Fourier operators in applied harmonic analysis John J. Benedetto - - PowerPoint PPT Presentation

Fourier operators in applied harmonic analysis

John J. Benedetto

Norbert Wiener Center Department of Mathematics University of Maryland, College Park http://www.norbertwiener.umd.edu

Outline

1

Waveform design and optimal ambiguity function behavior on Z/NZ

2

MIMO and a vector-valued DFT on Z/NZ

3

Finite Gabor sums on R

4

Balayage on LCAGs, and Fourier frames and non-uniform sampling on Rd

5

STFT frame inequalities on Rd

6

ΦDO frame inequalities on Rd

Outline

1

Waveform design and optimal ambiguity function behavior on Z/NZ

2

MIMO and a vector-valued DFT on Z/NZ

3

Finite Gabor sums on R

4

Balayage on LCAGs, and Fourier frames and non-uniform sampling on Rd

5

STFT frame inequalities on Rd

6

ΦDO frame inequalities on Rd

Frames

Let H be a separable Hilbert space, e.g., H = L2(Rd), Rd, or Cd. F = {xn} ⊆ H is a frame for H if ∃ A, B > 0 such that ∀ x ∈ H, Ax2 ≤

• |x, xn|2 ≤ Bx2.

Theorem If F = {xn} ⊆ H is a frame for H then ∀x ∈ H, x =

• x, S−1xnxn =
• x, xnS−1xn,

where S : H → H, x → x, xnxn is well-defined. Frames are a natural tool for dealing with numerical stability,

• vercompleteness, noise reduction, and robust representation

problems.

Fourier frames, goal, and a litany of names

Definition E = {xn} ⊆ Rd, Λ ⊆

• Rd. E is a Fourier frame for L2(Λ) if

∃A, B > 0, ∀F ∈ L2(Λ), A ||F||2

L2(Λ) ≤

• n

| < F(γ), e−2πixn·γ > |2 ≤ B ||F||2

L2(Λ).

Goal Formulate a general theory of Fourier frames and non-uniform sampling formulas parametrized by the space M(Rd) of bounded Radon measures. Motivation Beurling theory (1959-1960). Names Riemann-Weber, Dini, G.D. Birkhoff, Paley-Wiener, Levinson, Duffin-Schaeffer, Beurling-Malliavin, Beurling, H.J. Landau, Jaffard, Seip, Ortega-Cert` a–Seip.

Balayage and the theory of generalized Fourier frames

Balayage

Let M(G) be the algebra of bounded Radon measures on the LCAG G. Balayage in potential theory was introduced by Christoffel (early 1870s) and Poincar´ e (1890). Definition (Beurling) Balayage is possible for (E, Λ) ⊆ G × G, a LCAG pair, if ∀µ ∈ M(G), ∃ν ∈ M(E) such that ˆ µ = ˆ ν on Λ. We write balayage (E, Λ). The set, Λ, of group characters is the analogue of the original role of Λ in balayage as a collection of potential theoretic kernels. Kahane formulated balayage for the harmonic analysis of restriction algebras.

Balayage and the theory of generalized Fourier frames

Spectral synthesis

Definition (Wiener, Beurling) Closed Λ ⊆ G is a set of spectral synthesis (S-set) if ∀µ ∈ M(G), ∀f ∈ Cb(G), supp( f) ⊆ Λ and ˆ µ = 0 on Λ = ⇒

• G f dµ = 0.

(∀T ∈ A′( G), ∀φ ∈ A( G), supp(T) ⊆ Λ and φ = 0 on Λ ⇒ T(φ) = 0.) Ideal structure of L1(G) - the Nullstellensatz of harmonic analysis T ∈ D′( Rd), φ ∈ C∞

c (

Rd), and φ = 0 on supp(T) ⇒ T(φ) = 0, with same result for M( Rd) and C0( Rd). S2 ⊆ R3 is not an S-set (L. Schwartz), and every non-discrete G has non-S-sets (Malliavin). Polyhedra are S-sets. The 1

3-Cantor set is an S-set with

non-S-subsets.

Balayage and the theory of generalized Fourier frames

Strict multiplicity

Definition Γ ⊆ G is a set of strict multiplicity if ∃ µ ∈ M(Γ)\{0} such that ˇ µ vanishes at infinity in G. Riemann and sets of uniqueness in the wide sense. Menchov (1916): ∃ closed Γ ⊆ R/Z and µ ∈ M(Γ)\{0}, |Γ| = 0 and ˇ µ(n) = O((log |n|)−1/2), |n| → ∞. 20th century history to study rate of decrease: Bary (1927), Littlewood (1936), Salem (1942, 1950), Ivaˇ sev-Mucatov (1957), Beurling. Assumption ∀ γ ∈ Λ and ∀ N(γ), compact neighborhood, Λ ∩ N(γ) is a set of strict multiplicity.

