Structure of shift-invariant subspaces and their bases for the - - PowerPoint PPT Presentation

Structure of shift-invariant subspaces and their bases for the Heisenberg group

Azita Mayeli

City University of New York Queensborough College

Texas A&M University

July 17, 2012 Concentration week - Larsonfest 2012

This is a joint work with Brad Curry and Vignon Oussa of St. Louis University

Why do we study shift-invariant subspaces?

• Shift-invariant spaces are used as models for spaces of signals

and images in math and engineering applications.

• The scales of shift-invariant subspaces are "good"

approximation spaces of any function, in particular, potential functions, such as Sobolev functions.

• They are core spaces of MRAs.
• These spaces are used in sampling theory.

Historical comments

• In 1994, De Boor, De Vore, and Ron characterized finitely

generated shift invariant subspaces of L2(Rn) in terms of range functions introduced by Helson (1964).

• In 2000, Bownik extended the results to countably generated

shift invariant subspaces of L2(Rn), and characterized frame and Reisz families.

• Kamyabi, and et al. (2008) and Cabrelli and et al. (2009) studied

shift-invariant subspaces in the context of locally compact abelian groups.

• Aldroubi, Cabrelli, Heil, Hernandez, Kornelson, Molter, Speegle,

Sun, Weiss, Wilson, · · ·

The Heisenberg group The Heisenberg group N ≡ R2 × R. We let the group product (p, q, t)(p′, q′, t′) = (p + p′, q + q′, t + t′ + pq′), (p, q, t)−1 = (−p, −q, −t + pq) [(p, q, t), (p′, q′, t′)] = (0, 0, pq′ − qp′). We fix the standard Euclidean measure on R3.

The Heisenberg group The Heisenberg group N ≡ R2 × R. We let the group product (p, q, t)(p′, q′, t′) = (p + p′, q + q′, t + t′ + pq′), (p, q, t)−1 = (−p, −q, −t + pq) [(p, q, t), (p′, q′, t′)] = (0, 0, pq′ − qp′). We fix the standard Euclidean measure on R3. Schrödinger representation. For λ ∈ R/{0} πλ : N → U(L2(R)) πλ(p, q, t)f(x) = e2πiλte−2πiqλxf(x − p) = e2πiλtMqλlpf(x)

The Heisenberg group The Heisenberg group N ≡ R2 × R. We let the group product (p, q, t)(p′, q′, t′) = (p + p′, q + q′, t + t′ + pq′), (p, q, t)−1 = (−p, −q, −t + pq) [(p, q, t), (p′, q′, t′)] = (0, 0, pq′ − qp′). We fix the standard Euclidean measure on R3. Schrödinger representation. For λ ∈ R/{0} πλ : N → U(L2(R)) πλ(p, q, t)f(x) = e2πiλte−2πiqλxf(x − p) = e2πiλtMqλlpf(x) Dual space. N ≡ R/{0} and we simply show λ ∈ N.

The Heisenberg group Fourier transform. For f ∈ L1(N) ∩ L2(N) and λ ∈ N ˆ f(λ) =

• n∈N

f(n)πλ(n) dn.

The Heisenberg group Fourier transform. For f ∈ L1(N) ∩ L2(N) and λ ∈ N ˆ f(λ) =

• n∈N

f(n)πλ(n) dn. Definition: f, g, h ∈ L2(R) (f ⊗ g)(h) = f, hg

The Heisenberg group Fourier transform. For f ∈ L1(N) ∩ L2(N) and λ ∈ N ˆ f(λ) =

• n∈N

f(n)πλ(n) dn. Definition: f, g, h ∈ L2(R) (f ⊗ g)(h) = f, hg Plancherel transform. F : L2(N) − → ⊕

• N

L2(R) ⊗ L2(R) |λ|dλ f − → f = { f(λ)}

N

f =

• N

ˆ f(λ)2

HS |λ|dλ.

Notation: HS := HS(L2(R)) = L2(R) ⊗ L2(R).

The Heisenberg group

Lemma

L2(N) ∼ = ⊕

(0,1]

l2(Z, HS) dα.

