LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two...

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LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES Christopher Heil Georgia Tech [email protected] http://www.math.gatech.edu/heil

Transcript of LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two...

Page 1: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES

Christopher Heil

Georgia Tech

[email protected]

http://www.math.gatech.edu/∼heil

Page 2: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

NOTATION

Time-frequency shift: gλ(x) = ga,b(x) = e2πibx g(x − a), λ = (a, b) ∈ R2

Gabor system: G(g, Λ) = {gλ}λ∈Λ

Page 3: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

STATEMENT OF THE CONJECTURE

Time-frequency shift: gλ(x) = ga,b(x) = e2πibx g(x − a), λ = (a, b) ∈ R2

Gabor system: G(g, Λ) = {gλ}λ∈Λ

HRT Conjecture (1996)

If:

(a) 0 <

∫ ∞

−∞|g(x)|2 dx < ∞,

(b) Λ = {(ak, bk)}Nk=1 is a finite set of distinct points in R2,

then G(g, Λ) is linearly independent. That is,

N∑

k=1

ck gak,bk = 0 a.e. ⇐⇒ c1 = · · · = cN = 0.

Page 4: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

(PUBLISHED?) REFERENCES

1. H/R/T, PAMS, 1996.

2. Linnell, Von Neumann algebras . . . , PAMS, 1999.

3. Rzeszotnik, Four points, 2004.

4. Balan, A noncommutative Wiener Lemma . . . , TAMS, 2008.

5. Balan/Krishtal, almost periodic . . . Wiener’s lemma, JMAA, 2010.

6. Bownik/Speegle, . . . Parseval wavelets, Ill. J. Math., 2010.

7. Demeter, . . . special configurations, MRL, 2010.

8. Demeter/Gautam, . . . lattice Gabor systems, PAMS, 2012.

9. Demeter/Zaharescu, . . . (2, 2) configurations, JMAA, 2012.

10. Bownik/Speegle, . . . faster than exponential decay, Bull. LMS, 2013.

11. Benedetto/Bourouihiya, . . . behaviors at infinity, J. Geom. Anal., 2014

12. Grochenig, . . . independence of time-frequency shifts?, preprint, 2014.

Talk by Eric Weber: “My failed attempts at the HRT Conjecture.”

Page 5: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

FALSE FOR THE AFFINE GROUP

Time-scale shifts of any compactly supported scaling function are

dependent.

0.5 1 1.5 2 2.5 3

-0.25

0.25

0.5

0.75

1

1.25

D4(x) = 1+√

34

D4(2x) + 3+√

34

D4(2x − 1) + 3−√

34

D4(2x − 2) + 1−√

34

D4(2x − 3).

0.5 1.0 1.5 2.0 2.5 3.0

-1.0

-0.5

0.5

1.0

1.5

2.0

Page 6: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

The Devil’s Staircase (Cantor–Lebesgue Function)

1 2

1

ϕ(x) = 12ϕ(3x) + 1

2ϕ(3x − 1) + ϕ(3x − 2) + 1

2ϕ(3x − 3) + 1

2ϕ(3x − 4).

Page 7: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

ZERO DIVISOR CONJECTURE

The group algebra of a group G is

CG =

{

g∈G

cgg : cg ∈ C with only finitely many cg 6= 0

}

.

Finite order elements yield zero divisors: If gn = e, then

(g − e)(gn−1 + · · · + g + e) = 0.

Page 8: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

ZERO DIVISOR CONJECTURE

The group algebra of a group G is

CG =

{

g∈G

cgg : cg ∈ C with only finitely many cg 6= 0

}

.

Finite order elements yield zero divisors: If gn = e, then

(g − e)(gn−1 + · · · + g + e) = 0.

Zero Divisor Conjecture (c. 1940, still open)

If G is a torsion-free group and α, β ∈ CG, then

α 6= 0 and β 6= 0 =⇒ αβ 6= 0.

Attributed to Higman, Kaplansky, . . .

Higman (1940): True if G is a locally indicable group

Page 9: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

HEISENBERG GROUP

H ={

zMbTa : z ∈ T, a, b ∈ R}

,

CH =

{ N∑

k=1

ckMbkTak: N > 0, ck ∈ C, ak ∈ R, bk ∈ R

}

.

Theorem (Speegle)

CH has no zero divisors.

Proof idea: If

α =

M∑

j=1

zjMbjTaj, β =

N∑

k=1

wkMdkTck,

then

αβ =

M∑

j=1

N∑

k=1

tjkMbj+dkTaj+ck.

Order the translations and then the modulations, obtain a “highest-

order term” . . .

Page 10: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

A similar proof works for the affine group. This is essentially the

“locally indicable group” argument (assume every finitely generated

subgroup can be mapped homomorphically onto Z)

Corollary: The set of time-frequency shift operators

{MbTa : a, b ∈ R}

is a finitely linearly independent set of operators:

N∑

k=1

ckMbkTak= 0 =⇒ c1 = · · · = cN = 0.

