,π)<∞. b
H2(G×G) (x,π)
b ZG×Gb
It can be checked that H2(G×G) becomes a Hilbert space, with the inner product
bhF,Ki = tr[K(x,π)∗F(x,π))]dσ(x,π).
H2(G×G)
b ZG×Gb
Let L2(G,B(H )) be the Banach space of all measurable functions φ:G→B(H ) with
π π
kφk2 = kφ(x)k2dx<∞,
L2(G,B(Hπ))
ZG
where for each x ∈ G by kφ(x)k we mean the operator norm of φ(x). More explanations about spaces related to
functions with values in a Banach space can be found in [17]. For each ψ ∈ L2(G) and π ∈ G, the modulation of ψ
with respect to π, is an operator valued mapping defined almost every where on G by ψ (x) =ψ(x)π(x). The linear
π
transformation M : L2(G) →L2(G,B(H )) defined by ψ 7→ ψ is called the π-modulation opberator. Clearly, M is
π π π π
an isometry, because for each ψ ∈L2(G) we have
kM ψk2 = kM ψ(x)k2dx
π L2(G,B(Hπ)) π
ZG
= kψ(x)π(x)k2dx=kψk2
L2(G)
ZG
Our definition for modulation coincides with the usual definition of modulation as multiplication by a character in
the abelian group case. Indeed, when G is abelian, each irreducible representation of G is one-dimensional and the
corresponding representation space H is isomorphic to C. For f ∈ L2(G) and φ ∈ L2(G,B(H )), let hf,φi be the
π π π
bounded operator on H defined by
π
hf,φi = f(y)φ(y)∗dy.
π
ZG
We interpret this operator valued integralin the weak sense. That is, for any z ∈H we define hf,φi z by specifying
π π
its inner product with an arbitrary v ∈H via
π
(3.1) hhf,φi z,vi= f(y)hφ(y)∗z,vidy.
π
ZG
Since the map y 7→ hφ(y)∗z,vi belongs to L2(G), the right hand side integral (3.1) is the ordinary integral of a
function in L1(G). It is not difficult to see that |hhf,φi z,vi| ≤ kzkkvkkfk kφk , therefore hf,φi
π L2(G) L2(G,B(Hπ)) π
defines a bounded linear operator on H , whose norm salsifies khf,φi k≤kfk kφk . We can consider
π π L2(G) L2(G,B(Hπ))
h.,.i :L2(G)×L2(G,B(H ))→B(H )
π π π
as a separately continuous sesqulinear map with values in B(H ). When G is an abelian group, for each π ∈ G, this
π
sesqulinear map coincides with the usual inner product of L2(G).
b
CONTINUOUS GABOR TRANSFORM FOR A CLASS OF NON-ABELIAN GROUPS 5
Definition 3.1. Let ψ be a window function (a fixed nonzero function in L2(G)) and f ∈ C (G). We define the
c
continuous Gabor transform of f with respect to the window function ψ as a measurablefields of operatorson G×G
by
b
(3.2) G f(x,π):= f(y)ψ(x−1y)π(y)∗dy.
ψ
ZG
Before studying the basic properties of our extension for the continuous Gabor transform we verify ambiguous points
of this definition. First, note that we consider the operator-valued integral (3.2) in the weak sense. In other words,
for each (x,π)∈G×G and ζ,ξ ∈H we have
π
hG f(x,π)ζ,ξi= f(y)ψ(x−1y)hπ(y)∗ζ,ξidy.
b ψ
ZG
Since the map y 7→ hπ(y)∗ζ,ξi is a bounded continuous function on G, the right hand side integral is the ordinary
integral of a function in L1(G). It is clear that |hG f(x,π)ζ,ξi| ≤ kζkkξkkfk kψk , so G f(x,π) is indeed a
ψ L2(G) L2(G) ψ
bounded linear operator on H such that kG f(x,π)k≤kfk kψk .