Balayage and the theory of generalized Fourier frames

A theorem of Beurling

Definition E = {xn} ⊆ Rd is separated if ∃ r > 0, ∀m, n, m = n ⇒ ||xm − xn|| ≥ r. Theorem Let Λ ⊆ Rd be a compact S-set, symmetric about 0 ∈ Rd, and let E ⊆ Rd be separated. If balayage (E,Λ), then E is a Fourier frame for L2(Λ). Equivalent formulation in terms of PWΛ = {f ∈ L2(Rd) : supp( ˆ f) ⊆ Λ}. ∀F ∈ L2(Λ), F =

x∈E < F, S−1(ex) >Λ ex in L2(Λ).

For Rd and other generality beyond Beurling’s theorem in R, the result above was formulated by Hui-Chuan Wu and JB (1998), see Landau (1967).

Balayage and the theory of generalized Fourier frames

Outline

1

Waveform design and optimal ambiguity function behavior on Z/NZ

2

MIMO and a vector-valued DFT on Z/NZ

3

Finite Gabor sums on R

4

Balayage on LCAGs, and Fourier frames and non-uniform sampling on Rd

5

STFT frame inequalities on Rd

6

ΦDO frame inequalities on Rd

Short time Fourier transform (STFT)

STFT Vgf(x, ω) =

• f(t)g(t − x)e−2πit·ω dt.

g2 = 1. Vector-valued inversion, f = Vgf(x, ω)eωτxg dωdx′ Vgf(x, ω) = e−2πix·ωVGF(ω, −x), f = F and g = G.

• Vgf
• L2(R2d) = g2 f2.

Quantum mechanics and Moyal’s formula (1949) for the cross-Wigner distribution W(f, g)(x, ω). STFT as (X, µ)-frames. See Ali, Antoine, and Gazeau (1993 and 2000) , Gabardo and Han (2003), and Fornasier and Rauhut (2005).

The Feichtinger algebra

Let g0(x) = 2d/4e−πx2. Then G0(γ) = g0(γ) = 2d/4e−πγ2 and g02 = 1. The Feichtinger algebra, S0(Rd), is S0(Rd) = {f ∈ L2(Rd): fS0 = Vg0f1 < ∞}. The Fourier transform of S0(Rd) is an isometric isomorphism

• nto itself, and, in particular, f ∈ S0(Rd) if and only if F ∈ S0(

Rd). Feichtinger On a new Segal algebra 1981; Wiener amalgam spaces; modulation spaces; Feichtinger and Gr¨

• chenig, W. Sun,

Walnut, Zimmermann; Gr¨

• chenig Foundations of

Time-Frequency Analysis 2001.

Norbert Wiener Center Balayage and short time Fourier transform frames

Gr¨

• chenig’s non-uniform Gabor frame theorem

Theorem Given any g ∈ S0(Rd). There is r = r(g) > 0 such that if E = {(sn, σn)} ⊆ Rd × Rd is a separated sequence satisfying

∞

• n=1

B((sn, σn), r(g)) = Rd × Rd, then the frame operator, S = Sg,E, defined by Sg,E f = ∞

n=1f, τsneσng τsneσng,

is invertible on S0(Rd). Further, if f ∈ S0(Rd), then f = ∞

n=1f, τsneσngS−1 g,E(τsneσng),

where the series converges unconditionally in S0(Rd). E depends on g.

Norbert Wiener Center Balayage and short time Fourier transform frames

Balayage and a non-uniform Gabor frame theorem

Theorem Let E = {(sn, σn)} ⊆ Rd × Rd be a separated sequence; and let Λ ⊆ Rd × Rd be an S-set of strict multiplicity that is compact, convex, and symmetric about 0 ∈ Rd × Rd. Assume balayage is possible for (E, Λ). Given g ∈ L2(Rd), such that g2 = 1. Then ∃ A, B > 0, such that ∀f ∈ S0(Rd), for which supp( Vgf) ⊆ Λ, A f2

2 ≤

∞

n=1|Vgf(sn, σn)|2 ≤ B f2 2 .

Norbert Wiener Center Balayage and short time Fourier transform frames

Balayage and a non-uniform Gabor frame theorem (continued)

Theorem Consequently, the frame operator, S = Sg,E, is invertible in L2(Rd)–norm on the subspace of S0(Rd), whose elements f have the property, supp ( Vgf) ⊆ Λ. Further, if f ∈ S0(Rd) and supp( Vgf) ⊆ Λ, then f = ∞

n=1f, τsneσngS−1 g,E(τsneσng),

where the series converges unconditionally in L2(Rd). E does not depend on g.

