Wavelets in abstract Hilbert space
Mathieu Sablik Mathematiques,
ENS Lyon, 46 allee d’Italie, F69364 LYON Cedex 07,
France
∗June-July 2000
Introduction
The purpose of my training period has been to provide some generalization of the wavelet theory from L2to an abstract Hilbert spaceH. Indeed, a great deal of properties of the wavelet transform can be generalized to abstract Hilbert space based on its fundamental properties alone. In particuar, the aim of this work is to generalize what constitutes the success of wavelets, that is to say to find a vector ψ called the ”mother wavelet” and a group G of unitary operators such that{g.ψ : g ∈ G} is a basis of H or better, an orthonormal basis.
1 A first attempt: Multiresolution analysis
The first idea is to generalize multiresolution analysis, so we want to find the minimal hypothesis to generalize the multiresolution analysis used for L2(R, dµ).
Definition 1.1. Let H be a Hilbert space. A multiresolution analysis on H (MRA) consists of a sequence of closed subspaces Vj, j∈ Z, a unitary operator γ and a group G of unitary operators satisfying:
1. Vj ⊂ Vj+1 for all j∈ Z
2. f ∈ Vj if and only if γ f∈ Vj+1 for all j∈ Z 3. T
jVj={0}
4. S
jVj=H
5. ∃ϕ ∈ V0 such that {g ϕ : g ∈ G} is an orthonormal basis for V0, ϕ is called a scaling function of the given MRA.
∗Work done while visiting Uppsala University, advised by Prof. Kyril Tintarev
Thanks to (2) and (5) we have by induction that {γjg ϕ : g∈ G} is an orthonormal basis for Vj for all j ∈ Z. Then we define by induction closed subspaces Wj such that∀j ∈ Z Vj+1 = Vj⊕ Wj, so we have H =L
jWj and to find a mother wavelet of H it suffices only to find a mother wavelet of W0 associated to G.
Proposition 1.2.
Wj+1= γ Wj ∀j ∈ Z Proof. We have:
u∈ Wj+1 ⇐⇒ u∈ Vj+2 and PVj+1u= 0
⇐⇒ γ−1u∈ Vj+1 and PVjγ−1u= 0 since γ is an unitary
⇐⇒ γ−1u∈ Wj
⇐⇒ u∈ γ Wj
We make now further assumptions, namely that G w Z (that is to say that an enumeration of G exists such that for all k, l in Z we have gkgl= gk+l) and that for all k ∈ Z we have gkγ = γ g2k ⇐⇒ γ−1gk = g2kγ−1. In this case H is separable. All these assumptions are met when H = L2, γ is the binary dilation operator and gk is the translation by k, i.e. when we have the original construction of orthonormal wavelets developed by Meyer and Mayat.
Theorem 1.3. We have γ−1ϕ ∈ V−1 ⊂ V0 so ∃(αk)k∈Z ∈ CZ such that γ−1ϕ = P
kαkgkϕ. In this case, ψ = P
k∈Z(−1)kα1−kγ gkϕ is a mother wavelet associated to the group G0={(γj, g) : j∈ Z, g ∈ G}.
Proof. First of all, for all l∈ Z we have:
glψ=X
k∈Z
(−1)kαkγ g2l+1−k ϕ
To verify the theorem, it suffices to prove that < gkψ, glϕ > for all (k, l)∈ Z2, spank∈Z{gkψ} ⊕ V0= V1 and < gk, gl>= δk,l for all (k, l)∈ Z2.
• For all j ∈ Z we have:
< gjψ, ϕ > = <P
k(−1)kαkγ g2j+1−kϕ,P
lαkγ glϕ >
= P
(k,l)(−1)kαkαl < g2j+1−kϕ, glϕ >
= P
k(−1)kαkα2j+1−k
= P
kα2kα2j+1−2k − P
kα2k+1α2j−2k
= P
kα2kα2j+1−2k − P
k0α2j+1−2k0α2k0 with k0 = j− k
< gjψ, ϕ > = 0
So for all (k, l)∈ Z2 we have < gkψ, glϕ >=< gk−lψ, ϕ >= 0.