The Heisenberg group

Lemma

L2(N) ∼ = ⊕

(0,1]

l2(Z, HS) dα. Techniques: Plancherel transform and periodization.

The Heisenberg group

Lemma

L2(N) ∼ = ⊕

(0,1]

l2(Z, HS) dα. Techniques: Plancherel transform and periodization. Sketch of proof. Given f ∈ L2(N). f2 = f2 =

• N
• f(λ)2

HS |λ| dλ =

1

• j∈Z

(α + j)1/2 f(α + j)2

HS dα

The Heisenberg group

Lemma

L2(N) ∼ = ⊕

(0,1]

l2(Z, HS) dα. Techniques: Plancherel transform and periodization. Sketch of proof. Given f ∈ L2(N). f2 = f2 =

• N
• f(λ)2

HS |λ| dλ =

1

• j∈Z

(α + j)1/2 f(α + j)2

HS dα

Let ˆ fα(j) := (α + j)1/2ˆ f(α + j).

The Heisenberg group

Lemma

L2(N) ∼ = ⊕

(0,1]

l2(Z, HS) dα. Techniques: Plancherel transform and periodization. Sketch of proof. Given f ∈ L2(N). f2 = f2 =

• N
• f(λ)2

HS |λ| dλ =

1

• j∈Z

(α + j)1/2 f(α + j)2

HS dα

Let ˆ fα(j) := (α + j)1/2ˆ f(α + j). Define T : f → Tf Tf : (0, 1] → l2(Z, HS); α → ˆ fα := {ˆ fα(j)}j∈Z.

The Heisenberg group

Lemma

L2(N) ∼ = ⊕

(0,1]

l2(Z, HS) dα. Techniques: Plancherel transform and periodization. Sketch of proof. Given f ∈ L2(N). f2 = f2 =

• N
• f(λ)2

HS |λ| dλ =

1

• j∈Z

(α + j)1/2 f(α + j)2

HS dα

Let ˆ fα(j) := (α + j)1/2ˆ f(α + j). Define T : f → Tf Tf : (0, 1] → l2(Z, HS); α → ˆ fα := {ˆ fα(j)}j∈Z. We show T is isometric isomorphism and f2 = Tf2 = 1 Tf(α)2

l2(Z,HS) dα.

Shift-invariant spaces Recall: N = R2 × R. Fix Γ = (aZ × bZ) × Z, a, b > 0.

Shift-invariant spaces Recall: N = R2 × R. Fix Γ = (aZ × bZ) × Z, a, b > 0. Shift-invariant space. We say a closed subspace V ≤ L2(N) is shift-invariant if lγ f = f(γ−1·) ∈ V ∀ f ∈ V, ∀ γ ∈ Γ

Shift-invariant spaces Recall: N = R2 × R. Fix Γ = (aZ × bZ) × Z, a, b > 0. Shift-invariant space. We say a closed subspace V ≤ L2(N) is shift-invariant if lγ f = f(γ−1·) ∈ V ∀ f ∈ V, ∀ γ ∈ Γ A version of range function. Given a Hilbert space H, a measurable map J J : α ∈ (0, 1] → the family of closed subspaces of H J : α → J(α) ≤ H

• Fiber. We say {J(α)}α is fiber of the range function J.

Shift-invariant spaces Recall: N = R2 × R. Fix Γ = (aZ × bZ) × Z, a, b > 0. Shift-invariant space. We say a closed subspace V ≤ L2(N) is shift-invariant if lγ f = f(γ−1·) ∈ V ∀ f ∈ V, ∀ γ ∈ Γ A version of range function. Given a Hilbert space H, a measurable map J J : α ∈ (0, 1] → the family of closed subspaces of H J : α → J(α) ≤ H

• Fiber. We say {J(α)}α is fiber of the range function J.

Lemma

The map is unitary.

• πλ : N → U(l2(Z, HS)),

( πλ(n)h) (j) = πλ+j(n) ◦ h(j)

Structure of shift-invariant spaces

Theorem

Given V ≤ L2(N) and Γ = (aZ × bZ) × Z, the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := JV such that J(α) ≤ l2(Z, HS), and

• πα(Γ1)(J(α)) ⊆ J(α),

a.e. α ∈ (0, 1] and V = T−1 ⊕

(0,1]

J(α)dα

• .