Norbert Wiener Center Balayage and short time Fourier transform frames

The support hypothesis

To show supp( Vgf) ⊆ Λ. We compute that if f, g ∈ L1(Rd) ∩ L2(Rd), then

• Vgf(ζ, z) = e−2πiz·ζ f(−z) ˆ

g(−ζ). Let d = 1, Λ = [−Ω, Ω] × [−T, T] ⊆ R × R, g ∈ PW[−Ω,Ω], g ∈ L1(R), ˆ g(t) = ˆ g(−t). For this window g, we take any even f ∈ L2(R) that is supported in [−T, T].

Norbert Wiener Center Balayage and short time Fourier transform frames

Outline

1

Waveform design and optimal ambiguity function behavior on Z/NZ

2

MIMO and a vector-valued DFT on Z/NZ

3

Finite Gabor sums on R

4

Balayage on LCAGs, and Fourier frames and non-uniform sampling on Rd

5

STFT frame inequalities on Rd

6

ΦDO frame inequalities on Rd

DFT Frames

Definition Let N ≥ d and let s : Z/dZ → Z/NZ be injective. The rows {Em}N−1

m=0

• f the N × d matrix
• e2πims(n)/N

m,n

form an equal-norm tight frame for Cd which we call a DFT frame.

DFT FUNTFs

Let N ≥ d and form an N × d matrix using any d columns of the N × N DFT matrix (e2πijk/N)N−1

j,k=0. The rows of this N × d matrix, up to

multiplication by 1 √ d , form a FUNTF for Cd.

John J. Benedetto and Jeffrey J. Donatelli Frames and a vector-valued ambiguity function

Vector-valued DFT

Definition Let {Ek}N−1

k=0 be a DFT frame for Cd. Given u : Z/NZ → Cd, we define

the vector-valued discrete Fourier transform of u by ∀ n ∈ ZN, F(u)(n) = u(n) =

N−1

• m=0

u(m) ∗ E−mn, where ∗ is pointwise (coordinatewise) multiplication. We have that F : ℓ2(Z/NZ × Z/dZ) → ℓ2(Z/NZ × Z/dZ) is a linear operator.

Vector-valued Fourier inversion

Theorem (Andrews, Benedetto, Donatelli) The vector valued Fourier transform is invertible if and only if s, the function defining the DFT frame, has the property that ∀n ∈ Z/dZ, (s(n), N) = 1. The inverse is given by ∀ m ∈ Z/NZ, u(m) = F −1 u(m) = 1 N

N−1

• n=0
• u(n) ∗ Emn.

In this case we also have that F ∗F = FF ∗ = NI where I is the identity

• perator.

STFT and ambiguity function

Short time Fourier transform – STFT The narrow band cross-correlation ambiguity function of v, w defined on R is A(v, w)(t, γ) =

• R

v(s + t)w(s)e−2πisγds. A(v, w) is the STFT of v with window w. The narrow band radar ambiguity function A(v) of v on R is A(v)(t, γ) =

• R

v(s + t)v(s)e−2πisγds = eπitγ

• R

v

• s + t

2

• v
• s − t

2

• e−2πisγds, for (t, γ) ∈ R2.

John J. Benedetto and Jeffrey J. Donatelli Frames and a vector-valued ambiguity function

Goal

Let v be a phase coded waveform with N lags defined by the code u. Let u be N-periodic, and so u : ZN − → C, where ZN is the additive group of integers modulo N. The discrete periodic ambiguity function Ap(u) : ZN × ZN − → C is Ap(u)(m, n) = 1 N

N−1

• k=0

u(m + k)u(k)e−2πikn/N. Goal Given a vector valued N-periodic code u : ZN − → Cd, construct the following in a meaningful, computable way: Generalized C-valued periodic ambiguity function A1

p(u) : ZN × ZN −

→ C Cd-valued periodic ambiguity function Ad

p(u) : ZN × ZN −

→ Cd The STFT is the guide and the theory of frames is the technology to

• btain the goal.

John J. Benedetto and Jeffrey J. Donatelli Frames and a vector-valued ambiguity function

Multiplication problem

Given u : ZN − → Cd. If d = 1 and en = e2πin/N, then Ap(u)(m, n) = 1 N

N−1

• k=0

u(m + k), u(k)enk. Multiplication problem To characterize sequences {Ek} ⊆ Cd and multiplications ∗ so that A1

p(u)(m, n) = 1

N

N−1

• k=0

u(m + k), u(k) ∗ Enk ∈ C is a meaningful and well-defined ambiguity function. This formula is clearly motivated by the STFT.

