Lecture 2 Wavelets on Groups

A matured mathematical theory looks like a tree. There is a solid trunk which supports all branches and leaves but could not be alive without them. In the case of group approach to wavelets the trunk of the theory is a construction of wavelets from a square integrable representation [38], [5, Chap. 8]. We begin from this trunk which is a model for many different generalisations and will continue with some smaller “generalising” branches later.

2.1 Wavelet Transform on Groups

Let G be a group with a left Haar measure dµ and let ρ be a unitary irreducible representation of a group G by operators ρ_g, g ∈ G in a Hilbert space H.

The wavelet transform can be defined as a mapping from H to a space of functions over G via its representational coefficients (also known as matrix coefficients):

Proof.[Answer] v(n)=⟨ v,e_n ⟩.

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Figure 2.1: The Gaussian function e−x2/2.

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Proof. We have:

Wavelet transform maps vectors of H to functions on G. We can consider a map in the opposite direction sends a function on G to a vector in H.

Proof.[Answer]

1. For Exercises 4: v=∑_−∞^∞ v(n) e_n.
2. For Exercises 5:

   v(x) =

   1 2π i

   ∫

   ℝ+2

   v(a,b) x−(b−ia)

   da db a 3 2

   .

Proof. We have:

where the scalar product in the first line is on H and in the last line is on L₂(G). Now the result follows from the totality of coherent states w_g in H.

Proof. We have:

where h′=g⁻¹h.

An important particular case of such an invariant subspace is Gårding space.

The following proposition explain the usage of the name “inverse” (not “adjoint” as it could be expected from Lemma 10) for M.

Proof. It follows from Propositions 6 and 11 that operator MW: H → H intertwines ρ with itself. Then Corollaries 7 and 12 imply that the image MW is a ρ-invariant subspace of H containing w₀. From irreducibility of ρ by Schur’s Lemma [159, § 8.2] one concludes that MW=cI on C for a constant c∈ℂ.

We have similarly

2.2 Square Integrable Representations

So far our consideration of wavelets was mainly algebraic. Usually in analysis we wish that the wavelet transform can preserve an analytic structure, e.g. values of scalar product in Hilbert spaces. This accomplished if a representation ρ possesses the following property.

Square integrable representations of groups have many interesting properties (see [85, § 14] for unimodular groups and [87], [5, Chap. 8] for not unimodular generalisation) which are crucial in the construction of wavelets. For example, for a square integrable representation all functions ⟨ ρ(g)v₁,v₂ ⟩ with an admissible vector v₁ and any v₂∈ H are square integrable on G; such representation belong to dicrete series; etc.

Proof.[Hint] The set of all admissible vectors is an ρ-invariant subspace of H.

For an admissible vector w we take its normalisation w₀=||w||/c w to obtain:

Such a w₀ as a vacuum state produces many useful properties.

Proof. We already knew that MW=cI for a constant c∈ ℂ. Then (6) exactly says that c=1. Because W and M are adjoint operators it follows from MW=I on H that:

which is exactly the isometry of W  (7). Finally condition (6) is a partticular case of general isometry of W for vector w₀.

Wavelets from square integrable representation closely related to the following notion:

Proof. Again we have a simple application of the previous formulas:

where transformation (12) is due to  (7).

Proof.[Hint] Use that an left invariant subspace of L₂(G) is in fact an right ideal in convolution algebra, see Lemma 36.

We consider only fundamentals of the wavelet construction here. There are much results which can be stated in an abstract level. To avoid repetition we will formulate it later on together with an interesting examples of applications.

The construction of wavelets from square integrable representations is general and straightforward. However we can not use it everywhere we may wish:

1. Some important representations are not square integrable.
2. Some groups, e.g. ℍⁿ, do not have square representations at all.
3. Even if representation is square integrable, some important vacuum vectors are not admissible, e.g. the Gaussian e^−x2 in 3.
4. Sometimes we are interested in Banach spaces, while unitary square integrable representations are acting only on Hilbert spaces.

To be vivid the trunk of the wavelets theory should split into several branches adopted to particular cases and we describe some of them in the next lectures.

2.3 Fundamentals of Wavelets on Homogeneous Spaces

Let G be a group and H be its closed normal subgroup. Let X=G/H be the corresponding homogeneous space with a left invariant measure dµ. Let s: X → G be a Borel section in the principal bundle of the natural projection p: G → G/H. Let ρ be a continuous representation of a group G by invertible unitary operators ρ(g), g ∈ G in a Hilbert space H.

For any g∈ G there is a unique decomposition of the form g=s(x)h, h∈ H, x=p(g)∈ X. We will define r: G → H: r(g)=h=(s(p(g)))⁻¹g from the previous equality. Then there is a geometric action of G on X → X defined as follows

Let ρ: G → L(V) be a unitary representation of the group G by operators in a Hilbert space V.

Note that mapping h → χ(h) from (13) defines a character of the subgroup H. The condition (14) can be easily achieved by a renormalisation w₀ as soon as we sure that the integral in the left hand side is finite and non-zero.

The wavelet transform (similarly to [eq:wavelet-transform]the group case) can be defined as a mapping from V to a space of bounded continuous functions over G via representational coefficients

Due to (13) such functions have simple transformation properties along H-orbits:

Thus the wavelet transform is completely defined by its values indexed by points of X=G/H. Therefore we prefer to consider so called induced wavelet transform.

There is a natural representation of G in W(X). It can be obtained if we first lift functions from X to G, apply the left regular representation Λ and then pul them back to X. The result defines a representation λ(g): W(X) → W(X) as follow

We recall that χ(h) is a character of H defined in (13) by the vacuum vector w₀. Of course, for the case of trivial H={e} (17) becomes the left regular representation Λ(g) of G.

Proof. We have with obvious adjustments in comparison with Proposition 6:

We again introduce a transform adjoint to W.

Proof. We have:

where x′=g⁻¹ · x.

The following proposition explain the usage of the name for M.

Proof. It follows from Propositions 37 and 42 that operator MW: H → H intertwines ρ with itself. Then Corollaries 38 and 43 imply that the image MW is a ρ-invariant subspace of H containing w₀. Because of MWw₀=w₀ we conclude that MW is a projection.

From irreducibility of ρ by Schur’s Lemma [159, § 8.2] one concludes that MW=cI on H for a constant c∈ℂ. Particularly

From the condition (14) it follows that ⟨ cw₀,w₀ ⟩=⟨ MW w₀,w₀ ⟩=⟨ w₀,w₀ ⟩ and therefore c=1.

We have similar

Proof. From the definition of the wavelet transform:

Since the wavelet transform is an isometry we conclude that | f(x) |≤ c||f|| for c=||w₀||, which implies the assertion about two types of convergence.

We denote by W^*: W^*(X) → H and M^*: H → W^*(X) the adjoint (in the standard sense) operators to W and M respectively.

Proof. We show the equality in the first form (23) (but we will apply it often in the second one):

Proof. Again we have a simple application of the previous formulas:

where transformation (25) is due to (23).