Similarly, the set of time-scale shifts is finitely linearly independent.

But HRT asks: Is there a g 6= 0 such that

N∑

k=1

ckMbkTakg = 0?

Page 11: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

STATEMENT OF THE CONJECTURE

Time-frequency shift: gλ(x) = ga,b(x) = e2πibx g(x − a), λ = (a, b) ∈ R2

Gabor system: G(g, Λ) = {gλ}λ∈Λ

HRT Conjecture (1996)

If:

(a) 0 <

∫ ∞

−∞|g(x)|2 dx < ∞,

(b) Λ = {(ak, bk)}Nk=1 is a finite set of distinct points in R2,

then G(g, Λ) is linearly independent. That is,

N∑

k=1

ck gak,bk = 0 a.e. ⇐⇒ c1 = · · · = cN = 0.

Notation:

HRT(Λ) = HRT for that Λ (and all g)

HRT(g) = HRT for that g (and all Λ; implicitly g 6= 0)

Page 12: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

METAPLECTIC TRANSFORMS

If T is a 2 × 2 matrix with | det(T )| = 1, then there exists a unitary

metaplectic transform U = UT on L2(R) such that

G(Ug, TΛ) = G(g, Λ).

Corollary

If T is an area-preserving linear transformation on R2, then

HRT(Λ) is true ⇐⇒ HRT(TΛ) is true.

(Similar result if T is a translation operator.)

This result does not generalize to linear transformations in higher

dimensions!

Page 13: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

LINES

Λ ⊂ {0} × R is easy:

N∑

k=1

ck e2πibkxg(x) = 0 a.e. ⇐⇒ c1 = · · · = cN = 0.

(Exponentials are independent on any set of positive measure.)

Corollary

HRT(Λ) is true if Λ ⊂ line.

Proof. Rotation is an area-preserving linear transformation.

Page 14: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

LATTICES

A lattice in R2 is the integer span of two independent vectors.

Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ × bZ.

Proof. Every (full-rank) lattice in R2 is the image of aZ × bZ under

an area-preserving linear transformation.

Theorem

HRT is true for Λ ⊂ aZ × bZ.

ab = 1 is easy via Zak transform.

ab 6= 1 is not easy. Proofs by:

Linnell (von Neumann algebras)

Bownik/Speegle (shift-invariant spaces)

Demeter/Gautam (spectral theory/random Schrodinger operators)

These results do not generalize to higher dimensions!

Page 15: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

FOUR POINTS

Four points in R2 need not lie on a translate of a lattice.

Theorem (Rzeszotnik)

HRT(Λ) is true for Λ ={

(0, 0), (1, 0), (0, 1), (√

2, 0)}

.

HRT Subconjecture (currently open)

HRT(Λ) is true for Λ ={

(0, 0), (1, 0), (0, 1), (√

2,√

2)}

.

(Open even if we assume g is continuous—or Schwartz!)

Some special four point cases

Demeter/Zahrescu: HRT true if Λ = vertices of a trapezoid.

Demeter: HRT true if Λ ⊂ two parallel lines.

Benedetto/Bourouihiya: Conditions on g

Page 16: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

CONDITIONS ON g

HRT(g) is true if:

Theorem (H/R/T, 1996)

(a) supp(g) is contained in [a,∞) or (−∞, a], or

(b) g(x) = p(x)e−x2

, where p is a polynomial.

Theorem (Bownik/Speegle, 2013)

(a) limx→∞

|g(x)| ecx2

= 0 for all c > 0, or

(b) limx→∞

|g(x)| ecx log x = 0 for all c > 0.

Still open: limx→∞

|g(x)| ecx = 0 for all c > 0.

Theorem (Benedetto/Bourouihiya)

(a) g is ultimately positive and b1, . . . , bN are independent over Q, or

(b) |Λ| = 4, g is ultimately positive, g(x) and g(−x) are ultimately

decreasing.

Page 17: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

PERTURBATIONS

Theorem (H/R/T)

(a) If HRT(g) is true, then there exists an ε = εg > 0 such that

‖h − g‖2 < ε =⇒ HRT(h) is true.

(b) If HRT(Λ) is true, then there exists an ε = εΛ > 0 such that

‖Λ′ − Λ‖∞ < ε =⇒ HRT(Λ′) is true.

Corollary

HRT(g) is true for an open, dense set of g ∈ L2(R).

Proof. HRT(g) is true for all compactly supported g;

perturb g by εg.

Page 18: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

Theorem

If HRT(g) is true, then there exists an εg > 0 such that

‖h − g‖2 < εg =⇒ HRT(h) is true.

