shannon-type wavelet on the heisenberg group

Shannon-type wavelet on the Heisenberg group

Azita Mayeli

Stony Brook University

May 26, 2009

Azita Mayeli (Stony Brook University) Shannon-type wavelet on the Heisenberg group May 23, 2009 1 / 19

Outline

1 Motivation: Frame multiresolution analysis on Rn (FMRA)

2 The Heisenberg Group

3 FMRA on the Heisenberg group

4 Shannon-type wavelet


Motivation: Frame multiresolution analysis on Rn (FMRA)

Frame Multiresolution Analysis on R (FMRA)

Definition: A sequence of closed subspaces {Vj}j∈Z of L2(R) is aFMRA of L2(R) if

{0} ⊆ · · · ⊆ Vj ⊆ Vj+1 ⊆ · · · ⊆ L2(R),⋃Vj = L2(H),⋂Vj = {0},

f ∈ Vj ⇔ f (2−j·) ∈ V0 (scaling property),V0 is left shift-invariant, and∃φ ∈ V0, called the scaling function, such that {φ(· − k)}k∈Z is aframe for V0, i.e., there exists 0 < α ≤ β <∞ such that

α ‖ f ‖2≤∑

k

| 〈f , φ(· − k)〉 |2≤‖ f ‖2 ∀f ∈ V0.

Note: The construction of {φ,V0} is usually the start point for an FMRA.



ObservationsLet Wj be the complement orthogonal subspace of Vj in Vj+1. Then

Vj+1 = Vj ⊕Wj

= Vj−1 ⊕Wj−1 ⊕Wj

...

= Vk ⊕(⊕

k≤m≤jWm

).

Therefore

Vj+1 =⊕

m≤jWm, & L2(R) =

⊕m∈Z

Wm.

Moreover,

f ∈ W0 ⇔ f (2−j·) ∈ W0.

Main issue: Construction of a ψ ∈ W0 such that {ψ(· − k)}k∈Z is aframe for W0, and hence obtain the frame{2j/2ψ(2j · −k) : j ∈ Z, k ∈ Zn} for L2(R).Azita Mayeli (Stony Brook University) Shannon-type wavelet on the Heisenberg group May 23, 2009 4 / 19


The notion of an FMRA was introduced by Benedetto / Li (’93), whichis an extension of an MRA.

They characterize {φ,V0} such that there exists a ψ ∈ W0 for which{ψ(· − k)}k∈Z is a frame for W0.

Their method is based on the construction of periodic functions on theFourier side.



Our objective

To generalized the notion of FMRA to the Heisenberg group andconstruct frame wavelet.

We need to replace

Z→ ? (1)D2j → ?. (2)

Our approach for the construction of a concrete {φ,V0} will therepresentation theory of the group, but different from the method in theclassical case.

Historical comment: Lemarié (’89) constructed an MRA and obtains anONB of spline wavelets on stratified Lie groups. (A complicatedapproach.)


The Heisenberg Group

The Heisenberg group H

H = {x = (p, q, t) : p, q, t ∈ R}.

(p, q, t)(p′, q′, t′) =(p + p′, q + q′, t + t′ + (pq′ − qp′)/2

)

The Haar measure on H is the usual Lebesgue measure.Translation

TωF(ν) = F(ω−1ν) ν, ω ∈ HDilation a > 0

AaF(ω) = a−2F(a−2ω)

where for ω = (p, q, t) ∈ H ax = (ap, aq, a2t). (a ∈ Aut(H).)

For d ∈ N, define

Γ := Γd = {(m, dk, l + 1/2dmk) : m, k, l ∈ Z}.




H = {x = (p, q, t) : p, q, t ∈ R}.


)The Haar measure on H is the usual Lebesgue measure.

TranslationTωF(ν) = F(ω−1ν) ν, ω ∈ H

Dilation a > 0AaF(ω) = a−2F(a−2ω)


For d ∈ N, define

Γ := Γd = {(m, dk, l + 1/2dmk) : m, k, l ∈ Z}.




