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DIMENSION FUNCTIONS OF RATIONALLY DILATED
GMRAS AND WAVELETS
MARCIN BOWNIK AND KENNETH HOOVER
Abstract. In this paper we study properties of generalized multiresolution analyses (GM- RAs) and wavelets associated with rational dilations. We characterize the class of GMRAs associated with rationally dilated wavelets extending the result of Baggett, Medina, and Merrill. As a consequence, we introduce and derive the properties of the dimension func- tion of rationally dilated wavelets. In particular, we show that any mildly regular wavelet must necessarily come from an MRA (possibly of higher multiplicity) extending Auscher’s result from the setting of integer dilations to that of rational dilations. We also character- ize all 3 interval wavelet sets for all positive dilation factors. Finally, we give an example of a rationally dilated wavelet dimension function for which the conventional algorithm for constructing integer dilated wavelet sets fails.
- Introduction The wavelet dimension function is an important subject in the theory of wavelets. For a given orthonormal wavelet Ψ = {ψ^1 ,... , ψL} ⊂ L^2 (RN^ ) associated with an integer expansive dilation A ∈ MN (Z), its dimension function is defined as
(1.1) DΨ(ξ) =
∑^ L
`=
∑^ ∞
j=
k∈ZN
|̂ψ`((Aᵀ)j^ (ξ + k))|^2.
Initially, the wavelet dimension function was introduced in the one dimensional dyadic case by Lemari´e-Rieusset [31, 32] to prove that all compactly supported wavelets are associated with a multiresolution analysis (MRA). After that Gripenberg [28] and Wang [38] used it to characterize all wavelets arising from an MRA. A further application of the wavelet dimension function is due to Auscher [2] who proved two fundamental results: (i) there are no regular wavelet bases for the Hardy space H^2 (R), and (ii) any mildly regular wavelet basis of L^2 (RN^ ) must arise from an MRA (possibly of higher multiplicity). Furthermore, Auscher established that the wavelet dimension function DΨ describes dimensions of certain subspaces of `^2 (ZN^ ), and hence it is integer-valued. A systematic study of properties of the wavelet dimension function (often called multiplicity function) was initiated by Baggett, Medina, and Merrill [3, 6] who introduced the concept of a generalized multi-resolution analysis (GMRA), and studied its relation to wavelets. In
Date: June 23, 2008. 2000 Mathematics Subject Classification. Primary: 42C40. Key words and phrases. wavelet, GMRA, rational dilation, wavelet dimension function. The first author was partially supported by the NSF grant DMS-0653881. 1
2 MARCIN BOWNIK AND KENNETH HOOVER
particular, they established the consistency equation for a multiplicity function
(1.2)
ω∈[(Aᵀ)−^1 ZN^ /ZN^ ]
D(ξ + ω) = L + D(Aᵀξ) for a.e. ξ ∈ RN^ ,
where we use the convention that [G/H] denotes a transversal (a set of representatives of distinct cosets) of a quotient group G/H. In a related development, Bownik, Rzeszotnik, and Speegle [22] characterized all possible integer-valued functions which are dimension functions of a wavelet. It turns out that every wavelet dimension function must satisfy an additional condition,
(1.3)
k∈ZN
(^1) ∆(ξ + k) ≥ DΨ(ξ) for a.e. ξ ∈ RN^ ,
where ∆ =
ξ ∈ RN^ : DΨ
(Aᵀ)−j^ ξ
≥ 1 for all j ∈ N ∪ { 0 }