Key ingredient: If HRT(g) is true, then G(g, Λ) is a Riesz basis for its

span. The value of εg is determined by the lower frame bound.

Fundamental Problem

Determine the lower frame bound of G(g, Λ) in terms of properties of

g or Λ.

(Some results by Christensen et al. exist, but they are not strong

enough.)

Page 19: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

SPECTRUM

HRT(Λ) is true if and only if the point spectrum of

T =N

k=1

ckMbkTak

is empty (for c1, . . . , cN not all zero; note I = M0T0).

Theorem (Balan, 2008)

T cannot have an isolated eigenvalue of finite multiplicity.

Therefore, if the point spectrum of T is not empty, then it can only

contain eigenvalues of infinite multiplicity, or eigenvalues that also

belong to the continuous spectrum of T.

Page 20: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

ASYMPTOTICS

Modulation space condition:∫∫

|〈g, MbTag〉| v(a, b) da db < ∞.

The weight v is submultiplicative and satisfies GRS.

Theorem (Grochenig, 2014)

If G(g, Λ) is a frame but not a Riesz basis for L2(R), then the lower

Riesz bound An of G(g, Λn) decays like

An ≤ C sup|λ|>n

v(λ)−2,

where Λn = Λ ∩ Bn(0).

Remark: An = 0 ⇐⇒ G(g, Λn) is dependent.

Page 21: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

Polynomial weight v(a, b) = (1 + |a| + |b|)s:

An = O(n−2s).

Subexponential weight v(z) = ec(|a|+|b|)s:

An = O(e−cns

).

“From a numerical point of view even small sets of time-frequency

shifts may look linearly dependent. [This] illustrates the spectacular

difference between a conjectured mathematical truth and a compu-

tationally observable truth.”

Page 22: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

JUST THREE POINTS

We sketch a proof that

{

g(x), g(x − a), g(x)e2πix}

is linearly independent.

Suppose that

c1 g(x) + c2 g(x − a) + c3 g(x)e2πix = 0, all x ∈ R.

Rewriting,

g(x − a) = m(x) g(x),

where m(x) = −c1

c2− c3

c2e2πix is 1-periodic.

Page 23: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

g(x − a) = m(x) g(x) for all x.

Iterate:

g(x − 2a) = g((x − a) − a)

= m(x − a) g(x − a)

= m(x − a) m(x) g(x).

g(x − ka) =

k−1∏

j=0

m(x − ja)

g(x).

Page 24: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

|g(x − ka)| =

k−1∏

j=0

|m(x − ja)|

|g(x)|.

ln |g(x − ka)| =k−1∑

j=0

ln |m(x − ja)| + ln |g(x)|.

ln |g(x − ka)| =k−1∑

j=0

p(x − ja) + ln |g(x)|

where p is 1-periodic.

Page 25: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

1

kln |g(x − ka)| =

1

k

k−1∑

j=0

p(x − ja) +1

kln |g(x)|.

The value1

kp(x− ja) is the area of the box with base centered at (the

fractional part of) x − ja and height p(x − ja) and width 1/k:

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Since p is 1-periodic, these boxes are scattered around the interval

[0, 1].

Page 26: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

1

k

k−1∑

j=0

p(x − ja)

The sum is the sum of the areas in the boxes:

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

Page 27: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

1

k

k−1∑

j=0

p(x − ja)

The sum is the sum of the areas in the boxes:

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

It’s like a Riemann sum approximation to

∫ 1

0

p(x) dx, except the boxes

are “randomly distributed” — at least, if a is irrational.

Ergodic Theorem:1

k

k−1∑

j=0

p(x − ja) →∫ 1

0

p(x) dx as k → ∞.

Page 28: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

1

k

k−1∑

j=0

p(x − ja) ≈∫ 1

0

p(x) dx = C

k−1∑

j=0

ln |m(x − ja)| ≈ Ck

k−1∏

j=0

|m(x − ja)| ≈ eCk

|g(x − ka)| =

(k−1∏

j=0

|m(x − ja)|)

|g(x)| ≈ eCk |g(x)|

If C > 0 then g is growing as x → −∞, contradicting

∫ ∞

−∞|g(x)|2 dx < ∞.

Symmetric argument if C < 0. C = 0 takes more work.

Page 29: LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES · A lattice in R2 is the integer span of two independent vectors. Lemma: HRT for Λ ⊂ lattice ⇐⇒ HRT for Λ ⊂ aZ×bZ. Proof.

CHALLENGE

HRT Subconjecture (currently open . . . but . . . ?)

HRT(Λ) is true for Λ ={

(0, 0), (1, 0), (0, 1), (√

2,√

2)}

,

even if g is Schwartz.

A survey article is available

http://www.math.gatech.edu/∼heil/papers/hrtnotes.pdf

A newer survey by H. and Speegle will be available shortly.

THANK YOU.