H = {x = (p, q, t) : p, q, t ∈ R}.


)The Haar measure on H is the usual Lebesgue measure.Translation

TωF(ν) = F(ω−1ν) ν, ω ∈ H

Dilation a > 0AaF(ω) = a−2F(a−2ω)


For d ∈ N, define

Γ := Γd = {(m, dk, l + 1/2dmk) : m, k, l ∈ Z}.




H = {x = (p, q, t) : p, q, t ∈ R}.






For d ∈ N, define

Γ := Γd = {(m, dk, l + 1/2dmk) : m, k, l ∈ Z}.




H = {x = (p, q, t) : p, q, t ∈ R}.






For d ∈ N, define

Γ := Γd = {(m, dk, l + 1/2dmk) : m, k, l ∈ Z}.Azita Mayeli (Stony Brook University) Shannon-type wavelet on the Heisenberg group May 23, 2009 7 / 19


Frame: Let a > 0. We say ψ ∈ L2(H) is a (wavelet) frame for L2(H) iffor any f ∈ L2(H)

α ‖ f ‖2≤∑

γ∈Γ,j∈Z| 〈f ,T2−jγA2−jψ〉 |2≤‖ f ‖2 .

α = β, the frame is called tight frame.

α = β = 1, the frame is called Parseval frame.

For tight frame the reconstruction formula holds:

f = 1/α∑

γ∈Γ,j∈Z〈f ,T2−jγA2−jφ〉 T2−jγA2−jφ.



Fourier Analysis on H

Schrödinger representation: λ ∈ R∗, ρλ : H→ U(L2(R)) is definedby

ρλ(p, q, t)g(x) = eiλteiλ(px+ 12 (pq))g(x + q) ∀ g ∈ L2(R)

Fourier transformation: For f ∈ L1(R)

f (λ) :=F(f )(ρλ) =∫

Hf (ω)ρλ(ω)dω

〈f (λ)g1, g2〉 =∫

Hf (ω)〈ρλ(ω)g1, g2〉dω ∀ g1, g2 ∈ L2(R).



Plancherel Theorem: The Fourier transformation F can be extendedfrom L1 ∩ L2(H) to L2(H), and for any f ∈ L2(H)

‖ f ‖2=∫

R∗‖ f (λ) ‖2

H.S | λ | dλ.

The basic properties of the Fourier transform remain valid forf , g ∈ L2(H):

(i) (f ∗ g)(λ) = f (λ)g(λ), where

f ∗ g(ω) =∫

Hf (ν)g(ν−1ω)dν,



(ii) (Lωf )(λ) = ρλ(ω)f (λ), for ω ∈ H

(iii) (f )(λ) = f (λ)∗ where f (ω) = f (ω−1).

Lemma: For any f ∈ L2(H)

f (a·)(λ) = a−4Da−1f (a−2λ)Da.


FMRA on the Heisenberg group

Construction of a sicn-type function:Once step towards the FMRA

We do not start with construction of a scaling function φ.

We start with the construction of a band-limited function S, instead.

Definition: We call function S band-limited if supp(S) is a boundedsubset of R∗.

Theorem: There exists a S ∈ L2(H) such thatS = S∗ and S ∗ S = S.supp(S) ⊂ I0 = [− 1

2d ,1

2d ] where d ∈ N.Sketch of proof: Pick an ONB {ei}i∈N0 in L2(R). For any λ ∈ R∗, defineeλi := D1/

√|λ|ei.

Write I0 = ∪kIk0 where Ik

0 = [− 122k+1d ,−

122k+3d ) ∪ ( 1

22k+3d ,1

22k+1d ].



Sinc-type function - continued

For any λ 6= 0 define

S(λ) =

{∑22k

i=0

(eλi ⊗ eλi

)if λ ∈ Ik

0, for some k ∈ N0,

0 otherwise.

Therefore ∫R∗‖ S(λ) ‖2

H.S | λ | dλ <∞.

and by Plancherel theorem, S has a pre-image in S ∈ L2(H).