. A similar result in the context of GMRAs was obtained by Baggett and Merrill [7], and then further generalized by Bownik and Rzeszotnik in [19]. In addition, the wavelet dimension function was extensively studied by a number of other authors including Behera [8], Paluszy´nski, Siki´ˇ c, Weiss, and Xiao [33], Ron and Shen [36], and Weber [39]. As a result, the wavelet dimension function found a natural interpretation using the theory of shift-invariant spaces. However, up to the present time, the wavelet dimension function has been studied exclu- sively for the class of integer dilations. The only exception is the work of Bownik and Speegle [23], who introduced an analogue of the wavelet dimension function for real dilations factors in one dimension. The goal of this paper is to initiate the study of the wavelet dimension func- tions in higher dimensions. We will concentrate on the class of rational expansive dilations A ∈ MN (Q), since most non-rational dilations admit only minimally supported frequency (MSF) wavelets due to results of Chui and Shi [25] and Bownik [14]. For every rational expansive dilation A there exists a plenty of non-MSF wavelets by a result of Bownik and Speegle [23]. Moreover, in one dimension Auscher [1] constructed rationally dilated wavelets with nice localization and smoothness properties. On the other hand, every MSF wavelet (or more generally combined MSF wavelet) has its space of negative dilates invariant under all translations. Hence, one can easily define the wavelet dimension function for such wavelets. However, this is rarely done since more information is carried by the wavelet set itself than by the wavelet dimension function. There are some significant differences in the theory of dimension functions between the already well-understood case of integer dilations and that of rational dilations. The most prominent manifestation of that is a surprising gain of shift-invariance of GMRAs associated with wavelets. Namely, any GMRA associated with rationally dilated wavelet Ψ must be Γ-SI with Γ = AZN^ + ZN^. Note that in the classical case of A ∈ MN (Z), this self-improvement is non-existent. In addition, the corresponding multiplicity function must satisfy an analogue of the consistency equation (1.2). This leads to a characterization of GMRAs associated with rationally dilated wavelets extending the earlier result of Baggett, Medina, and Merrill [6]. As a consequence of this result we establish the formula for the wavelet dimension function as
(1.4) DΨ(ξ) =
∑^ L
`=
∑^ ∞
j=
k∈Γ∗
|̂ψ`((Aᵀ)j^ (ξ + k))|^2 ,
4 MARCIN BOWNIK AND KENNETH HOOVER
Parseval frame for L^2 (RN^ ), i.e.,
||f ||^2 =
∑^ L
`=
j∈Z
k∈ZN
|〈f, ψj,k` 〉|^2 for all f ∈ L^2 (RN^ ),
and different scales of A(Ψ) are mutually orthogonal, i.e.,
〈ψj,k, ψ
′ j′,k′^ 〉^ = 0^ for all^ ,
′ (^) = 1,... , L, j 6 = j′ (^) ∈ Z, k, k′ (^) ∈ ZN (^).
2.1. Shift-invariant spaces and the spectral function. The general properties of SI spaces were studied by a number of authors, see [11, 13, 34]. Here, we only list the results that will be used later on.
Definition 2.2. Suppose that Γ is a (full rank) lattice, i.e, Γ = P ZN^ , where P ∈ MN (R) is an N × N invertible matrix. We say that a closed subspace V ⊂ L^2 (RN^ ) is shift invariant (SI) with respect to the lattice Γ, if
f ∈ V =⇒ Tγ f ∈ V for all γ ∈ Γ.
Given a countable family Φ ⊂ L^2 (RN^ ) and the lattice Γ we define the Γ-SI system EΓ(Φ) by
EΓ(Φ) = {Tγ ϕ : ϕ ∈ Φ, γ ∈ Γ}.
When Γ = ZN^ , we often drop the superscript Γ, and we simply say that V is SI. Likewise, E(Φ) means EZ
N (Φ). The spectral function of SI spaces, which was introduced by Bownik and Rzeszotnik in [19], can be defined in several equivalent ways using range functions or dual Gramians. The following result, see [20, Lemma 2.5], can also serve as a definition.
Lemma 2.3. If V ⊂ L^2 (RN^ ) is Γ-SI and Φ ⊂ V is a countable family such that EΓ(Φ) is a Parseval frame for V , then its spectral function is
(2.1) σΓ V (ξ) =
|RN^ /Γ|
ϕ∈Φ
| ϕˆ(ξ)|^2 ,
where |RN^ /Γ| is the Lebesgue measure of the fundamental domain of RN^ /Γ, i.e., |RN^ /Γ| = | det P | if Γ = P ZN^. In particular, (2.1) does not depend on the choice of Φ as long as EΓ(Φ) is a Parseval frame for V.
