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)(p0 , q0 , t0 ) = (p + p0 , q + q0 , t + t0 + pq0 ),
(p, q, t)−1 = (−p, −q, −t + pq)
[(p, q, t), (p0 , q0 , t0 )] = (0, 0, pq0 − qp0 ).
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)(p0 , q0 , t0 ) = (p + p0 , q + q0 , t + t0 + pq0 ),
(p, q, t)−1 = (−p, −q, −t + pq)
[(p, q, t), (p0 , q0 , t0 )] = (0, 0, pq0 − qp0 ).
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) = e2πiλt e−2πiqλx f (x − p)
= e2πiλt Mqλ lp f (x)
The Heisenberg group
The Heisenberg group N ≡ R2 × R. We let the group product
(p, q, t)(p0 , q0 , t0 ) = (p + p0 , q + q0 , t + t0 + pq0 ),
(p, q, t)−1 = (−p, −q, −t + pq)
[(p, q, t), (p0 , q0 , t0 )] = (0, 0, pq0 − qp0 ).
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) = e2πiλt e−2πiqλx f (x − p)
= e2πiλt Mqλ lp f (x)
b ≡ R/{0} and we simply show λ ∈ N.
b
Dual space. N
The Heisenberg group
b
Fourier transform. For f ∈ L1 (N) ∩ L2 (N) and λ ∈ N
Z
f̂ (λ) =
f (n)πλ (n) dn.
n∈N
The Heisenberg group
b
Fourier transform. For f ∈ L1 (N) ∩ L2 (N) and λ ∈ N
Z
f̂ (λ) =
f (n)πλ (n) dn.
n∈N
Definition: f , g, h ∈ L2 (R)
(f ⊗ g)(h) = hf , hig
The Heisenberg group
b
Fourier transform. For f ∈ L1 (N) ∩ L2 (N) and λ ∈ N
Z
f̂ (λ) =
f (n)πλ (n) dn.
n∈N
Definition: f , g, h ∈ L2 (R)
(f ⊗ g)(h) = hf , hig
Plancherel transform.
F : L2 (N) −→
Z
⊕
L2 (R) ⊗ L2 (R) |λ|dλ
b
N
f −→ bf = {bf (λ)}Nb
Z
kf k =
kf̂ (λ)k2HS |λ|dλ.
b
N
Notation: HS := HS(L2 (R)) = L2 (R) ⊗ L2 (R).
The Heisenberg group
Lemma
L (N) ∼
=
2
Z
⊕
(0,1]
l2 (Z, HS) dα.
The Heisenberg group
Lemma
L (N) ∼
=
2
Z
⊕
l2 (Z, HS) dα.
(0,1]
Techniques: Plancherel transform and periodization.
The Heisenberg group
Lemma
L (N) ∼
=
2
Z
⊕
l2 (Z, HS) dα.
(0,1]
Techniques: Plancherel transform and periodization.
Sketch of proof. Given f ∈ L2 (N).
Z
Z 1X
kf k2 = kbf k2 =
kbf (λ)k2HS |λ| dλ =
k(α + j)1/2bf (α + j)k2HS dα
b
N
0
j∈Z
The Heisenberg group
Lemma
L (N) ∼
=
2
Z
⊕
l2 (Z, HS) dα.
(0,1]
Techniques: Plancherel transform and periodization.
Sketch of proof. Given f ∈ L2 (N).
Z
Z 1X
kf k2 = kbf k2 =
kbf (λ)k2HS |λ| dλ =
k(α + j)1/2bf (α + j)k2HS dα
b
N
Let
f̂α (j) := (α + j)1/2 f̂ (α + j).
0
j∈Z
The Heisenberg group
Lemma
L (N) ∼
=
2
Z
⊕
l2 (Z, HS) dα.
(0,1]
Techniques: Plancherel transform and periodization.
Sketch of proof. Given f ∈ L2 (N).
Z
Z 1X
kf k2 = kbf k2 =
kbf (λ)k2HS |λ| dλ =
k(α + j)1/2bf (α + j)k2HS dα
b
N
Let
0
j∈Z
f̂α (j) := (α + j)1/2 f̂ (α + j).
Define T : f → Tf
Tf : (0, 1] → l2 (Z, HS); α 7→ f̂α := {f̂α (j)}j∈Z .
The Heisenberg group
Lemma
L (N) ∼
=
2
Z
⊕
l2 (Z, HS) dα.
(0,1]
Techniques: Plancherel transform and periodization.
Sketch of proof. Given f ∈ L2 (N).
Z
Z 1X
kf k2 = kbf k2 =
kbf (λ)k2HS |λ| dλ =
k(α + j)1/2bf (α + j)k2HS dα
0
b
N
Let
j∈Z
f̂α (j) := (α + j)1/2 f̂ (α + j).
Define T : f → Tf
Tf : (0, 1] → l2 (Z, HS); α 7→ f̂α := {f̂α (j)}j∈Z .
