1.3 Wavelet Bases on Bounded Domains
2.1.2 Gelfand Frames
As we have seen in Chapter 1, wavelet Riesz bases Ψ are almost always constructed inL2(Ω) first. Then in a second step, utilizing approximation and smoothness prop-erties of the corresponding approximating spacesSj and ˜Sj, it is shown that appro-priately rescaled versions of the system Ψ are also Riesz bases in other smoothness spaces.
It is now our aim to generalize this very strategy towards the case of frames.
Namely, we shall introduce a special class of frames in a Hilbert spaceV that, when rescaling the frame elements appropriately, gives rise to a frame for another Hilbert space H. We will have to investigate whether the expansion (2.1.9) also converges in other topologies than in that ofV.
To this end, we shall recall first a powerful concept from [86, 89]. Given a separa-ble and reflexive Banach spaceB with normed dual B′, a system G ={gθ}θ∈Θ ⊂B′ with associated Banach sequence space b is called a Banach frame for B, if the coefficient operator
G:B →b, f 7→ hf,Gi⊤B×B′ (2.1.13) is bounded with the norm equivalence
kGfkb hkfkB, f ∈B, (2.1.14) and there exists a bounded left–inverseR:b→B ofG, the so–calledreconstruction operator. In fact, the concept of Banach frames generalizes that of Hilbert frames, as is easily shown in the following lemma.
Lemma 2.1. Any frame Ψ = {ψλ}λ∈J for a separable Hilbert space V is also a Banach frame for V, with associated sequence space ℓ2 =ℓ2(J).
Proof. Given a frame Ψ for V, let G := ˜Ψ ⊂ V be the canonical dual frame.
IdentifyingV andV′ via the Riesz map, we can assume thatG ={h·,ψ˜λiV}λ∈J ⊂V′. The operatorGfrom (2.1.13) then turns out to be the dual analysis operator ˜F from (2.1.8), so that we can choose the reconstruction operatorR =F∗, which is a left–
inverse of ˜F.
For the discretization of elliptic operator equations, we will have to work with expansions both in the Hilbert space V and in a densely embedded Banach space B ֒→V. More precisely, we shall work in a Gelfand triple situation (B, V, B′) where
B ⊂V ≃V′ ⊂B′, (2.1.15)
and we assume explicitly that the right inclusion is also dense. In particular, this holds if B is also a Hilbert space.
A special class of frames tailored to this Gelfand frame situation has been intro-duced in [51]. Namely, a given frame Ψ forV with canonical dual frame ˜Ψ is called aGelfand frame for the Gelfand triple (B, V, B′), if Ψ⊂B, ˜Ψ⊂B′ and there exists a Gelfand triple b, ℓ2(J), b′
of sequence spaces such that
F∗ :b →B, F∗c=c⊤Ψ and F˜ :B →b, F f˜ = hf,ψ˜λiB×B′
λ∈J (2.1.16) are bounded operators. By duality arguments in the Gelfand triple (B, V, B′), we can infer from the identity ˜F∗F =F∗F˜ = idV that for a Gelfand frame Ψ, also the operators
F˜∗ :b′ →B′, F˜∗c=c⊤Ψ and˜ F :B′ →b′, F f = hf, ψλiB′×B
λ∈J (2.1.17) are bounded, cf. [92]. For a visualization of the various mappings in a Gelfand frame, we refer the reader to Figure 2.1.
B //
Fe
V //B′ F
b //
F∗
OO
ℓ2 //b′ Fe∗
OO
Figure 2.1: Mapping diagram for a Gelfand frame Ψ
The next result clarifies the relations between Gelfand and Banach frames:
Proposition 2.2. If Ψ is a Gelfand frame for (B, V, B′) with the Gelfand triple of sequence spaces (b, ℓ2(J), b′), then Ψ˜ and Ψ are Banach frames for B and B′, respectively.
Proof. We only show that ˜Ψ is a Banach frame forB, since the second claim follows by an analogous argument. By definition of a Gelfand frame, it is ˜Ψ⊂B′, and the coefficient operator G: B → b is given by Gf = ˜F f = hf,Ψ˜i⊤B×B′. Since ˜Ψ is the canonical dual of Ψ, we have by (2.1.9) the representation f = hf,Ψ˜iVΨ for each f ∈V, with convergence inV. But forf ∈B, we havehf,ψ˜λiV =hf,ψ˜λiB×B′ by the definition of the dual pairing. Since, moreover, F∗F f˜ ∈ B by the boundedness of F∗ and ˜F, the series P
λ∈Jhf,ψ˜λiB×B′ψλ =F∗F f˜ =f also converges inB. Finally, this yields the injectivity ofG= ˜F
kfkB =X
λ∈J
hf,ψ˜λiB×B′ψλ
B =kF∗F f˜ kB .kF f˜ kb .kfkB. A bounded reconstruction operatorR is of course given byF∗ :b→B.