Clearly, if V ⊂ L^2 (RN^ ) is Γ-SI, then V is also Γ′-SI for any lattice Γ′^ ⊂ Γ. Hence, one can also talk about the spectral function σΓ ′ V with respect to a sparser lattice Γ ′. Nevertheless,
by a result of Bownik and Rzeszotnik [20, Corollary 2.7] both of these spectral functions coincide.
Theorem 2.4. Suppose Γ′^ ⊂ Γ are two lattices and V ⊂ L^2 (RN^ ) is Γ-SI. Then,
σΓ V (ξ) = σΓ
′ V (ξ)^ for a.e.^ ξ^ ∈^ R
N .
Consequently, we can drop the dependence of the spectral function of V on a lattice Γ and simply write σV instead of σΓ V. This shows that the spectral function is a very fundamental notion of “size” of SI spaces which is independent of the underlying lattice Γ. In contrast, when working with the dimension function of a shift-invariant space, one needs to use the formula (2.2) when considering a sublattice Γ′^ ⊂ Γ, see [17, Lemma 2.4].
DIMENSION FUNCTIONS OF RATIONALLY DILATED GMRAS AND WAVELETS 5
We recall that dimΓ V : RN^ → N∪{ 0 , ∞} is the multiplicity function of the projection-valued measure coming from the representation of Γ on V via translations by Stone’s Theorem [3, 6]. Alternatively, one can define dimΓ V (ξ) = dim span{( ˆϕ(ξ + k))k∈Γ∗ : ϕ ∈ Φ}, where Φ ⊂ V is a countable set of generators of V , i.e., V = spanEΓ(Φ). Lemma 2.5. Suppose Γ′^ ⊂ Γ are two lattices and V ⊂ L^2 (RN^ ) is Γ-SI. Then, (2.2) dimΓ
′ V (ξ) =^
ω∈[(Γ′)∗/Γ∗]
dimΓ V (ξ + ω) for a.e. ξ ∈ RN^.
Here, Γ∗^ = {x ∈ RN^ : 〈x, k〉 ∈ Z for all k ∈ Γ} is the dual lattice of Γ, and [(Γ′)∗/Γ∗] is a transversal of (Γ′)∗/Γ∗. The following theorem summarizes properties of the spectral function. Theorem 2.6. Let S be the collection of all possible SI subspaces of L^2 (RN^ ), i.e., V ∈ S if and only if there exists a lattice Γ such that V is Γ-SI. The spectral function satisfies the following properties: (V, W ∈ S)
(a) σV : RN^ → [0, 1] is a measurable function, (b) V =
⊕N
i=1 Vi, where^ Vi^ ∈^ S^ =⇒^ σV^ (ξ) =^
∑N
i=1 σVi (ξ), (c) V =
i∈N Vi, where^ Vi^ is^ Γ-SI for a fixed lattice^ Γ =⇒^ σV^ (ξ) =^
i∈N σVi (ξ), (d) V ⊂ W =⇒ σV (ξ) ≤ σW (ξ), (e) V ⊂ W =⇒ (V = W ⇐⇒ σV (ξ) = σW (ξ)), (f ) σV (ξ) = (^1) E (ξ) ⇐⇒ V = Lˇ^2 (E), where E ⊂ RN^ is a measurable set. (g) V ⊂ Lˇ^2 (E), where E = supp σV , (h) σDAV (ξ) = σV ((Aᵀ)−^1 ξ), where A ∈ MN (R) is invertible, (i) if V is Γ-SI then dimΓ V (ξ) =
k∈Γ∗^ σV^ (ξ^ +^ k). Proof. The proof of Theorem 2.6 can be found in [19, 20] with the exception of (g). To see (g), let Φ be the same as in Lemma 2.3. Take any f ∈ V. Then, fˆ is an L^2 limit of functions of the form
ϕ∈Φ rϕ(ξ) ˆϕ(ξ), where only finitely many of the Γ
∗-periodic trigonometric polynomials