• Let W0= spank{gkψ}, we have V0⊕ W0⊂ V1, so we want to show that V0⊕ W0 ⊃ V1 that is to say γ gkϕ∈ V0⊕ W0 ∀k since {γ gkϕ: k ∈ Z} is a
By hypotheses, we can write that gjϕ=P
kαkγ g2j+kϕ=P
kαk−2jγ gkϕ and gjψ = P
k(−1)kαkγ g2j−k+1ϕ = P
k(−1)2j+1−kα2j+1−kγ gkϕ. So for some k∈ Z ,we deduce:
PV0γ gkϕ = P
jαk−2jgjϕ = P
(i,j)αk−2jαi−2jγ giϕ PW0γ gkϕ = P
j(−1)2j−k+1α2j−k+1gjψ = P
(i,j)(−1)i+kα2j−k+1α2j+1−iγ giϕ So we have:
PV0γ gkϕ+ PW0γ gkϕ=X
i
X
j
αk−2jαi−2j+X
j
(−1)i+kα2j−k+1α2j+1−i
γ giϕ
If i + k∈ 2N:
P
jαk−2jαi−2j+P
j(−1)i+kα2j−k+1α2j+1−i = P
jαk+2jαi+2j+P
jα2j−k+1α2j+1−i
= P
jαk+jαi+j
= < giγ−1ϕ, gkγ−1ϕ, >
= δi,k
If i + k∈ 2N + 1:
P
jαk−2jαi−2j+P
j(−1)i+kα2j−k+1α2j+1−i = P
jαk−2jαi−2j−P
jα2j−k+1α2j+1−i
= P
jαk−2jαi−2j−P
j0αi−2j0αk−2j0
with − 2j0= 2j + 1− i − k
= 0 So PV0γ gkϕ+ PW0γ gkϕ= γ gkϕfor all k∈ Z
• For all j ∈ Z we have:
< gjψ, ψ > = <P
k(−1)kαkγ g2j+1−kϕ,P
l(−1)lαlγ g1−lϕ
= P
(k,l)(−1)k+lαkαl < g2j+1−kϕ, g1−lϕ >
= P
k(−1)2k−2jαkαk−2j
= P
(k,l)αlαk−2j < gkϕ, glϕ >
= P
(k,l)αlαk < glϕ, gk+2jϕ >
= <P
lαlγ−1glϕ,P
kαkγ−1gk+2jϕ >
= < γ−1ϕ, gjγ−1ϕ >
< gjψ, ψ > = δj,0
So for all (k, l)∈ Z2 we have < gkψ, glψ >=< gk−lψ, ψ >= δj,k.
This construction uses too specific assumptions, such as gkγ= γ g2k ⇐⇒
γ−1gk = g2kγ−1 so perhaps it is better to consider another approach: the wavelet transform.
2 Some notions of the group theory
This section is to remind, without proof, some results about Haar measure on a locally compact group and about representation of groups on Hilbert spaces which are used in construction of the abstract wavelet transform.
2.1 The Haar measure
Theorem 2.1. Let G be a locally compact group. There is a measure µ on the Borelian σ-algebra, B, called the left-invariant measure such as:
1. µ(K) <∞ if K is compact 2. µ(U ) > 0 if U6= ∅ is open
3. µ(xB) = µ(B) for all B∈ B and x ∈ G
4. µ(U ) = sup{µ(K), K compact, K ⊂ U}, for all U open 5. µ(B) = inf{µ(U), U open, U ⊃ B} for all B ∈ B
Moreover if another measure ν has the same proprieties, then ∃c > 0 such that ν= cµ.
Theorem 2.2. L2(G, dµ) is a Hilbert space.
