lecture 4 structure theorems for gabor framesmbvajiac/conferences/chapman_lec04... · 2017. 11....

Lecture 4 – Structure theorems for Gaborframes

David WalnutDepartment of Mathematical Sciences

George Mason UniversityFairfax, VA USA

Chapman Lectures, Chapman University, Orange, CA6-10 November 2017

Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames

Outline

Basic properties of Gabor framesGabor frames with compactly supported windowsExistence of Gabor framesZak transform methods and characterizations of framesJanssen’s representation of the Gabor frame operatorDensity theorems for Gabor framesThe Wexler-Raz and Ron-Shen duality


Time and frequency shifts

Definition

Given a, b ∈ Rd recall that the time-shift and frequency-shiftoperators Ta and Mb (respectively) on L2(Rd ) are given byTaf (x) = f (x − a) and Mbf (x) = e2πi(b·x) f (x). Note that(Mbf )∧(γ) = f̂ (γ − b).

Definition

Given a function g ∈ L2(R), the window function, and a discreteset Λ = {(λ, µ) : λ, µ ∈ Rd}, the collection

G(g,Λ) = {Tλ Mµg}

is called a Gabor system. If Λ has the form Λ = αZ× βZ,α, β ∈ R, then we denote G(g,Λ) ⊆ L2(R) by G(g, α, β). Theconstants α and β are called the frame parameters.


Definition

If the Gabor system G(g,Λ) is a frame for L2(Rd ) it is referred toas a Gabor frame. Hence there are constants 0 < A ≤ B suchthat for all f ∈ L2(RD),

A ‖f‖2 ≤∑

(λ,µ)∈Λ

|〈f ,Tλ Mµg〉|2 ≤ B ‖f‖2 .

Associated to a Gabor frame is the Gabor frame operator Sgiven by

Sf =∑

(λ,µ)∈Λ

〈f ,Tλ Mµg〉Tλ Mµg.

In the lattice case, G(g, α, β) has particularly niceproperties and we will concentrate on this case in thesequel.


The dual frame

LemmaThe Gabor frame operator associated to a Gabor frameG(g, α, β) commutes with the operators Tαk and Mβn for allk , n ∈ Z.

Proof:

(S ◦ Tα)f =∑k ,n

〈Tαf ,Tαk Mβng〉Tαk Mβng

=∑k ,n

〈f ,Tα(k−1) Mβng〉Tαk Mβng

=∑k ,n

〈f ,Tαk Mβng〉Tα(k+1) Mβng = (Tα ◦ S)f .

And similarly (S ◦Mβ)f = (Mβ ◦ S)f .


Since S−1 also commutes with Tαk and Mβn, the dualframe associated to the Gabor frame G(g, α, β) has theform

G(S−1g, α, β).

We call the function γ◦ = S−1g the canonical dual windowfor the Gabor frame.Recall that any f ∈ L2(R) can be written as

f =∑

k

∑n

〈f ,Tαk Mβng〉Tαk Mβnγ◦

=∑

k

∑n

〈f ,Tαk Mβnγ◦〉Tαk Mβng.

There are other “dual windows” that can replace γ◦ in theabove expansions.


Because of these properties of the frame operator, wedefine for g, γ ∈ L2(R), the operator Sg,γ by

Sg,γ f =∑k ,n

〈f ,TαkMβng〉TαkMβnγ

The frame operator for G(g, α, β) is in this notation Sg,g .S∗g,γ = Sγ,g so that Sg,γ is not in general self-adjointHowever, if γ is dual to g, then Sg,γ = S∗g,γ = I.


Invariance of Gabor frames

G(g, α, β) is a frame for L2(R) if and only if G(ĝ, β, α) is aframe for L2(R), and each frame has the same framebounds.G(g, α, β) is a frame for L2(R) if and only if G(Dag, α′, β′) isa frame for L2(R) where Da is the dilation operatorDag(x) = a1/2g(ax), with a = α/α′ = β′/β.Since α′β′ = αβ, this shows that the existence of Gaborframes for given α, β depends only on the product αβ.


Compactly supported windows

Let g(x) = α−1/21[0,α], α > 0.

If αβ = 1, then {e2πiβmt}m∈Z is an o.n. basis for L2[0, α].Therefore G(g, α, β) is an orthonormal basis for L2(R).



If α > 1/β, {e2πiβmt}m∈Z is incomplete on L2[0, α].If αβ > 1 then G(g, α, β) is incomplete.



If α < 1/β, {e2πiβmt}m∈Z is overcomplete on L2[0, α]. Itremains complete if elements are removed.If αβ < 1 then G(g, α, β) is overcomplete.



If α′ > α then G(g, α′, β) is incomplete no matter what β issince shifts of g do not cover R.This shows that density considerations alone are notsufficient to guarantee a frame.



Theorem

Let g ∈ L2(R) and α, β > 0 be such that:0 < a ≤

∑n

|g(x − nα)|2 ≤ b 0 and

supp(g) ⊂ I ⊂ R, where I is some interval of length 1/β.Then G(g, α, β) is a Gabor frame for L2(R) with frame boundsa/β, b/β.