Structure of shift-invariant spaces

Theorem

Given V ≤ L2(N) and Γ = (aZ × bZ) × Z, the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := JV such that J(α) ≤ l2(Z, HS), and

• πα(Γ1)(J(α)) ⊆ J(α),

a.e. α ∈ (0, 1] and V = T−1 ⊕

(0,1]

J(α)dα

• .

Sketch of proof. “(ii) ⇒ (i)” We need to show that if φ ∈ V, then lγ1γ0φ ∈ V for any γ1γ0 ∈ Γ.

Structure of shift-invariant spaces

Theorem

Given V ≤ L2(N) and Γ = (aZ × bZ) × Z, the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := JV such that J(α) ≤ l2(Z, HS), and

• πα(Γ1)(J(α)) ⊆ J(α),

a.e. α ∈ (0, 1] and V = T−1 ⊕

(0,1]

J(α)dα

• .

Sketch of proof. “(ii) ⇒ (i)” We need to show that if φ ∈ V, then lγ1γ0φ ∈ V for any γ1γ0 ∈ Γ. This follows by T(lγ1γ0φ)(α) = e2πiαγ0 πα(γ1)(Tφ(α)).

Structure of shift-invariant spaces

Theorem

Given V ≤ L2(N) and Γ = (aZ × bZ) × Z, the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := JV such that J(α) ≤ l2(Z, HS), and

• πα(Γ1)(J(α)) ⊆ J(α),

a.e. α ∈ (0, 1] and V = T−1 ⊕

(0,1]

J(α)dα

• .

Sketch of proof. “(i) ⇒ (ii)” Recall that L2(N) ≡ 1

0 l2(Z, HS) dα.

Structure of shift-invariant spaces

Theorem

Given V ≤ L2(N) and Γ = (aZ × bZ) × Z, the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := JV such that J(α) ≤ l2(Z, HS), and

• πα(Γ1)(J(α)) ⊆ J(α),

a.e. α ∈ (0, 1] and V = T−1 ⊕

(0,1]

J(α)dα

• .

Sketch of proof. “(i) ⇒ (ii)” Recall that L2(N) ≡ 1

0 l2(Z, HS) dα.

Take {em}m ONB for l2(Z, HS).

Structure of shift-invariant spaces

Theorem

Given V ≤ L2(N) and Γ = (aZ × bZ) × Z, the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := JV such that J(α) ≤ l2(Z, HS), and

• πα(Γ1)(J(α)) ⊆ J(α),

a.e. α ∈ (0, 1] and V = T−1 ⊕

(0,1]

J(α)dα

• .

Sketch of proof. “(i) ⇒ (ii)” Recall that L2(N) ≡ 1

0 l2(Z, HS) dα.

Take {em}m ONB for l2(Z, HS). Define gm,p(α) := e2πiαpem, p ∈ Z.

Structure of shift-invariant spaces

Theorem

Given V ≤ L2(N) and Γ = (aZ × bZ) × Z, the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := JV such that J(α) ≤ l2(Z, HS), and

• πα(Γ1)(J(α)) ⊆ J(α),

a.e. α ∈ (0, 1] and V = T−1 ⊕

(0,1]

J(α)dα

• .

Sketch of proof. “(i) ⇒ (ii)” Recall that L2(N) ≡ 1

0 l2(Z, HS) dα.

Take {em}m ONB for l2(Z, HS). Define gm,p(α) := e2πiαpem, p ∈ Z. Then {gm,p}m,p is an ONB for 1

0 l2(Z, HS) dα.

Structure of shift-invariant spaces

Theorem

Given V ≤ L2(N) and Γ = (aZ × bZ) × Z, the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := JV such that J(α) ≤ l2(Z, HS), and

• πα(Γ1)(J(α)) ⊆ J(α),

a.e. α ∈ (0, 1] and V = T−1 ⊕

(0,1]

J(α)dα

• .

Sketch of proof. “(i) ⇒ (ii)” Recall that L2(N) ≡ 1

0 l2(Z, HS) dα.