John J. Benedetto and Jeffrey J. Donatelli Frames and a vector-valued ambiguity function

A1

p(u) for cross product frames

Take ∗ : C3 × C3 − → C3 to be the cross product on C3 and let {i, j, k} be the standard basis. i ∗ j = k, j ∗ i = −k, k ∗ i = j, i ∗ k = −j, j ∗ k = i, k ∗ j = −i, i ∗ i = j ∗ j = k ∗ k = 0. {0, i, j, k, −i, −j, −k, } is a tight frame for C3 with frame constant 2. Let E0 = 0, E1 = i, E2 = j, E3 = k, E4 = −i, E5 = −j, E6 = −k. The index operation corresponding to the frame multiplication is the non-abelian operation • : Z7 × Z7 − → Z7, where 1 • 2 = 3, 2 • 1 = 6, 3 • 1 = 2, 1 • 3 = 5, 2 • 3 = 1, 3 • 2 = 4, 1 • 1 = 2 • 2 = 3 • 3 = 0, n • 0 = 0 • n = 0, 1 • 4 = 0, 1 • 5 = 6, 1 • 6 = 2, 4 • 1 = 0, 5 • 1 = 3, 6 • 1 = 5, 2 • 4 = 3, 2 • 5 = 0, etc. The three ambiguity function assumptions are valid and so we can write the cross product as u × v = u ∗ v = 1 22

6

• s=1

6

• t=1

u, Esv, EtEs•t. Consequently, A1

p(u) can be well-defined.

John J. Benedetto and Jeffrey J. Donatelli Frames and a vector-valued ambiguity function

Inverse 4D Quaternion Julia Set

Epilogue

If (G, •) is a finite group with representation ρ : G − → GL(Cd), then there is a frame {En}n∈G and bilinear multiplication, ∗ : Cd × Cd − → Cd, such that Em ∗ En = Em•n. Thus, we can develop Ad

p(u) theory in this setting.

Analyze ambiguity function behavior for (phase-coded) vector-valued waveforms v : R − → Cd, defined by u : ZN − → Cd as v =

N−1

• k=0

u(k)✶[kT,(k+1)T), in terms of Ad

p(u). (See Figure)

John J. Benedetto and Jeffrey J. Donatelli Frames and a vector-valued ambiguity function

Outline

1

Waveform design and optimal ambiguity function behavior on Z/NZ

2

MIMO and a vector-valued DFT on Z/NZ

3

Finite Gabor sums on R

4

Balayage on LCAGs, and Fourier frames and non-uniform sampling on Rd

5

STFT frame inequalities on Rd

6

ΦDO frame inequalities on Rd

The HRT Conjecture

g ∈ L2(R), Λ = {(αk, βk)}N

k=1 ⊆ R2 distinct,

G(g, Λ) = {e2πiβkxg(x − αk) : k = 1, 2, . . . , N}. The HRT conjecture: G(g, Λ) is linearly independent in L2(R). Heil, Ramanathan, Topiwala (1996) Linnell (1999), Kutyniok (2002), Rzeszotnik (2005), Balan (2008), Demeter (2010), Demeter and Zaharescu (2010), Bownik and Speegle (2012) B and B (2007)

Kronecker’s theorem (1884)

Theorem Let {β1, β2, . . . , βN} ⊆ R be a linearly independent set over Q, and let θ1, . . . , θN ∈ R. If U, ǫ > 0, then there exist p1, . . . , pN ∈ Z and u > U such that ∀k = 1, . . . , N, |βku − pk − θk| < ǫ, and ∀k = 1, . . . , N, |e2πiβku − e2πiθk | < 4πǫ. Bohr compactification proof (Spectral Synthesis, Section 3.2.12) Compact E ⊆ Γ, a LCAG, is a Kronecker set if ∀ǫ > 0 and ∀φ ∈ C(E), |φ| = 1 on E, ∃x ∈ G such that ∀γ ∈ E, |φ(γ) − (γ, x)| < ǫ. Varopoulos theorem: E Kronecker ⇒ M(E) = A′(E). Spectral synthesis problems.

The HRT conjecture for positive functions

Using Kronecker’s theorem, we have the following. Theorem g ∈ L2(R) ultimately positive. Λ = {(αk, βk)}N

k=0, {β0, . . . , βN} linearly

independent over Q. The HRT conjecture holds for G(g, Λ). Theorem g ∈ L2(R) such that g(x) and g(−x) are ultimately positive and ultimately strictly decreasing. Assume card(Λ) = 4. The HRT conjecture holds for G(g, Λ).