Theorem 2.3. If T ∈ Lp(G, dµ), p≥ 1 and U ∈ L1(G, dµ) then we can define:
T∗ U(x) = Z
G
T(y)U (y−1x) dµ(y) And we have: kT ∗ Ukp≤ kT kpkUk1
We can also define the convolution product for two functions in L2(G, dµ).
2.2 Representations of groups
LetH be a Hilbert space and let U(H) be the group of unitary operators acting inH, a homomorphism π from a group G to U(H) is called a unitary represen- tation.
Definition 2.4. A representation is called irreducible if and only if the only closed invariant subspaces are the trivial ones {0} and H, or, equivalently if every vector g6= 0 is cyclic (that is to say that span{π(x)g : x ∈ G} = H).
Definition 2.5. A unitary representation π of a group G in a Hilbert spaceH is square integrable if there exists ψ∈ H, ψ 6= 0, such that (x −→< π(x)ψ, ψ >
)∈ L2(G, dµ). ψ is called an admissible vector.
2.3 Schur’s lemma and one application
Definition 2.6. Let π1 and π2 be two representations of the same group G in H1 and H2. A bounded operator, A : H1 −→ H2, is called an interwining operator iff the following diagram is commutative:
H1 −−−−→ Hπ1(x) 1
yA
yA H2 −−−−→ Hπ2(x) 2
A◦ π1(x) = π2(x)◦ A
Schur’s lemma 2.7. Let π1and π2 be two unitary representations of the same group G in H1 and H2 such that π1(x) is surjective for every x ∈ G. Let A : H1 −→ H2 be a bounded and interwining operator. So A is a multiple of an isometry.
Schur’s lemma 2.8. Let A be a bounded operator and let π be an irreducible unitary representation of G onH such that ∀x ∈ G π(x)◦A = A◦π(x). Then A = λ IdH.
Proposition 2.9. Let G be a group, letH and H0 be two Hilbert spaces, let π be a unitary irreducible representation of G inH, let τ be a unitary representation of G in H0 and let T be a closed operator from H to H0 with domain D ⊂ H dense in H and stable under π. If T π(x) = τ(x)T for every x in G, we have T is a multiple of an isometry and T can be extended toH.
Proof. We denote by (., .) and (., .)0 the scalar product inH and H0, and byk.k andk.k0 the associated norms.
Consider onD the scalar product (g, f)T = (f, g)+(T f, T g)0and the assicu- ated normk.kT. Since T is closed,D equipped with the scalar product (., .)T is a Hilbert space. Since
(kT gk0)2 kgk2T
= (kT gk0)2
(kT gk0)2+kgk2 ≤ 1 ∀g ∈ H T is bounded fromD to H.
Thenkπ(x)gk2T =kgk2+(kτ(x)k0)2=kgk2T for every x∈ G, so π is a unitary representation of G in D. Moreover, for every g ∈ G we have g = π(x)π(x−1)g, so, since D is stable under π(x−1), π(x) is surjective for all x ∈ G. By the Schur’s lemma we have that T is a multiple of an isometry fromD to H0, so for every g inD we have:
(kT gk0)2= λkgkT2 = λkgk2+ λ(kT gk0)2
We deduce that T is a multiple of an isometry fromD to H0 and consequently we can extend T to a multiple of an isometry fromH to H0.
3 Wavelet transform in abstract Hilbert space
Let G be a locally compact group, with left Haar measure dµ (in fact we can do the same thing for the right measure) and π a irreducible, continuous and square integrable unitary representation of G on a Hilbert spaceH.
3.1 Definition of the wawelet transform
Definition 3.1. Let ψ be an admissible vector, letD be the set of vectors f ∈ H such that < π(x)ψ, f >∈ L2(G, dµ), we can define the wavelet transform:
Aψ: D −→ L2(G, dµ)
f −→ (x → < π(x)ψ, f >) We can deduce the following properties:
• Thanks to the Cauchy-Schwarz inegality, we have:
∀x ∈ G | Aψf(x)| ≤ kψk kfk
• Since π is square integrable, D 6= ∅, moreover D is invariant under π; as π is irreducible we haveD = H.