Take {em}m ONB for l2(Z, HS). Define gm,p(α) := e2πiαpem, p ∈ Z. Then {gm,p}m,p is an ONB for 1

0 l2(Z, HS) dα.

Let P := PS be the orthogonal projector of 1

0 l2(Z, HS) dα onto T(V).

Structure of shift-invariant spaces

Theorem

Given V ≤ L2(N) and Γ = (aZ × bZ) × Z, the following are equivalent: (i) V is shift-invariant. (ii) There is a unique range function J := JV such that J(α) ≤ l2(Z, HS), and

• πα(Γ1)(J(α)) ⊆ J(α),

a.e. α ∈ (0, 1] and V = T−1 ⊕

(0,1]

J(α)dα

• .

Sketch of proof. “(i) ⇒ (ii)” Recall that L2(N) ≡ 1

0 l2(Z, HS) dα.

Take {em}m ONB for l2(Z, HS). Define gm,p(α) := e2πiαpem, p ∈ Z. Then {gm,p}m,p is an ONB for 1

0 l2(Z, HS) dα.

Let P := PS be the orthogonal projector of 1

0 l2(Z, HS) dα onto T(V).

We prove that (ii) holds for J(α) := span{P(gm,p)(α)}.

Frames/Bessel/Riesz sequences

• Definition. A countable family {fn} ⊂ V is called a frame if there exist

positive constants A and B such that Af2 ≤

• n

|f, fn|2 ≤ Bf2 ∀f ∈ V. Frame is tight or Parseval if A = B. – {fn} is called a Bessel sequence for V if the upper bound condition holds.

• Definition. The family {fn} is a Riesz sequence if there are two

constants 0 < A ≤ B such that for any finite sequence {cn} ∈ l2 A

• |cn|2 ≤
• cnfn2 ≤ B
• |cn|2.

– A Riesz sequence is complete in its spanned space.

Shift frames/Bessel/Riesz sequences Let A ⊂ L2(N) be countable. Let Γ = (aZ × bZ) × Z. Let V := span E(A) where E(A) := {lγφ : φ ∈ A, γ ∈ Γ}. ab ∈ Z, then V is shift-invariant.

Shift frames/Bessel/Riesz sequences Let A ⊂ L2(N) be countable. Let Γ = (aZ × bZ) × Z. Let V := span E(A) where E(A) := {lγφ : φ ∈ A, γ ∈ Γ}. ab ∈ Z, then V is shift-invariant.

Theorem

The following are equivalent. (i) The system E(A) is a frame/Bessel/Riesz basis with constants A and B (ii) {Mαqlp(Tφ)(α) : (p, q) ∈ aZ × bZ, φ ∈ A} ⊂ J(α) is a frame/ Bessel/Riesz basis for almost α. (Here, Mαq and lp are the standard modulation and translation

• perators, respectively.)

Shift frames/Bessel/Riesz sequences Let A ⊂ L2(N) be countable. Let Γ = (aZ × bZ) × Z. Let V := span E(A) where E(A) := {lγφ : φ ∈ A, γ ∈ Γ}. ab ∈ Z, then V is shift-invariant.

Theorem

The following are equivalent. (i) The system E(A) is a frame/Bessel/Riesz basis with constants A and B (ii) {Mαqlp(Tφ)(α) : (p, q) ∈ aZ × bZ, φ ∈ A} ⊂ J(α) is a frame/ Bessel/Riesz basis for almost α. (Here, Mαq and lp are the standard modulation and translation

• perators, respectively.)

Note that Mαqlp(Tφ)(α) = πα(p, q)T(φ)(α) = T(l(p,q,0)φ)(α)

Shift frames/Bessel/Riesz sequences

Corollary

Let A = {φ}. Then the following are equivalent. (i) The system E(A) is an ONB. (ii) {T(lkφ)(α) : k ∈ αZ × βZ} is orthogonal and Tφ(α) = 1.

Shift frames/Bessel/Riesz sequences

Corollary

Let A = {φ}. Then the following are equivalent. (i) The system E(A) is an ONB. (ii) {T(lkφ)(α) : k ∈ αZ × βZ} is orthogonal and Tφ(α) = 1.

Thanks for your attention.