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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
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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
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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 .
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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.
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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).
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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Compactly supported windows
If α > 1/β, {e2πiβmt}m∈Z is incomplete on L2[0, α].If αβ >
1 then G(g, α, β) is incomplete.
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Compactly supported windows
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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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Compactly supported windows
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.
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Compactly supported windows
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.
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If in addition
0 < infx∈I|g(x)| ≤ sup
x∈I|g(x)|
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Compactly supported windows
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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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.
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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)|
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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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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‖∞.
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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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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).
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Frames at critical density
Theorem
Let g ∈ L2(R), α = β = 1.Z ◦ Sg,g f = |Zg|2 Zf .G(g, α, β) is a
frame if and only if 0 < a ≤ |Zg(x , ω)|2 ≤ b.In this case, the
frame is a Riesz basis.G(g, α, β) is an orthonormal basis if and
only if|Zg(x , ω)| = 1 on Q.
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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.
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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.
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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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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.
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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.
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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.
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Janssen considered g(x) = 1[0,c] and came up with “the tie.”
43Walnut (GMU) Lecture 4 – Structure theorems for Gabor
frames
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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.
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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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames
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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.
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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.
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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 .
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(⇐=) 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 .
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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.
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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.
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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.
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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.
Walnut (GMU) Lecture 4 – Structure theorems for Gabor frames