• We prove now that Aψ is closed: Let (fn)n∈N be a sequence from D converging to f inH and such that Aψfn converges in L2(G, dµ) to ϕ ∈ L2(G, dµ). By continuity of the scalar product, Aψfn(x) converges to
< π(x)ψ, f > for every x in G, we deduce that < π(.)ψ, f >= ϕ almost everywhere. So < π(x)ψ, f >∈ L2(G, dµ) that is to say f ∈ D and Aψfn converges to Aψf in L2(G, dµ). So Aψ is closed.
• We define for all x in G:
τ(x) : L2(G, dµ) −→ L2(G, dµ)
T −→ (y → T (x−1y))
τ is a unitary representation of G in the Hilbert space L2(G, dµ). As Aψπ(x)f (y) = < π(y)ψ, π(x)f >=< π(x−1y)ψ, f >, we have the follow- ing commutative diagram:
H −−−−→π(x) H
yAψ
yAψ L2(G, dµ) −−−−→ Lτ(x) 2(G, dµ)
Aψ◦ π(x) = τ(x) ◦ Aψ
We have the conditions of the proposition 2.9, so we deduce the following theorem:
Theorem 3.2. Aψ can be extanded to a multiple of an isometry from H to L2(G, dµ).
Corollary 3.3. If π is square integrable and irreducible, the subset A of all admissible vectors is dense in H.
Proof. Indeed,A is a subspace invariant under π, since π is irreducible, we have A = {0} which is impossible or A = H.
3.2 The inversion formula
Proposition 3.4. The adjoint operator of Aψ is:
A∗ψ : L2(G, dµ) −→ H
T −→ R
GT(x) π(x) ψ dµ(x)
Remark. In fact this is just a notation which means that this is the vector associated by the Riesz’s theorem to the bounded anti-linear functional f −→
R
G< f, T(x) π(x) ψ > dµ(x).
Proof. Let f ∈ H and T ∈ L2(G, dµ), we have:
< Aψf, T >L2(G,dµ) = R
G< π(x)ψ, f > T (x) dµ(x)
= R
G < f, π(x)ψ > T (x) dµ(x)
= < f,R
GT(x) π(x) ψ dµ(x) >
= < f, A∗ψT >
Theorem 3.5. Let ψ1 and ψ2 be two admissible vectors and let h∈ H be such that khk = 1, we define cψ1,ψ2 = < Aψ1h, Aψ2h >L2(G,dµ).
We have A∗ψ
2Aψ1 = cψ1,ψ2IdH.
Proof. We have Aψ1◦ π(x) = τ(x) ◦ Aψ1. In the same way, we can show that A∗ψ2◦ τ(x) = π(x) ◦ A∗ψ2. So we obtain π(x)◦ A∗ψ2◦ Aψ1 = A∗ψ2◦ Aψ1◦ π(x).
By the second version of the Schur’s lemma, we deduce that A∗ψ2◦ Aψ1 = λ IdH. The value of the constant is obtained by considering < Aψ1h, Aψ2h >L2(G,dµ)
and in fact it is independent of h.
Corollary 3.6. For all f and g in H we have < Aψ1f, Aψ2g >= cψ1,ψ2 <
f, g >.
Corollary 3.7. Let ψ1, ψ2 and φ inA and f ∈ H, we have:
Aψ1f ∗ Aφψ2(x) = Z
G
Aψ1f(y) Aφψ2(y−1x) dµ(y) = cψ1,ψ2Aφf(x)
3.3 Study of the image
Proposition 3.8. The image of Aψ1 is exactly the functions T ∈ L2(G, dµ) such that for all x∈ G T (x) = cψ1,ψ2−1Aψ1A∗ψ2T(x).
