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An Abstract Harmonic Analysis Approach

to Gabor Analysis

Vignon Oussa

Bridgewater State University, MA

March 2014

A Starting Point

I One reason why Gabor theory has attracted so muchattention is because of its practical applications.

I One recent (not so popular) application of Gabor theoryis found in the analysis of certain nilpotent Lie groups.

Sampling on the Heisenberg group

Theorem

(H. Führ, B. Currey, and A. Mayeli) Let H be the3-dimensional Heisenberg group. Then there exist an integerlattice subgroup Γ, and a left-invariant closed subspace H ofL2 (H) consisting of continuous functions such that for allf ∈ H,

1. ∑γ∈Γ |f (γ)|2 = ‖f ‖2H .

2. There exists a function S ∈ H such that

f (x) = ∑γ∈Γ

f (γ) S(γ−1x

)with convergence both in L2 (H)-norm and uniformly.

Sampling on the Heisenberg group

Theorem

(B. Currey, and A. Mayeli) The Heisenberg group admitssampling spaces with the interpolation property.

Theorem

(O) Let N be a simply connected, connected nilpotent Liegroup with Lie algebra n such that n = z⊕ b⊕ a, z is thecenter of n, b = RYd ⊕RYd−1 ⊕ · · · ⊕RY1,a =RXd ⊕RXd−1 ⊕ · · · ⊕RX1, z⊕ b is a maximal commutativeideal of n, [a, b] ⊆ z, and det ([Xi ,Yj ])1≤i ,j≤d is a non-trivial

homogeneous polynomial de�ned over the ideal [n, n] . Thereexist band-limited spaces which are sampling subspaces ofL2(N) with respect to some discrete set Γ.

The proofs of all these results mentioned, rely on thewell-known Density Condition of Gabor systems.

Question: Can we use abstract harmonic analysis tools tostudy time-frequency analysis?

Some Background

Larry Baggett decomposed the Stone-von Neumannrepresentation of the discrete Heisenberg group acting onL2 (R) into a direct integral of representations. As a result, heproved:

Theorem

(L. Baggett, 1990) If αβ > 1, there is no φ ∈ L2 (R) such that{e2πiαjφ (t − βk) : (j , k) ∈ Z2

}spans L2 (R) .

Tight Weyl-Heisenberg Frames with Integer

Oversampling

Let L be an integer greater than one. LetΓ = Z×Z× (Z/LZ) with group law(

n, k , `) (

n′, k ′, `′)=(n+ n′, k + k ′, `+ `′ + k ′n

).

Then Γ admits a unitary representation π acting in L2 (R)such that

π(n, k , `

)= e2πi`/LTkMn/L

where Tk is a translation operator and Mn/L is a modulationoperator. De�ne abelian normal subgroup

Γ1 = LZ×Z× (Z/LZ) .

Theorem

(H. Führ, 2002) The representation π admits the followingirreducible direct integral decomposition:∫ ⊕

[0,1)×[0, 1L)IndΓ

Γ1 (χω) dω.

Using this decomposition, H. Führ characterized functionsg ∈ L2 (R) such that π (Γ) g is a Parseval frame.

Central Theme

I will present a generalization of Führ's ideas to higherdimensions with some applications.

A Few Relevant Papers

I The content of this talk is taken from

I V. Oussa, An Abstract Harmonic Analysis Approach

to Gabor Theory, preprint (2014).

I The content of this talk extends results in

I H. Führ, Abstract harmonic analysis of continuous

wavelet transforms. Lecture Notes in Mathematics,1863. Springer-Verlag, Berlin, 2005

I A. Mayeli, V. Oussa, Regular Representations ofTime Frequency Groups, Math. Nachr. 1�21 (2014) /DOI 10.1002/mana.201300019

I L. Baggett, Processing a radar signal and

representations of the discrete Heisenberg group,Colloq. Math. 60/61 (1990), no. 1, 195�203.

Unitary Representations

I Let G be a locally compact group. A unitary

representation of G is a homomorphism π from G intothe group of unitary operators U (H) such that

x 7−→ π (x) u, u ∈ H

is strongly continuous, and H is a Hilbert space.

I If π admits an invariant subspace that is non-trivial, thenπ is reducible, otherwise π is irreducible.

Induced Representations Part 1

I Let G be a locally compact group, and let K be a closedsubgroup.