π ψ L2(G) L2(G)
Second, since f ∈ C (G) and ψ ∈ L2(G) we have f.L ψ ∈ J1(G) for each x ∈ G. Plancherel theorem, Theorem
c x
\
7.44 of [6], guarantee that f.L ψ(π) is a Hilbert-Schmidt operator for almost every where π ∈G. Thus, for σ-almost
x
every (x,π) in G×G we have G f(x,π)∈H . Indeed,
ψ (x,π)
b
Gψbf(x,π)= f(y)ψ(x−1y)π(y)∗dy
ZG
= f(y)ψ(x−1y)π(x)π(yx)∗dy
ZG
=π(x) f(y)ψ(x−1y)π(yx)∗dy
(cid:18)ZG (cid:19)
=π(x) f(yx−1)ψ(x−1yx−1)π(y)∗dy =π(x)F Rx−1(f.Lxψ) (π).
(cid:18)ZG (cid:19)
(cid:0) (cid:1)
In the next proposition we state some worthwhile properties of our definition. First, let us recall that when G is
unimodular and 1≤p<∞, the involution for g ∈Lp(G) is g(x)=g(x−1).
Proposition 3.2. Let ψ be a window function and f ∈C (G). Then, for each (x,π)∈G×G we have
c
\ e
(1) G f(x,π)=Lψ(f)(π), where Lψ(f) in L1(G) is defined for a.e. y in G by f(y)ψ(x−1y).
ψ x x b
^
(2) G f(x,π)∗ =F Lψ(f) (π).
ψ x
(cid:18) (cid:19)
(3) G f(x,π)=hf,M (L ψ)i .
ψ π x π
We call (1) the Fourier representation form of the continuous Gabor transform.
Proof. (1) and (3) follow immediately from the definitions. Note that, Ho¨lder’s inequality guarantee that Lψ(f)
x
^
belongstoL1(G). Let(x,π)∈G×Gandζ ∈H . For(2),usingunimodularityofGandtheidentityLψ(f)=f.L ψ,
π x x
we get
hG f(x,π)∗ζ,bζi=hζ,G f(x,π)ζi e g
ψ ψ
= hζ,f(y)ψ(x−1y)π(y)∗ζidy
ZG
= hf(y)L ψ(y)π(y)ζ,ζidy
x
ZG
^
= hf(y)L ψ(y)π(y)∗ζ,ζidy = F Lψ(f) (π)ζ,ζ .
x x
ZG (cid:28) (cid:18) (cid:19) (cid:29)
e g
6 ARASHGHAANIFARASHAHIANDRAJABALIKAMYABI-GOL
(cid:3)
ThenexttheoremshowsthatthecontinuousGabortransformpreservestheenergyofthesignal. Morepreicisly,we
prove that for each window function ψ, the linear operator G :C (G)→H2(G×G) given by f 7→G f is a multiple
ψ c ψ
of an isometry.
b
Theorem 3.3. Let ψ be a window function. Then, for each f ∈C (G) we have
c
kG fk =kfk kψk .
ψ H2(G×G) L2(G) L2(G)
b
Proof. Using Proposition 3.2, Theorem 2.1 of [14], Fubini’s theorem and also unimodularity of G we have
kG fk2 = kG f(x,π)k2 dσ(x,π)
ψ H2(G×G) ψ (x,π)
b ZG×Gb
= tr[G f(x,π)∗G f(x,π)]dσ(x,π)
ψ ψ
ZG×Gb
= tr[G f(x,π)∗G f(x,π)]dπdx
ψ ψ
ZGZGb
\
^ \
= tr[Lψ(f)(π)Lψ(f)(π)]dπdx
x x
ZGZGb
\ ∗\
= tr[Lψ(f)(π) Lψ(f)(π)]dπdx
x x
ZGZGb
= Lψ(f)(y)Lψ(f)(y)dydx=kfk2 kψk2 .
x x L2(G) L2(G)
ZGZG
(cid:3)
According to Theorem 3.3, the continuous Gabor transform G : C (G) → H2(G×G) defined by f 7→ G f is a
ψ c ψ
multiple an isometry. So, we can extend G uniquely to a bounded linear operatorfrom L2(G) into a closed subspace
ψ
of H2(G×G) which we still use the notation G for this extension and this extension forbeach f ∈L2(G) satisfies
ψ
kG fk =kfk kψk .
b ψ H2(G×G) L2(G) L2(G)
b
We call G f the continuous Gabor transform of f ∈ L2(G) with respect to the window function ψ, which can be
ψ
considered as the sesquilinear map (f,ψ)7→G f on L2(G)×L2(G) into H2(G×G).