If in addition

0 < infx∈I|g(x)| ≤ sup

x∈I|g(x)|


If in the above theorem g is continuous on R, G(g, α, β) aGabor frame implies that αβ < 1.If αβ > 1 the Gabor system is incomplete in L2(R).If αβ = 1 then at best the Gabor system will be completebut will lack a lower frame bound.In these considerations we see shades of the Balian-LowTheorem.


Necessary conditions

Theorem

Let g ∈ L2(R) and α, β > 0.If G(g, α, β) is a frame, it is necessary but not sufficient thatfor some 0 < a ≤ b,

a ≤∑

n

|g(x − nα)|2 ≤ b.

If G(g, α, β) is a frame, it is necessary but not sufficient thatαβ ≤ 1.If αβ > 1 then G(g, α, β) is not a frame, and in fact isincomplete.

We will see proofs of some of these facts later.


Representation of the frame operator

We begin with a basic decay assumption on our window g,by assuming that g ∈W (L∞, `1) = W (R), i.e.

‖g‖W =∑n∈Z

supx∈[0,1]

|g(x − n)|

Given α, β > 0, n ∈ Z, g, γ ∈W (R) define the correlationfunction

Gn(x) =∑

k

g(x − n/β − αk) γ(x − αk).

LemmaIf g, γ ∈W (R) then∑

n

‖Gn‖∞ 0, limβ→0+

∑n 6=0‖Gn‖∞ = 0.


Sufficient condition

Theorem (Walnut representation)

If g, γ ∈W (R) and α, β > 0, then the operator Sg,γ is given by

Sg,γ f (x) =1β

∑n

Gn(x) f (x − n/β).

In particular, Sg,γ is bounded with∥∥Sg,γ∥∥ ≤ 1β∑n ‖Gn‖∞.


TheoremLet g ∈W (R), α > 0 and suppose that there are constantsa, b > 0 such that

a ≤∑

k

|g(x − αk)|2 ≤ b.

Then there is a β0 > 0 such that G(g, α, β) is a Gabor frame forall 0 < β ≤ β0.


Zak transform methods

Definition

Recall that the Zak transform is defined for f ∈ L2(R) by

Zf (t , ω) =∑

n

f (x − n) e2πinx .

Zf (x + 1, ω) = e2πiωZf (x , ω), Zf (x , ω + 1) = Zf (x , ω).‖Zf‖L2(Q) = ‖f‖2. Here Q = [0,1]× [0,1].

Z (TkMnf )(x , ω) = e2πinx e2πikωZf (x , ω).


Frames with integer oversampling

Theorem

Let g ∈ L2(R) and let α = 1, β = 1/N, N ∈ N.

Z (TkMn/Ng)(x , ω) = e2πinx/N e−2πikωZg(x , ω −nN

).

Z ◦ Sg,g f =( N−1∑

j=0

|Zg(x , ω + jN

)|2)

Zf .

G(g, α, β) is a frame if and only if

0 < a ≤N−1∑j=0

|Zg(x , ω + jN

)|2 ≤ b.


Frames with rational oversampling

Definition

For f ∈ L2(R) and α = 1, β = p/q, p < q ∈ N. We define thevector-valued Zak transform by

(Zf (x , ω))j = Zf (x +jp, ω)

for j = 0, . . . , p − 1, (x , ω) ∈ Q1/p = [0,1/p]× [0,1].

The operator Z is a unitary operator from L2(R) onto the spaceof vector-valued functions

L2(Q1/p)× · · · × L2(Q1/p)︸︷︷︸p times.


We can therefore write for r = 0, . . . , p − 1

Z ◦ Sg,γ f(

x +rp, ω

)=

p−1∑s=0

Ar ,s(x , ω) Zf(

x +rp, ω

)where

Ar ,s(x , ω) =q−1∑j=0

Zg(

x +sp, ω − jp

q

)Zγ(

x+sp, ω− jp

q

)e2πij(r−s)/q.


Zibulski-Zeevi (1997)

The formulaZ ◦ Sg,γ f = A(x , ω)Zf

is called the Zibulskii-Zeevi representation.Leads to a characterization of frames in the case ofrational oversampling.

Theorem (Zibulskii-Zeevi)

Let g ∈ L2(R), and α = 1, β = p/q, p < q ∈ N. Then G(g, α, β)is a frame for L2(R) if and only if det A(x , ω) 6= 0 for a.e.(x , ω) ∈ Q1/p.


Further existence results

Seip and Wallstén (1992) showed that if g(x) = e−πx2,

then G(g, α, β) is a frame for L2(R) whenever αβ < 1.Janssen (1996) showed that if g(x) = e−x1[0,∞)(x), thenG(g, α, β) is a frame for L2(R) whenever αβ ≤ 1, and that ifg(x) = e−|x | then G(g, α, β) is a frame for L2(R) wheneverαβ < 1Janssen and Strohmer (2002) showed that ifg(x) = (ex + e−x )−1 then G(g, α, β) is a frame for L2(R)whenever αβ < 1.