I Let q : G → G/K be the canonical quotient map, let ϕbe a unitary representation of the group K acting in aHilbert space H.

I Let

K1 =

f : G → H : f is contq (support f ) is compact

f (gk) = ϕ (k)−1 f (g) where g ∈ G , k ∈ K

.

Induced Representations Part 2

I G acts on the space K1 by left translation.

I Assume G/K admits an invariant measure d (gK ).We construct a unitary representation of G by endowingK1 with⟨f , f ′

⟩=∫G/K

⟨f (g) , f ′ (g)

⟩Hd (gK ) for f , f ′ ∈ K1.

I Let K be the Hilbert completion of the space K1. Thetranslation operators extend to unitary operators on K

inducing a unitary representation: IndGK (ϕ)

IndGK (ϕ) (x) f (g) = f

(x−1g

)for f ∈ K.

Direct Integrals

I Let {Hα}α∈A be a family of separable Hilbert spaces. Thedirect integral of this family the {Hα}α∈A with respect toµ is the space of functions f de�ned on A such that f (α)is an element of Hα and∫

A‖f (α)‖2Hα

dµ (α) < ∞

with additional measurability conditions.I f : A→

⋃α

Hα such that f (α) ∈ Hα is called a vector

�eld on A.I The direct integral of the spaces Hα with respect to the

measure µ is denoted by∫ ⊕A

Hαdµ (α) .

Properties of Induction

I Inducing in stages: If G1 is a subgroup of G containing asubgroup K then

IndGK (ϕ) ' IndG

G1

(IndG1

K (ϕ)).

I The induction of a direct integral is an direct integral ofinduced representations

IndGK

(∫ ⊕A

ϕx dx

)'∫ ⊕A

[IndG

K (ϕx )]dx .

Gabor Representation

Let B be a non-singular matrix with real entries. De�ne theshift and frequency-shift operators acting in L2

(Rd)

Tk f (t) = f (t − k) for k ∈ Zd

Ml f (t) = e−2πi〈l ,t〉f (t) for l ∈ BZd

and de�ne Γ < U(L2(Rd))

Γ =⟨Tk ,Ml : (k , l) ∈ Zd × BZd

⟩.

Some Facts

I Γ is a nilpotent group with 2d generators (existence oflower central series of �nite length).

I [Γ, Γ] ={e2πi〈l ,Bk〉 ∈ T : (k , l) ∈ Zd × BZd

}⊂ T.

I If B has irrational entries, [Γ, Γ] is an in�nite centralsubgroup of Γ.

I If B has rational entries then [Γ, Γ] is a �nite subgroup ofthe torus.

Gabor Representation

Assume B is a rational matrix. So, [Γ, Γ] is isomorphic tosome �nite cyclic group Zm. Then, there is a semi-directproduct group

(Zm × BZd

)o Zd such that the

representation

π :(

Zm × BZd)o Zd → Γ

(j ,Bl , k) 7→ e2πjim MBlTk

is unitary, faithful and acts in L2(Rd).

A Special Subgroup

Let B? denote the inverse transpose of the matrix B . Put

Γ0 =

⟨[Γ, Γ] ,Ml ,Tk :

l ∈ BZd , k ∈ B?Zd ∩Zd

⟩' Zm × BZd ×

(B?Zd ∩Zd

).

I Γ0 is a normal abelian subgroup of Γ of �nite index.

I There exists an invertible matrix A such thatAZd = B?Zd ∩Zd .

Left Regular Representation

The left regular representation is a unitary representationwhich acts in l2 (Γ) as follows:

L(γ)a(α) = a(γ−1α)

for any sequence a ∈ l2 (Γ) .

Theorem

Assume that B is a rational matrix. The left regularrepresentation of

(Zm × BZd

)o Zd is decomposed as

follows:

L ' ⊕m−1k=0

∫ ⊕Rd

B?Zd ×Rd

A?Zd

IndΓΓ0χ(k,t) dt. (1)

Moreover, the measure dt in (1) is a Lebesgue measure class,and in general (1) is not an irreducible decomposition.

Proof.

Let e be the identity element in Γ.

L ' IndΓΓ0

(IndΓ0{e} (1)

)' IndΓ

Γ0

(∫ ⊕Γ̂0

χt dt

)'∫ ⊕

Γ̂0IndΓ

Γ0 (χt) dt.