ψ
We can conclude the following orthogonality relation for the continuous Gabor transform.
b
Corollary 3.4. Let ψ,ϕ be two window functions. The continuous Gabor transform satisfies the orthogonality rela-
tion; hG f,G gi = hϕ,ψi hf,gi , for each f,g ∈ L2(G). Moreover, the normalized Gabor transform
ψ ϕ H2(G×G) L2(G) L2(G)
b
kψk−1 G is an isometry from L2(G) onto a closed subspace of H2(G×G).
L2(G) ψ
Proof. By using the polarization identity, the result follows. (cid:3)
b
Remark 3.5. When G is a compact group automatically G is unimodular and also type I and so that we can use the
definition (3.2) of G f for all f,ψ ∈ L2(G), which first we define the continuous Gabor transform of f ∈ C (G) and
ψ c
then we extend it for elements of L2(G). But independent of (3.2) since G is a compact group we can consider for a
window function ψ in L2(G) and each f ∈L2(G) we can define the continuous Gabor transform of f with respect to
the window function ψ as a Hilbert-Schmidt operator valued function on G×G by
(3.3) Gψf(x,π)= f(y)ψ(x−1y)π(y)∗dy. b
ZG
CONTINUOUS GABOR TRANSFORM FOR A CLASS OF NON-ABELIAN GROUPS 7
In this case by compactness of G, it is guaranteed that for each (x,π)∈G×G we have G f(x,ω)∈HS(H ). In the
ψ π
sequel corollary we state the Plancheral formula for the case G is compact. It should be noted that although it is a
corollary of the preceding Theorem but it can be proved separately. b
Corollary 3.6. Let G be a compact group and ψ be a window function. Then, for each f ∈L2(G) we have
kG f(x,π)k2 dx=kfk2 kψk2 .
ψ HS L2(G) L2(G)
ZG[πX]∈Gb
To prove an inversion formula, we need some notations. Let ψ be a window function and K ∈ H2(G×G). The
conjugate linear functional
b
ℓK(g)= tr[K(y,π)G g(y,π)∗]dσ(y,π),
ψ ψ
ZG×Gb
is a bounded functional on L2(G). Indeed, using Cauchy-Schwartz inequality and Theorem 3.3 we have
|ℓK(g)|= tr[K(y,π)G g(y,π)∗]dσ(y,π)
ψ ψ
(cid:12)ZG×Gb (cid:12)
(cid:12) (cid:12)
≤(cid:12) |tr[K(y,π)G g(y,π)∗]|dσ(y,π(cid:12))
(cid:12) ψ (cid:12)
ZG×Gb
≤kKk kG gk =kKk kψk kgk .
H2(G×G) ψ H2(G×G) H2(G×G) L2(G) L2(G)
b b b
This shows that ℓK defines a unique element in L2(G), which we use the notation
ψ
tr[K(y,π)M (L ψ)]dσ(y,π),
π y
ZG×Gb
for this element of L2(G). According to this notation, for each g ∈L2(G) we have
tr[K(y,π)M (L ψ)]dσ(y,π),g = tr[K(y,π)G g(y,π)∗]dσ(y,π).
π y ψ
(cid:28)ZG×Gb (cid:29)L2(G) ZG×Gb
In the next theorem we prove an inversionformula.
Theorem 3.7. Let ψ,ϕ be two window functions such that hϕ,ψi 6=0. Then, for each f ∈L2(G) we have
L2(G)
f =hϕ,ψi−1 tr[G f(y,π)M (L ϕ)]dσ(y,π).