Proof. Let Pψ1,ψ2 = cψ1,ψ2−1Aψ1◦ A∗ψ2, we just need to show that Pψ1,ψ2 is a projector. Thanks to the inversion formula, we have:
Pψ2
1,ψ2 = cψ1,ψ2−2Aψ1◦ A∗ψ2◦ Aψ1◦ A∗ψ2
= cψ1,ψ2−1Aψ1◦ A∗ψ2
= Pψ1,ψ2
Moreover, Aψψ ∗ Aψψ = Aψψ, we deduce that the convolution operator T −→ T ∗ Aψψis the orthonormal projection from L2(G, dµ) onto Im(Aψ). So we have another way to obtain the image of Aψ.
3.4 Example
In this example we want to show the coherence between the abstract theory and the usual wavelet transform, so here H = L2(R).
Let G = {(a, b) : a 6= 0, b ∈ R} with the group law (a, b) (a0, b0) = (a a0, b+ a b0). G is locally compact and the left Haar measure on G is dµ = a−2da db. Then we define:
π: G −→ U(L2(R))
(a, b) −→ (f → (f(a,b): x → √1af(x−ba )))
In fact, to prove that π is square integrable and irreducible in view to apply the construction above, it requires a hard analytical argument that repeats much of the classical argument used to prove the same thing. The success of the classical wavelet transform consists in the fact that for some ψ∈ L2(R), if we consider the subgroup of G defined by H = {(2j, k2j) : j∈ Z and k ∈ Z}
then π(H)ψ is an orthonormal basis.
4 A way of discretisation: the frames
As we saw in the exampe above, the wavelet transform contains too much infor- mation, that is to say we do not need all the coefficients to recover the function.
At the same time it does not seem possible in general case to find a subgroup that provides an orthonormal basis.
So we want to find an admissible vector ψ and a subset H⊂ G (if possible discrete and better a discrete subgroup) with a new measure dν such that f = R
Hc(f )(h) π(h)ψ dν(h). Of course it is better if c(f ) has an easy formula. A first idea is to discretize the formula of AψA∗ψT, for T ∈ L2(G, dµ), using the Riemann sum, but the convergence is pointwise, so it is dificult to come back in H, and G has to be compact.
So we are going to use a method developed by H.G. Feichtinger and K.
Gr¨ochenig to construct atomic decomposition in Banach space, related to the integrable group representation in view to built frames in H.
4.1 Generalities about frames
Definition 4.1. A collection of elements Ψ = (ψi)i∈I is a frame if there exist constants a and b, 0 < a < b <∞, such that:
akfk2≤X
i∈I
| < ψi, f >|2≤ bkfk2 ∀f ∈ H
If a = b the frame is called tight, and if a = b = 1 we obtain an orthonormal basis.
Then we also define the frame operator:
F : H −→ l2(I)
f −→ {< ψ(i), f >}i
SinceF is linear and kFk ≤√
b,F has an adjoint:
F∗: l2(I) −→ H {ci}i∈I −→ P
iciψi
Let Sf = F∗Ff = P
I < ψi, f > ψi, for all f ∈ H we have akfk2 ≤
< f,Sf > ≤ bkfk2. This shows that S is a positive bounded invertible linear operator and b−1kfk2 ≤ < f, S−1f >≤ a−1kfk2.
Lemma 4.2. IfΨ is a frame on H, eΨ =S−1Ψ is also a frame forH called the dual frame.
Proof. We have:
P
i| < eψi, f >|2 = P
i| < ψj,S−1f >|2 = kFS−1fk22
= < (F∗F)(F∗F)f, (F∗F)f > = < f, S−1f >
So the result follows.