2.1. HILBERT AND GELFAND FRAMES 41 In the original definition of Gelfand frames, the sequence spaces b and ℓ2 are completely unrelated, up to the dense and continuous embedding b ֒→ ℓ2. This is mainly due to the fact thatb is in general an arbitrary Banach sequence space and therefore structurally different from the Hilbert space ℓ2. However, in applications we may assume that b has more stringent structural properties. As an example, when it comes to frame discretizations of elliptic differential operators as in Chapter 5, the role of b will be played by a weighted ℓ2 space. In view of these situations, for any strictly positive diagonal matrixW= diag(wλ)λ∈J >0 and 1< p <∞, let us introduce the weighted ℓp spaces
ℓp,W :={c:kckℓp,W :=kWckℓp <∞}. (2.1.18) Then we can immediately relateℓ2,W and ℓ2 by an isomorphism
ϕW :ℓ2,W →ℓ2, ϕWc=Wc (2.1.19) with structurally identical dual mapping
ϕ∗W :ℓ2 →ℓ2,W−1, ϕ∗Wc=Wc. (2.1.20) Both mappings ϕW and ϕ∗W have operator norm 1 in the respective topologies.
Given that the abstract sequence spaceb in the Gelfand frame definition is in fact such a weighted ℓ2 space, we can strengthen Proposition 2.2 substantially in the following way:
Proposition 2.3. Let H be a Hilbert space and Ψ = {ψλ}λ∈J be a Gelfand frame for (H, V, H′) with the Gelfand triple of sequence spaces (ℓ2,W, ℓ2, ℓ2,W−1), where w : J → R+ is a strictly positive weight function. Then the systems W−1Ψ = {wλ−1ψλ}λ∈J andWΨ =˜ {wλψ˜λ}λ∈J are (Hilbert) frames forH andH′, respectively.
Proof. We only prove that W−1Ψ is a Hilbert frame in H, since the other claim follows by analogy.
Firstly, Ψ being a Gelfand frame for (H, V, H′), we know that the operator F∗ : ℓ2,W → H, F∗c = c⊤Ψ, is bounded. The composition T := F∗ϕ−W1 : ℓ2 → H, being the synthesis operator of the system W−1Ψ ⊂ H, is also bounded, so that W−1Ψ is a Bessel system inH which is equivalent to the validity of the upper frame bound, see also [29].
Concerning the lower frame bound, we utilize from Proposition 2.2 that ( ˜Ψ, ℓ2,W) is a Banach frame forH, with bounded reconstruction operatorR =F∗ :ℓ2,W →H.
SinceF∗ is onto, this is also the case for the bounded operatorT :=F∗ϕ−W1 :ℓ2 →H already considered above. Hence, for the bounded pseudoinverse T† =ϕWF˜ :H→ ℓ2, we have f =T T†f, so that the lower frame bound follows from
kfk4H = hT T†f, fiH2
= X
λ∈J
(T†f)λw−λ1ψλ, f
H
2
≤ X
µ∈J
(T†f)µ
2X
λ∈J
hf, w−λ1ψλiH
2
≤ kT†k2L(H,ℓ2)kfk2H
X
λ∈J
hf, w−λ1ψλiH
2.
Remark 2.4. Proposition 2.3 also sheds some light on the slightly different settings in [51, 133]. In both papers, frames are considered for the numerical discretization of an elliptic operator equation Au=f with an isomorphism A:H →H′.
In[133], a frame Ψfor H is used without explicitly requiring that a rescaled ver-sion ofΨ is also anL2–frame. Moreover, the energy space H is implicitly identified with its normed dual H′ via the Riesz mapping. For the numerical approximation of u in H, this setting is sufficient since a frame Ψ for H is automatically H–
dense. The assumptions then cover also the case where the underlying frame is not of wavelet type, cf. [123, 124].
Conversely, the more restricted setting of Gelfand frames in [51], as introduced in this section, conforms to the well–known constructions of wavelet Riesz bases.
The results in loc. cit. make use of localization arguments that require the canonical dual frameΨ˜ to be contained and to be stable in L2, see Section 2.2. Moreover, it is assumed there to have a Gelfand triple setting where the middle spaceL2 is identified with its normed dual.