Outline

1

Waveform design and optimal ambiguity function behavior on Z/NZ

2

MIMO and a vector-valued DFT on Z/NZ

3

Finite Gabor sums on R

4

Balayage on LCAGs, and Fourier frames and non-uniform sampling on Rd

5

STFT frame inequalities on Rd

6

ΦDO frame inequalities on Rd

Norbert Wiener Center Department of Mathematics, University of Maryland, College Park

Discrete ambiguity functions

Let u : {0, 1, . . . , N − 1} → C. up : ZN → C is the N-periodic extension of u. ua : Z → C is an aperiodic extension of u: ua[m] = u[m], m = 0, 1, . . . , N − 1 0,

• therwise.

The discrete periodic ambiguity function Ap(u) : ZN × ZN → C of u is Ap(u)(m, n) = 1 N

N−1

• k=0

up[m + k]up[k]e2πikn/N. The discrete aperiodic ambiguity function Aa(u) : Z × Z → C of u is Aa(u)(m, n) = 1 N

N−1

• k=0

ua[m + k]ua[k]e2πikn/N.

Norbert Wiener Center Department of Mathematics, University of Maryland, College Park

CAZAC sequences

u : ZN → C is Constant Amplitude Zero Autocorrelation (CAZAC): ∀m ∈ ZN, |u[m]| = 1, (CA) and ∀m ∈ ZN \ {0}, Ap(u)(m, 0) = 0. (ZAC) Empirically, the (ZAC) property of CAZAC sequences u leads to phase coded waveforms w with low aperiodic autocorrelation A(w)(t, 0). Are there only finitely many non-equivalent CAZAC sequences?

”Yes” for N prime and ”No” for N = MK 2, Generally unknown for N square free and not prime.

Norbert Wiener Center Department of Mathematics, University of Maryland, College Park

Definition

Let N be a prime number. A Bj¨

• rck CAZAC sequence of length N is

u[k] = eiθN(k), k = 0, 1, . . . , N − 1, where, for N = 1 (mod 4), θN(k) = arccos

• 1

1 + √ N k N

• ,

and, for N = 3 (mod 4), θN(k) = 1 2 arccos 1 − N 1 + N

• [(1 − δk)

k N

• + δk].

δk is Kronecker delta and k

N

• is Legendre symbol.

Absolute value of Bjorck code of length 17

Absolute value of Bjorck code of length 53

Absolute value of Bjorck code of length 101

Absolute value of Bjorck code of length 503

Absolute value of Bjorck code of length 701

Bj¨

• rck CAZAC Discrete Narrow-band Ambiguity Function

Let up denote the Bj¨

• rck CAZAC sequence for prime p, and let Ap(up)

be the discrete narrow band ambiguity function defined on Z/pZ × Z/pZ. Theorem (J. and R. Benedetto and J. Woodworth) |Ap(up)(m, n)| ≤ 2 √p + 4 p for all (m, n) ∈ (Z/pZ × Z/pZ) \ (0, 0). The bound is more precise but not better than

2 √p depending on

whether p ≡ 1 (mod 4) or p ≡ 3 (mod 4). The proof is at the level of Weil’s proof of the Riemann hypothesis for finite fields and depends on Weil’s exponential sum bound. Elementary construction/coding and intricate combinatorial/geometrical patterns.

Mathematical problems, deterministic solutions, and sparsity applications

Estimate the minimal upper bound B(p) of the number of CAZACs of prime length p (number theory and analysis). Construct CAZACs of prime length p (algebraic geometry). For given CAZACs µp of prime length p, estimate minimal local behavior |A(µp)| (number theory and analysis).

T Q E T' Q' E' b b' Storage/ Transmission

Use these results in transform (T) based image compression via sparsity and GAs (e.g., OMP). Expand quantization (Q) technology with Σ∆ for noise reduction.

Theme, theory, and applications

Theme: Frame representations and sparse, non-uniform, non-linear sampling for numerical stability and noise reduction Theory: Number theoretic waveform design and finite Gabor frames, vector-valued harmonic analysis, the HRT conjecture, balayage, and STFT and ΦDO frame inequalities Applications at the NWC: Non-linear sampling and quantization, e.g.,Σ∆, frame potential energy optimization for classification, non-linear kernels and dimension reduction, Schr¨

• dinger operators with barrier potentials for classification

(Czaja and Ehler), HSI, LIDAR, data fusion