And, it turns that

S = S∗ &S ∗ S = S.



Scaling function

Theorem Define V0 = L2(H) ∗ S. Then there exists a φ ∈ V0 such that{Tγφ}γ∈Γ constitutes a Parseval frame for V0. (Recall that

Γd = {(m, dk, l + dmk/2) : m, k, l ∈ Z})

Proof: Let P : L2(H)→ V0 be the orthogonal projector. Then for anyf ∈ L2(H) one has

P(f )(λ) = f (λ) ◦ Sλ.

Let m := mV0 : R∗ → N0 ∪ {∞} be the associated multiplicity function toV0 defined by

m(λ) = rank(Pλ).

Then supp(m) ⊂ I0 and

m(λ) | λ | +m(λ− 1) | λ− 1 |≤ 1/d.

To complete the proof we need the following theorem:Azita Mayeli (Stony Brook University) Shannon-type wavelet on the Heisenberg group May 23, 2009 14 / 19


Scaling function - continued

Theorem: (Führ, ’05) Suppose H is a left-invariant subspace of L2(H)and {Pλ} be the associated projection filed such that for any f ∈ H

P(f )(λ) = f (λ)Pλ

where P is the orthogonal projector of L2(H) onto H.

Let mH be its associated multiplicity function defined bymH(λ) = rank(Pλ). Then there exists a Parseval frame of the form{Tγφ}γ∈Γ with an appropriate φ ∈ H if and only if the inequality

mH(λ) |λ|+ mH (λ− 1) |λ− 1| ≤ 1d. (3)

By this theorem, the proof is completed since mV0 satisfies theinequality (3).



Construction of an FMRA for H

Observations

For j ∈ Z, define Sj = 24jS(2j·). Then the following hold:

supp(Sj) ⊂ Ij := [− 22j

2d ,22j

2d ]S∗j = Sj and Sj ∗ Sj = Sj

S ∗ Sj = S ∀j > 0 and Sj ∗ S = Sj ∀j < 0,f ∗ Sj → 0 as j→ −∞ ∀f ∈ L2(H),f ∗ Sj → f as j→∞.



Construction of an FMRA for H - continued

Define Vj = L2(H) ∗ Sj. Then

For f ∈ Vj, f ∗ Sj = Sj (since Sj ∗ Sj = Sj)Vj ⊂ Vj+1 (since Sj ∗ Sk = Sj if j ≤ k)f ∈ Vj ⇔ f (2k−j·) ∈ Vk

∩Vj = {0} (since Pj(f ) = f ∗ Sj → 0 as j→ −∞)∪Vj = L2(H) (since f ∗ Sj → f as j→∞, and,{T2−jγA2−jφ} is a Parseval frame for Vj (since Vj = A2−jV0.)


Shannon-type wavelet


We use the following general approach to find the wavelet:

Recall that Wj denotes the orthogonal complement of Vj in Vj+1, i.e.,Vj+1 = Vj ⊕Wj for any j ∈ Z.

Evidently

Vj+1 =⊕k≤j

Wk,

and hence

L2(H) =⊕k∈Z

Wk,

Automatically, the scaling property for Wj also holds:

f ∈ Wj ⇔ f (2−j·) ∈ W0.




Theorem: Let Q0 be the orthogonal projector of L2(H) onto W0.Define ψ = Q0(A2−1φ). Then ψ ∈ W0 is band-limited and {T2−1γψ}γ is aParseval frame for W0.

Sketch of proof:

ψ is band-limited since P1 = P0 ⊕ Q0. More precisely,supp(ψ) ⊂ [−1/d, 1/d] \ [−1/(2d), 1/(2d)].

{T2−1γA2−1φ : γ ∈ Γ} is a Parseval frame for V1.

Q0 commutes with the translation operator. �

Corollary: {T2−jγA2−jψ : j ∈ Z, γ ∈ Γ} is a Parseval frame for L2(H).


shannon-type wavelet on the heisenberg group

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