Next, identifying Γ̂0 with Zm × Rd

B?Zd × Rd

A?Zd , it follows that

L ' ⊕m−1k=0

∫ ⊕Rd

B?Zd ×Rd

A?Zd

IndΓΓ0χ(k,t) dt.

Theorem

Let us suppose that B is a rational matrix. Then

π '∫ ⊕

Rd

Zd ×Rd

B?Zd

IndΓΓ0χ(1,σ) dσ.

This is an irreducible decomposition which is realized as actingin

L2

(Rd

Zd× Rd

B?Zd, l2(

ΓΓ0

)).

Sketching the Proof

1. Let Γ1 =⟨[Γ, Γ] ,Ml : l ∈ BZd

⟩.

2. π '∫ ⊕

Rd

Zd

[IndΓ

Γ1χ(1,−t)

]dt (the intertwining map here is

a periodization operator)

3.

IndΓΓ1χ(1,−t) ' IndΓ

Γ0

[IndΓ0

Γ1

(χ(1,−t)

)]' IndΓ

Γ0

[∫ ⊕Γ̂0Γ1

χ(1,−t,ξ) dξ

]

'∫ ⊕

Γ̂0Γ1

IndΓΓ0χ(1,−t,ξ)dξ.

4. π '∫ ⊕

Rd

Zd ×Rd

B?Zd

IndΓΓ0χ(1,σ)dσ.

Theorem

Assume that B is an rational matrix. L is equivalent to adirect sum of card ([Γ, Γ])-many non-equivalent highlyreducible representations such that

L 'card([Γ,Γ])−1⊕

k=0

Lk where Lk '∫ ⊕

Rd

B?Zd ×Rd

A?Zd

IndΓΓ0χ(k,t) dt.

Moreover, the representations Lk are disjoint from π wheneverk 6= 1, and the Gabor representation π is equivalent asubrepresentation of the subrepresentation L1 of L if and onlyif |detB | ≤ 1.

Application to Time-Frequency Analysis

Let π be a unitary representation of a locally compact groupG , acting in H. We say that π is admissible, if and only ifthere exists some vector φ ∈ H such that

W πφ : H → L2 (G ) , W π

φ ψ (x) = 〈ψ,π (x) φ〉

is an isometry of H into L2 (G ) .

Notice that if G has the discrete topology, then π isadmissible if and only if there exists some vector φ ∈ H suchthat π (G ) φ is a Parseval frame.

Application to Time-Frequency Analysis

Theorem

A representation of the group Γ is admissible if and only if therepresentation is equivalent to a subrepresentation of the leftregular representation of Γ.

Application to Time-Frequency Analysis

Theorem

Let B be a rational matrix. There exists a vector g ∈ L2(Rd)

such that the system{MlTkg : l ∈ BZd , k ∈ Zd

}is a

Parseval frame if and only if |detB | ≤ 1.

Application to Time-Frequency Analysis

There exists a unitary operator

A :∫ ⊕

Σ

[⊕`(σ)

k=1l2(

ΓΓ0

)]dσ→ L2

(Rd)

which intertwines the representations∫ ⊕Σ

[⊕`(σ)

k=1IndΓ

Γ0

(χ(1,σ)

)]dσ with π

Σ is a subset of a measurable transversal of Rd

B?Zd × Rd

A?Zd .

Characterization of Parseval frames

Theorem

Let us suppose that |detB | ≤ 1. Then

π '∫ ⊕

Σ

[⊕`(σ)

k=1IndΓ

Γ0

(χ(1,σ)

)]dσ

with ` (σ) ≤ |detA| dσ-almost everywhere. Moreover,π (Γ) f is a Parseval frame in L2

(Rd)if and only if

f = A(a (σ)σ∈Σ

)such that for dσ-almost every σ ∈ Σ, and

for i , j ∈ {1, · · · , ` (σ)} we have

‖a (σ) (i)‖l2(

ΓΓ0

) = 1 and 〈a (σ) (i) , a (σ) (j)〉l2(

ΓΓ0

) = 0.

Concluding Remarks

I For the case where B has some irrational entries, therepresentation theoretic tools used to obtain our resultsbreak down.

I One major problem is that the underlying group displayssome major pathological behavior (G. Folland gave a talkon these groups during the last FT)

Thank you for your attention