rϕ are non-zero. Consequently, fˆ (ξ) = 0 for all ξ 6 ∈ supp σV , which proves (g). 2.2. Generalized multiresolution analyses. The concept of a generalized multiresolution analysis (GMRA) was introduced by Baggett, Medina, and Merrill in their seminal work [6]. Its original formulation requires that the expansive dilation A preserves the underlying lattice Γ, i.e., AΓ ⊂ Γ. By a standard dilation argument this can be reduced to the case of the standard lattice ZN^ and an integer dilation A. In this work we are interested in a larger class of expansive dilations which do not necessarily preserve the lattice Γ, but nevertheless A ∩ AΓ is a (full rank) lattice. A reduction to the case of the standard lattice Γ = ZN^ corresponds precisely to the class of rational dilations A. While the definition below does not mention this assumption explicitly, all of our results involve only rationally dilated GMRAs. Definition 2.7. A sequence {Vj }j∈Z of closed subspaces of L^2 (RN^ ) is called a generalized multiresolution analysis (GMRA) if
DIMENSION FUNCTIONS OF RATIONALLY DILATED GMRAS AND WAVELETS 7
Definition 2.9. Suppose that Ψ = {ψ^1 ,... , ψL} ⊂ L^2 (RN^ ) and A is a rational dilation. We define the quasi-affine system associated with (Ψ, A) by
Aq(Ψ) =
j∈Z
OA
−j (^) ZN (Dj^ Ψ)
=E
j∈Z
θ∈Θj
|ZN^ /(ZN^ ∩ A−j^ ZN^ )|^1 /^2
TθDj^ Ψ
where Θj is a transversal of (ZN^ + A−j^ ZN^ )/ZN^.
Even though the orthogonality of an affine system is not preserved by the corresponding quasi-affine system, it turns out the Parseval frame property still carries over between affine and quasi-affine systems. This result was first established in [15, Theorem 3.4]. A different proof of Theorem 2.10 was given in [29, Theorem 2.17].
Theorem 2.10. Suppose that Ψ = {ψ^1 ,... , ψL} ⊂ L^2 (RN^ ). The affine system A(Ψ) is a Parseval frame if and only if its quasi-affine counterpart Aq(Ψ) is a Parseval frame.
We define the negative part of the quasi-affine system Aq(Ψ) as
Aq −(Ψ) =
j< 0
OA
−j (^) ZN (Dj^ Ψ)
=E
j< 0
θ∈Θj
|ZN^ /(ZN^ ∩ A−j^ ZN^ )|^1 /^2
TθDj^ Ψ
where Θj is the same as in Definition 2.9. We will need the following result about Aq −(Ψ).
Lemma 2.11. Suppose that Ψ = {ψ^1 ,... , ψL} ⊂ L^2 (RN^ ) is a rationally dilated semi-ortho- gonal wavelet such that its space of negative dilates
(2.6) V (Ψ) = span {ψj,k : j < 0 , k ∈ ZN^ , = 1,... , L}
is ZN^ -shift invariant. Then the system Aq −(Ψ) forms a Parseval frame for V (Ψ).
Proof. Define the lattice Γj for each j ∈ Z by Γj = ZN^ + A−j^ ZN^ and let Mj ∈ MN (Q) be such that Mj ZN^ = Γj. Then we can express the quasi-affine system as
Aq(Ψ) = { ψ˜j,k: j ∈ Z, k ∈ ZN^ , = 1,... , L},
where for each j ∈ Z, k ∈ ZN^ , and ` = 1,... , L, we define
ψ˜` j,k(x) =^
| det A|j/^2 |ZN^ /(ZN^ ∩ A−j^ ZN^ )|^1 /^2
ψ`(Aj^ (x + Mj k)).