We show T is isometric isomorphism and
Z 1
2
2
kf k = kTf k =
kTf (α)k2l2 (Z,HS) dα.
0
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 7→ 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 7→ J(α) ≤ H
Fiber. We say {J(α)}α is fiber of the range function J.
Lemma
The map is unitary.
π
eλ : N → U(l2 (Z, HS)),
(e
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
π
eα (Γ1 )(J(α)) ⊆ J(α),
and
V = T −1
Z
⊕
a.e. α ∈ (0, 1]
!
J(α)dα .
(0,1]
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
π
eα (Γ1 )(J(α)) ⊆ J(α),
and
V = T −1
Z
⊕
a.e. α ∈ (0, 1]
!
J(α)dα .
(0,1]
Sketch of proof. “(ii) ⇒ (i)”
We need to show that if φ ∈ V, then lγ1 γ0 φ ∈ V for any γ1 γ0 ∈ 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
π
eα (Γ1 )(J(α)) ⊆ J(α),
and
V = T −1
Z
⊕
a.e. α ∈ (0, 1]
!
J(α)dα .
(0,1]
Sketch of proof. “(ii) ⇒ (i)”
We need to show that if φ ∈ V, then lγ1 γ0 φ ∈ V for any γ1 γ0 ∈ Γ.
This follows by T(lγ1 γ0 φ)(α) = e2πiαγ0 π
eα (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
π
eα (Γ1 )(J(α)) ⊆ J(α),
a.e. α ∈ (0, 1]
and
V=T
−1
Z
⊕
!
J(α)dα .
(0,1]
Sketch of proof. “(i) ⇒ (ii)” Recall that L2 (N) ≡
R1
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
π
eα (Γ1 )(J(α)) ⊆ J(α),
a.e. α ∈ (0, 1]
and
V=T
−1
Z
⊕
!
J(α)dα .
(0,1]
Sketch of proof. “(i) ⇒ (ii)” Recall that L2 (N) ≡
Take {em }m ONB for l2 (Z, HS).
R1
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
π
eα (Γ1 )(J(α)) ⊆ J(α),
a.e. α ∈ (0, 1]
and
V=T
−1
Z
⊕
!
J(α)dα .
(0,1]
R1
Sketch of proof. “(i) ⇒ (ii)” Recall that L2 (N) ≡ 0 l2 (Z, HS) dα.
Take {em }m ONB for l2 (Z, HS). Define gm,p (α) := e2πiαp em , 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
π
eα (Γ1 )(J(α)) ⊆ J(α),
a.e. α ∈ (0, 1]
and
V=T
−1
Z
⊕
!
J(α)dα .
(0,1]
R1
Sketch of proof. “(i) ⇒ (ii)” Recall that L2 (N) ≡ 0 l2 (Z, HS) dα.
Take {em }m ONB for l2 (Z, HS). Define gm,p (α) := e2πiαp em , p ∈ Z.
R1
Then {gm,p }m,p is an ONB for 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
π
eα (Γ1 )(J(α)) ⊆ J(α),
a.e. α ∈ (0, 1]
and
V=T
−1
Z
⊕
!
J(α)dα .
(0,1]
R1
Sketch of proof. “(i) ⇒ (ii)” Recall that L2 (N) ≡ 0 l2 (Z, HS) dα.
Take {em }m ONB for l2 (Z, HS). Define gm,p (α) := e2πiαp em , p ∈ Z.
R1
Then {gm,p }m,p is an ONB for 0 l2 (Z, HS) dα.
R1
Let P := PS be the orthogonal projector of 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
π
eα (Γ1 )(J(α)) ⊆ J(α),
a.e. α ∈ (0, 1]
and
V=T
−1
Z
⊕
!
J(α)dα .
(0,1]
R1
Sketch of proof. “(i) ⇒ (ii)” Recall that L2 (N) ≡ 0 l2 (Z, HS) dα.
Take {em }m ONB for l2 (Z, HS). Define gm,p (α) := e2πiαp em , p ∈ Z.
R1
Then {gm,p }m,p is an ONB for 0 l2 (Z, HS) dα.
R1
Let P := PS be the orthogonal projector of 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
X
Akf k2 ≤
|hf , fn i|2 ≤ Bkf k2 ∀f ∈ V.
n
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
X
X
X
A
|cn |2 ≤ k
cn fn k2 ≤ 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) := {lγ φ : 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) := {lγ φ : 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) {Mαq lp (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
operators, 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) := {lγ φ : 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) {Mαq lp (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
operators, respectively.)
Note that
Mαq lp (Tφ)(α) = π
eα (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(lk φ)(α) : k ∈ αZ × βZ} is orthogonal and kTφ(α)k = 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(lk φ)(α) : k ∈ αZ × βZ} is orthogonal and kTφ(α)k = 1.
Thanks for your attention.