Theorem 4.3. Let(ψi)i∈I be a frame, we have:
f =X
i∈I
< ψi, f > eψi=X
i∈I
< eψi, f > ψi for all f ∈ H With convergence in H
Proof. Let < f, g >]=< f,S−1g > for f ,g in H, this defines an inner product onH equivalent to < ., . > and such thatP
i| < ψi, f >|2=kfk2]. So ψi is an orthonormal basis for < ., . >]and we have:
f =X
i
< ψi, f >]ψi=X
i
< eψi, f > ψi with convergence inH
We also have f = P
i < ψi, f > eψi with weak convergence. To prove the strong convergence, we take J⊂ I a finite set and we have:
kP
i∈J< ψi, f > eψik = supkgk=1 D P
i∈J < ψi, f > eψi, gE
≤ supkgk=1 P
i∈J
< ψi, f > 212 P
i∈J
< eψi, g > 212
≤ supkgk=1kgk] P
i∈J
< ψi, f > 212
4.2 Construction of frames
Definition 4.4. A family (xi)I∈ GI is called U-dense (U ⊂ G) ifS
ixiU = G.
It is called V -separated if for some relatively compact neighbourhood V of the identity, the sets (xiV)I are pairwise disjoint. And it is called relatively- separated if it is a finite union of V -separated families.
Let NU((xi)I) = supj∈Icard{i ∈ I : xi ∈ xjU}, if (xi)i∈I is relatively- separated and U is relatively compact, then NU((xi)I) is finite
Definition 4.5. Let U ⊂ G be a neighborhood of the identity and let (xi)I
be a U -dense and relatively-separated subset. A bounded uniform partition of unity of size U (U -BUPU) is a family Θ = (θi)I such that P
iθi = 1 almost everywhere, 0≤ θi ≤ 1 and supp θi⊂ xiU for all i∈ I.
Definition 4.6. For U a neighborhood of identity, the U -oscillation of Aψψis:
∀x ∈ H ΩUAψψ(x) = sup
u∈U|Aψψ(ux)− Aψψ(x)|
Proposition 4.7. Let T ∈ L2(G, dµ), (xi)i∈I be a relatively-separated family, U be a neighborhood of the identity and Θ be a U -BUPU associated. Then we can define a bounded operator acting on L2(G, dµ):
TΘT = X
i
< θi, T >L2(G,dµ) τxiAψψ with convergence in L2(G, dµ)
Proof. Let J⊂ I be a finite set, we have:
P
i∈J
R
GθiT dµ
2
≤ P
i
R
xiU|T | dµ2
≤ µ(U)P
i∈J
R
xiU|T |2dµ
by Cauchy-Schwarz
≤ µ(U) N((xi)I)kT k22 since (xi)I is relatively separated So (< θi, T >)i∈I ∈ l2(I), since τxi is an isometry of L2(G, dµ), we deduce
that TΘ is well defined on L2(G, dµ), moreover,we have:
kTΘTk2≤ µ(U) N((xi)I)kT k22kAψψk2 SoTΘ is bounded and its norm does not depend on Θ.
Theorem 4.8. Let (xi)i∈I, be a relatively-separated family, U be a neighbor- hood of the identity, Θ be a U -BUPU associated and ψ ∈ A such that Aψψ ∈ L1(G, dµ), ΩUAψψ ∈ L1(G, dµ), kΩUAψψk1 < 1 and cψ,ψ = 1. Then there exists(ei)i∈I∈ HI such that:
f =X
i∈I
< ei, f > π(xi)ψ with convergence in H
Proof. Let T ∈ Im(Aψ) and x∈ G, since T = T ∗ Aψψwe have:
|T (x) − TΘT(x)| = |R
GT(y)h P
iθi(y)
Aψψ(y−1x)− Aψψ(x−1i x)i dµ(y)|
≤ R
G|T (y)| P
iθi(y)
supu∈U|Aψψ(y−1x)− Aψψ(u y−1x)|dµ(y)
≤ |T (x)| ∗ ΩUAψψ(x) So we have:
kT − TΘTk2≤ k|T | ∗ ΩUAψψk2≤ kT k2kΩUAψψk1
So if F ∈ L2(G, dµ)∗ Aψψ= Im(Aψ) and if the neighborhood U and ψ are chosen such that kΩUAψψk1 <1 then|||Id − TΘ||| < 1, so TΘ is invertible on Im(Aψ).