In addition to Aq −(Ψ), we consider the positive part of the quasi-affine system
Aq +(Ψ) = { ψ˜j,k: j ≥ 0 , k ∈ ZN^ , = 1,... , L}.
We claim that
(2.7)
V (Ψ) =span Aq −(Ψ), (V (Ψ))⊥^ =span Aq +(Ψ).
8 MARCIN BOWNIK AND KENNETH HOOVER
Indeed, by the semi-orthogonality of Ψ we have
(2.8) (V (Ψ))⊥^ = span {ψj,k : j ≥ 0 , k ∈ ZN^ , = 1,... , L}.
Given any k 1 ∈ ZN^ , we have A−j^ k 1 ∈ A−j^ ZN^ ⊂ ZN^ + A−j^ ZN^ = Γj and so there is some k 2 ∈ ZN^ such that Aj^ Mj k 2 = k 1 , implying that ψj,k 1 = |ZN^ /(ZN^ ∩ A−j^ ZN^ )|^1 /^2 ψ˜j,k 2. Thus,
we obtain the inclusions ⊂ in (2.7). On the other hand, since ZN^ ⊂ M (^) j− 1 Γj = M (^) j− 1 ZN^ +
M (^) j− 1 A−j^ ZN^ , then there are γ, β ∈ ZN^ such that M (^) j− 1 γ + M (^) j− 1 A−j^ β = k 1 , implying that
ψ˜`j,k 1 =^
1 |ZN^ /(ZN^ ∩A−j^ ZN^ )|^1 /^2 Tγ^ ψ
` j,β. Since^ V^ (Ψ) is shift invariant, so is (V^ (Ψ)) ⊥. This yields that
ψ˜j,k 1 ∈^ V^ (Ψ) when^ j <^ 0, and ψ˜j,k 1 ∈^ (V^ (Ψ))
⊥ (^) when j ≥ 0. Thus, we obtain the reverse
inclusions ⊃ in (2.7). By our hypothesis, the affine system A(Ψ) is a Parseval frame for L^2 (RN^ ). This implies by Theorem 2.10 that Aq(Ψ) is also a Parseval frame. Since Aq(Ψ) = Aq −(Ψ) ∪ Aq +(Ψ), (2.7) implies that Aq −(Ψ) and Aq +(Ψ) are Parseval frames for V (Ψ) and (V (Ψ))⊥, respectively.
- Spectral function of wavelets The goal of this section is to establish the formula for the spectral function of rationally dilated wavelets. Hence, we wish to show the following generalization of a result of Bownik and Rzeszotnik [19, Theorem 4.2].
Theorem 3.1. Suppose that Ψ = {ψ^1 ,... , ψL} ⊂ L^2 (RN^ ) is a rationally dilated semi- orthogonal wavelet such that its space of negative dilates V (Ψ) is ZN^ -shift invariant. Then,
(3.1) σV (Ψ)(ξ) =
∑^ L
`=
∑^ ∞
j=
|̂ψ`((Aᵀ)j^ ξ)|^2.
In the proof of Theorem 3.1 we will need the following elementary result about rational lattices.
Lemma 3.2. Suppose that Γ is a rational lattice, i.e., Γ = P ZN^ for some invertible P ∈ MN (Q). Then, we have the isomorphism
(3.2) (ZN^ + Γ)/ZN^ ' Γ/(ZN^ ∩ Γ).
Moreover,
(3.3) |ZN^ /(ZN^ ∩ Γ)| = | det P | · |Γ/(ZN^ ∩ Γ)|.
Proof. To see (3.2) one can use the second isomorphism theorem for groups, whereas (3.3) follows from the elementary properties of the volume d(Γ) = | det P | of a lattice Γ = P ZN^. In particular, if Γ′^ ⊂ Γ are two lattices, then the quotient group Γ/Γ′^ has the order d(Γ′)/d(Γ), see [24].