L2(G) ψ π y
ZG×Gb
Proof. By Theorem 3.3, we have G f ∈H2(G×G). As we mentioned above, the integral
ψ
hϕ,ψi−L21(G) b tr[Gψf(y,π)Mπ(Lyϕ)]dσ(y,π),
ZG×Gb
is a well-defined function in L2(G), for convenience in calculations we use fϕ for this function. Using Corollary 3.4,
ψ
for each g ∈L2(G) we have
hfϕ,gi =hϕ,ψi−1 tr[G f(y,π)G g(y,π)∗]dσ(y,π)
ψ L2(G) L2(G) ψ ϕ
ZG×Gb
=hϕ,ψi−1 hG f,G gi
L2(G) ψ ϕ H2(G×G)
b
=hf,gi .
L2(G)
Which follows f =fϕ and so the inversion formula holds. (cid:3)
ψ
As an immediate consequence of Theorem 3.7 we have the following corollary.
8 ARASHGHAANIFARASHAHIANDRAJABALIKAMYABI-GOL
Corollary 3.8. Let ψ be a window function such that kψk =1. Then, for each f ∈L2(G) we have
L2(G)
f = tr[G f(y,π)M (L ψ)]dσ(y,π).
ψ π y
ZG×Gb
Also when G is compact the inversion theorem can be deduced more applicable in the following corollary.
Corollary 3.9. Let G be a compact group and ψ,ϕ be two window functions such that hϕ,ψi 6= 0. Then, for
L2(G)
each f ∈L2(G) we have
f(x)=hϕ,ψi−1 d tr[G f(y,π)M (L ϕ)(x)]dy for a.e x∈G.
L2(G) π ψ π y
ZG[πX]∈Gb
The following proposition gives us a useful relation of Gabor transform of two non-orthogonalwindow functions.
Proposition 3.10. For window functions ψ,ϕ with hϕ,ψi 6=0, we have G∗G =hϕ,ψi I .
L2(G) ϕ ψ L2(G) L2(G)
Proof. Let S :H2(G×G)→L2(G) be the bounded linear operator defined by
ϕ
b S (K)= tr[K(y,π)M (L ϕ)]dσ(y,π).
ϕ π y
ZG×Gb
ThenS is theadjointoperatorofG . Infact,usingProposition3.2,foreachf ∈L2(G) andK ∈H2(G×G)wehave
ϕ ϕ
hS (K),fi = tr[K(y,π)G f(y,π)∗]dσ(y,π) b
ϕ L2(G) ϕ
ZG×Gb
=hK,G fi =hG∗(K),fi .
ϕ H2(G×G) ϕ L2(G)
b
Now, Theorem 3.7 shows that G∗G =hϕ,ψi I . (cid:3)
ϕ ψ L2(G) L2(G)
Let φ,φ′ ∈L2(G,B(H )) and also let the operator φ⊗ φ′ :L2(G)→L2(G,B(H )) defined by
π π π
f 7→φ⊗ φ′(f)=hf,φ′i φ.
π π
Now Proposition (3.10) can be considered as the following continuous resolution of the identity operator for compact
groups.
Corollary 3.11. Let G be a compact group and ψ,ϕ window functions with hϕ,ψi 6=0. Then, we have
L2(G)
I =hϕ,ψi−1 d tr[M (L ϕ)⊗ M (L ψ)]dy,
L2(G) L2(G) π π y π π y
ZG[πX]∈Gb
where the right integral is a notation for the bounded linear operator defined on L2(G) by
f 7→ d tr[M (L ϕ)⊗ M (L ψ)(f)]dy.
π π y π π y
ZG[πX]∈Gb
4. Examples
As the first example we study the theory on the Heisenberg group. In quantum mechanics, Weyl-quantization
(phase-spacequantization)is a method for systematicallyassociatinga quantum mechanicalHermitian operatorwith
a classical kernel function in phase space (see [21, 22]). The Heisenberg group which is associated to n-dimensional
quantum mechanical systems plays an important role in the Weyl-quantization.