Gröchenig and Stöckler (2013) found a class of windows,g, for which G(g, α, β) is a frame whenever αβ < 1.A function, g, is said to be totally positive if for every pair ofsequences

x1 < x2 < · · · < xN , and y1 < y2 < · · · < yN ,

the N × N matrix [g(xi − yj)] has nonnegative determinant.Such a function has finite type M if

ĝ(γ) =M∏ν=1

(1 + 2πiδνγ)−1.

Theorem

If g ∈ L2(R) is a totally positive function of finite type M ≥ 2,then G(g, α, β) is a frame if and only if αβ < 1.


Janssen considered g(x) = 1[0,c] and came up with “the tie.”

43Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames

Janssen representation

Theorem

For g, γ ∈ L2(R) sufficiently regular,

Sg,γ = (αβ)−1∑k ,n

〈γ,Tk/βMn/αg〉Tk/βMn/α

= (αβ)−1∑k ,n

〈γ,Mn/αTk/βg〉Mn/αTk/β.

The Janssen representation realizes the frame operator asa superposition of time-frequency shifts.


The Janssen representation follows from the followinglemma.

Lemma

For g, γ ∈ L2(R) sufficiently regular and for each n,

Gn(x) = α−1∑

k

〈γ,Mk/αTn/βg〉e2πikx/α

where Gn is the nth correlation function.


Theorem

(Wexler-Raz Biorthogonality Relations) If g, γ ∈ L2(R) aresufficiently regular then the following are equivalent.(a) Sg,γ = Sγ,g = I. In other words, γ is dual to g.

(b) (αβ)−1 〈γ,Mn/αTk/βg〉 ={

1 if (k ,n) = (0,0)0 otherwise

(c) G(γ,1/β,1/α) is biorthogonal to G(g,1/β,1/α).

Finding such a γ for a given g, α and β together withshowing that Sg,g and Sγ,γ are bounded is sufficient toshow that G(g, α, β) is a frame.


Proof of Wexler-Raz

(=⇒)

Recall that Sg,γ f (x) =1β

∑n

Gn(x) f (x − n/β) where

Gn(x) =∑

k

g(x − n/β − αk) γ(x − αk).

If Sg,γ = I it is easy to show that1β

G0(x) = 1 and that if

n 6= 0, Gn(x) = 0.


Since1β

Gn(x) =1αβ

∑k

〈γ,Tn/βMk/αg〉e2πikx/α it follows

that if n = 0,

1αβ〈γ,Tn/βMk/αg〉 =

{1 k = 00 k 6= 0

and that if n 6= 0

1αβ〈γ,Tn/βMk/αg〉 = 0

for all k .


(⇐=) By Janssen’s representation,

Sg,γ f = (αβ)−1∑k ,n

〈γ,Tk/βMn/αg〉Tk/βMn/αf

=∑k ,n

δ(k ,n),(0,0) Tk/βMn/αf = f .


Ron-Shen duality

Theorem

(Ron-Shen Duality Principle) Given g ∈ L2(R), α, β > 0,G(g, α, β) is a frame for L2(R) if and only if the systemG(g,1/β,1/α) is a Riesz basis for its closed linear span inL2(R).

The proof relies on the detailed structure of the frameoperator Sg,g .There are several important theorems that follow fromWexler-Raz and Ron-Shen.


Proof of density theorems

Theorem

Let g ∈ L2(R) and α, β > 0. If G(g, α, β) is a frame then αβ ≤ 1.If G(g, α, β) is a Riesz basis then αβ = 1.

Let G(g, α, β) be a frame and γ◦ is the cannonical dual ofg. By Wexler-Raz, 〈g, γ◦〉 = αβ.By definition of the cannonical dual,

g =∑k ,n

〈g,TαkMβnγ◦〉TαkMβng.


Since also g =∑k ,n

δ(k ,n),(0,0) TαkMβng

∑k ,n

|〈g,TαkMβnγ◦〉|2 ≤∑k ,n

|δ(k ,n),(0,0)|2 = 1.

Finally note that

αβ = 〈g, γ◦〉 =∑k ,n

〈g,TαkMβnγ◦〉〈TαkMβng, γ◦〉

and that by Cauchy-Schwarz,∣∣∣∣∑k ,n

〈g,TαkMβnγ◦〉〈TαkMβng, γ◦〉∣∣∣∣2 ≤∑

k ,n

|〈g,TαkMβnγ◦〉|2 ≤ 1.


Ron-Shen says that G(g,1/β,1/α) is a frame for L2(R) ifand only if G(g, α, β) is a Riesz basis for its closed linearspan.Since this is the case, G(g,1/β,1/α) is a frame for L2(R).

By the density theorem,1β

1α≤ 1, or αβ ≥ 1. Since also

αβ ≤ 1, αβ = 1.


lecture 4 structure theorems for gabor framesmbvajiac/conferences/chapman_lec04... · 2017. 11....

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