Consequently for f ∈ H, Aψf = TΘTΘ−1Aψf = P
i < θi,TΘ−1Aψf >
τxiAψψ= Aψ P
i< θi,TΘ−1Aψf > π(xi)ψ
. Thanks to the inversion formula, we have:
f =X
i
< θi,TΘ−1Aψf > π(xi) ψ
For all i∈ I, f −→< θi,TΘ−1Aψf >is a bounded linear functional onH, so
∃ei∈ H such that < θi,TΘ−1Aψf >=< ei, f > ∀f ∈ H. Now, we have:
f =X
i
< ei, f > π(xi) ψ
We also have the dual formula f =P
i< π(xi)ψ, f > eiwith weak conver- gence. Of course we want to have convergence in H, that why we are going to try the same argument as for the frames. So let J⊂ I a finite set, we have:
kP
i∈J< π(xi)ψ, f > eik = supkgk=1
D P
i∈J < π(xi)ψ, f > ei, gE
≤ supkgk=1
P
i∈J
< π(xi)ψ, f > 212 P
i∈J
< ei, g > 212
≤ supkgk=1
P
i∈J
< π(xi)ψ, f > 212 P
i∈I
< θi,TΘ−1g > 212
≤ supkgk=1
P
i∈J
< π(xi)ψ, f > 212p
µ(U )N ((xi)I|||TΘ−1Aψ||| kgk
≤ p
µ(U )N ((xi)I |||TΘ−1Aψ||| P
i∈J
< π(xi)ψ, f > 212
Nevertheless, in this argument we must assume that (< π(xi)ψ, f >)i∈I ∈ l2(I). In fact, we can always find a such family, but it is difficult to understand within this abstract argument, why in the case of usual wavelets in L2(R), the subgroup H ={(2j, k2j : j∈ Z and k ∈ Z} is enough.
Moreover, by this argument we do not prove that we have a frame although we have a recovery of the initial vector.
Conclusion
Summarily, the wavelet transform makes a connection between an abstract Hilbert space and L2(G, dµ). Abstract harmonic analysis provides us power- ful tools on L2(G, dµ), like the abstract Fourier transform, and makes it is easier to study the abstract Hilbert space. In particular, if G is commutative, we have an extention of the Poisson’s formula and when ψ an admissible vector and H is a subgroup of G we obtain a necessary condition for H.ψ being an or- thonormal family. This gives a departure point for a more general construction of multiresolution analysis.
There are other ways to improve this generalization. Indeed, if we study more the representation theory, we can see that the section 3 can be generalized to Banach space since so can the Schur’s lemma. Moreover, in the section 4, we can look for a reconstruction of an element of coorbit of Banach space (If Y ⊂ H is a Banach space, its coorbit is CoY = {f ∈ H : Aψf ∈ Y }).
To finish, I conclude by saying that this training period has introduced me into an interesting subject, the abstract harmonic analysis through wavelet transform, which has links with a great deal of other mathematical subjects and which can be used in areas as different as geometry and dynamical systems.
Moreover, I thank Kyril Tintarev, the director of my tainning period, for having welcomed me and for the help that he gave me during my trainning period.
References
[1] E. Hern´endez, G. Weiss: A first course on Wavelets, CRC Press, (1995).
[2] Y. Meyer: Ondelettes et operateurs, Hermann, (1990).
[3] H. G. Feichtinger, K. Gr¨ochenig: Non orthogonal wavelet and Gabor expan- sions, and group representations, Jones and Bartlett Publishers, (1992).
[4] P. Auscher: Wavelet bases for L2(R) with rational dilation factor, Jones and Bartlett Publishers, (1992).
[5] A. Weil: L’integration dans les groupes topologiques, Hermann, (1965).
[6] M. Sugiura: Unitary representations and harmonic analysis, Halsted press, (1975)
[7] A. Kirillov: Elements of the theory of representation, Springer, Berlin, (1976)