CONTINUOUS GABOR TRANSFORM FOR A CLASS OF NON-ABELIAN GROUPS 9
4.1. Heisenberggroup. WestartwiththedefinitionsandsomedescriptionoftheHeisenberggroups. TheHeisenberg
groupHn is a Lie group with the underlying manifold Rd, where d=2n+1. We denote points in Hn by h=(t,q,p)
with t∈R, q,p∈Rn and define the group operation by
1
(t ,q ,p )(t ,q ,p )=(t +t + (p .q −p .q ),q +q ,p +p ).
1 1 1 2 2 2 1 2 1 2 2 1 1 2 1 2
2
Itiseasytojustifythatthisisagroupoperation,withtheidentityelement0=(0,0,0)andalsotheinverseof(t,q,p)
is given by (−t,−q,−p). It can be checked that Lebesgue measure on R2n+1 = Hn is left and right invariant under
the groupactiondefined as above. Hence, Lebesgue measure on R2n+1 gives the Haar measureon Hn, and this group
is unimodular. Taylor in [19] proved that the following map
̺(t,q,p)f(x)=ei(t+q.x+q.p/2)f(x+p), for each f ∈L2(Rn),
isanirreducibleunitaryrepresentationofHn onthe HilbertspaceL2(Rn). Now,foreachλ6=0,themapδ (t,q,p)=
λ
(λt,signλ|λ|1/2q,|λ|1/2p), is an automorphism of Hn and so π (h) := ̺(δ (h)) defines an irreducible representation
λ λ
of Hn on the Hilbert space L2(Rn). Each representation π is given explicitly on L2(Rn) by
λ
π (t,q,p)f(x)=ei(tλ+|λ|1/2signλq.x+λq.p/2)f(x+|λ|1/2p).
λ
Except for the infinite-dimensional irreducible representations above, there are also the following one-dimensional
irreducible representations of Hn; π (t,q,p)=ei(ξ.q+η.p). Stone-von Neumann theorem and Kirillov theory imply
(ξ,η)
thatthese twoclassesofthe representationsofHn,exhaustthe irreduciblerepresentationsofHn[see[1]]. Also,no two
different representations of these two classes are unitarily equivalent. Hence, we can say that
Hn ={π |λ∈R\{0}} {π |(ξ,η)∈Rn×Rn}.
λ (ξ,η)
We associate to a function f in L1(Hn)∩L2(Hn), the F[ourier transform
c
f(π )= f(t,q,p)e−i(ξ.q+η.p)d(t,q,p), f(π )= f(t,q,p)π (t,q,p)∗d(t,q,p).
(ξ,η) λ λ
ZHn ZHn
For conveniebnce, we write f(λ) instead of f(π ) and also f(ξ,ηb) instead of f(π ). Then, the Plancherel theorem
λ (ξ,η)
for the Heisenberg group Hn is given by
b b b +∞ b
|f(t,q,p)|2d(t,q,p)=(2π)−(n+1) kf(λ)k2 |λ|ndλ.
HS
ZHn Z−∞
Also, the following inversion Fourier transform for the Heisenberg group Hbn holds;
+∞
f(t,q,p)=(2π)−(n+1) tr[π (t,q,p)f(λ)]|λ|ndλ.
λ
Z−∞
Note that, the representationsof the form π(ξ,η) have no contributionto thbe Plancherelformula and Fourierinversion
transform because this set of representations has zero Plancherelmeasure. In other words, Plancherel measure dπ on
Hn is givenbydπ =|λ|ndλanddπ =0. Thatis,the measure(2π)−(n+1)|λ|n onR\{0}is the Plancherelmeasure
λ (ξ,η)
on Hn.
c
Example 4.1. Let ψ(t,q,p)=2d/4e−πk(t,q,p)k2 be the classical Gaussian window function on Hn.
c
The associated continuous Gabor transform of f ∈ C (Hn) with respect to the window function ψ at each
c
(t,q,p,π )∈Hn×Hn can be identified by
λ
hg,Gψf(t,q,p,πcλ)ki= Lψ(t,q,p)(f)(t′,q′,p′)hg,πλ(t′,q′,p′)∗kid(t′,q′,p′)
ZHn
= Lψ (f)(t′,q′,p′)ei(tλ+|λ|1/2signλq.x+λq.p/2)g(x+|λ|1/2p)k(x)dxd(t′,q′,p′),
(t,q,p)
ZHnZRn
for each g,k∈L2(Rn).
10 ARASHGHAANIFARASHAHIANDRAJABALIKAMYABI-GOL
Similarly, the continuous Gabor transform at each (t,q,p,π )∈Hn×Hn can be identified by
(ξ,η)
G f(t,q,p,π )= Lψ (f)(t′,q′,p′)π (t′,qc′,p′)∗d(t′,q′,p′)
ψ (ξ,η) (t,q,p) (ξ,η)
ZHn
= Lψ (f)(t′,q′,p′)e−i(ξ.q+η.p)d(t′,q′,p′),
(t,q,p)
ZHn
where Lψ(t,q,p)(f)(t′,q′,p′)=2d/4f(t′,q′,p′)e−π(|t+t′+12(p.q′−p′.q)|2+kq+q′k2+kp+p′k2).
Using Theorem 3.7, for each f ∈L2(Hn) we can reconstruct f via
hf,ki = tr[F Lψ (f) (π)G k(t′,q′,p′,π)]dπd(t′,q′,p′)
L2(G) (t′,q′,p′) ψ
ZHnZHcn (cid:16) (cid:17)
=(2π)−(n+1) tr[F Lψ (f) (λ)G k(t′,q′,p′,λ)]|λ|ndλd(t′,q′,p′),
(t′,q′,p′) ψ
ZHnZR\{0} (cid:16) (cid:17)
for each k ∈ L2(Hn). We recall that, for Lie group Rd and a window function ψ, the classical Gabor transform of f
in L2(Rd) is
G f(x,w)= e−2πiw.yf(y)ψ(y−x)dy.
ψ
ZRd
one can consider the difference between these two transforms.
In the sequel, we provide another example of the continuous Gabor transform on the non-abelian matrix group
SL(2,R). The special linear group SL(2,R) has significant role in the theory of linear canonical transformation.
The linear canonical transformationis a generalizationof the Fourier, fractional Fourier, Laplace, Gauss-Weierstrass,
Bargmann and also the Fresnel transforms in particular cases (see [11, 15]). Canonical transforms provide an ap-
propriate tool for the analysis of a class of differential equations which are applicable in Fourier optics and quantum
mechanics.
4.2. Matrix group SL(2,R). We recall that G = SL(2,R) is the group of 2×2 real matrices of determinant one.
This group is unimodular and it’s Haar integral which is right and left invariant is given by
+∞ +∞ +∞ +∞ x y
f dxdydzdt.
z t
Z−∞ Z−∞ Z−∞ Z−∞ (cid:18) (cid:19)
One can list continuous unitary representations of SL(2,R) in five classes. For more explanations see [6].
(1) The trivial representation ι, acting on C.
(2) The discrete series {δ± :n≥2}. For n≥2, let H+ be the space of holomorphic functions f on the upper half
n n
plane U ={z :ℑ(z)>0} such that
kfk2 = |f(x+iy)|2yn−2dxdy <∞.
[n]
Z ZU
The representation δ+ of SL(2,R) on H+ is defined by
n n
a b az−c
δ+ f(z)=(−bz+d)−nf for each f ∈H+.
n c d −bz+d n
(cid:18) (cid:19) (cid:18) (cid:19)
Similarly,H−isthespaceofanti-holomorphicfunctionsf ontheupperhalfplaneU satisfyingtheconditionkfk <∞
n [n]
and the representation δ− of SL(2,R) on H− is given the same as δ+. It can be checked that, the representations
n n n
{δ± :n≥2} are unitary and irreducible.
n
(3) The mock discrete series {δ±}. Let H+ be the space of holomorphic functions f on half plane U such that
1 1
+∞
kfk2 =sup |f(x+iy)|2dx<∞,
[1]
y>0Z−∞