Proof of Theorem 3.1. Using Lemmas 2.3 and 2.11, we can compute the spectral function as
(3.4) σZ N V (Ψ)(ξ) =^
|RN^ /ZN^ |
∑^ L
`=
j< 0
θ∈Θj
|ZN^ /(ZN^ ∩ A−j^ ZN^ )|
|(T̂θDj^ ψ`)(ξ)|^2 ,
10 MARCIN BOWNIK AND KENNETH HOOVER
- GMRAs and wavelets The following result is quite surprising, since it shows the self-improving property of GM- RAs associated with rationally dilated wavelets. Namely, the core space V 0 of such a GMRA enjoys more shift-invariance than the ordinary ZN^ -SI. We should mention here that the study of integer dilated wavelets with improved shift-invariance goes back to Weber [40], see also [9, 37]. Note that in the case when A is integer-valued, no such improvement exists. This might explain why this rather elementary phenomenon remained unnoticed until this work.
Lemma 4.1. Suppose a GMRA {Vj }j∈Z associated with an arbitrary real dilation A gives rise to a semi-orthogonal wavelet Ψ, i.e., V 0 is the space of negative dilates V (Ψ) of Ψ. Then, V 0 is Γ-SI, where Γ = AZN^ + ZN^.
One should note that for a general dilation A ∈ MN (R), Γ = AZN^ + ZN^ does not have to a lattice, that is a discrete subgroup of RN^. In this case, Lemma 4.1 says that V 0 is invariant under translations Ty by y ∈ Γ, and hence also by y ∈ Γ.
Proof. Since V 0 is ZN^ -SI, the space V 1 is A−^1 ZN^ -SI. Since
(4.1) V 1 = V 0 ⊕ W 0 , where W 0 = spanE(Ψ),
V 1 is ZN^ -SI as well. Hence, V 1 is (A−^1 ZN^ + ZN^ )-SI. Consequently, V 0 is (AZN^ + ZN^ )-SI.
Lemma 4.1 enables us to show the following extension of a theorem due to Baggett, Medina, and Merrill [6] to the case of rational dilations.
Theorem 4.2. Suppose that a dilation A ∈ MN (Q) and Ψ is a semi-orthogonal wavelet with L generators associated with a GMRA {Vj }j∈Z. Then,
(4.2) V 0 is Γ-SI, where Γ = AZN^ + ZN^ ,
and its Γ-dimension function D(ξ) = dimΓ V 0 (ξ) satisfies
(4.3) D(ξ) < ∞ for a.e. ξ,
and the consistency inequality
(4.4)
ω∈[(Aᵀ)−^1 ZN^ /Γ∗]
D(ξ + ω) ≤ L +
ω′∈[ZN^ /Γ∗]
D(Aᵀξ + ω′) for a.e. ξ.
In addition, if Ψ is a wavelet, then we have equality in (4.4), i.e.,
(4.5)
ω∈[(Aᵀ)−^1 ZN^ /Γ∗]
D(ξ + ω) = L +
ω′∈[ZN^ /Γ∗]
D(Aᵀξ + ω′) for a.e. ξ.
Conversely, if a GMRA {Vj }j∈Z satisfies (4.2), (4.3), and (4.4), then there exists a semi- orthogonal wavelet Ψ (with at most L generators) associated with this GMRA. In addition, if (4.5) holds, then Ψ is a wavelet with L generators.
Proof. Suppose that Ψ is semi-orthogonal wavelet with L generators which is associated with {Vj }j∈Z. Lemma 4.1 guarantees that V 0 is Γ-SI. On the other hand, Theorem 3.1 gives an explicit formula for the spectral function of V 0. Thus, Theorem 2.6(i) yields
(4.6) dimΓ V 0 (ξ) =
k∈Γ∗
σV 0 (ξ + k).
DIMENSION FUNCTIONS OF RATIONALLY DILATED GMRAS AND WAVELETS 11
Consequently,
∫
RN^ /Γ∗
dimΓ V 0 (ξ) =
RN
σV 0 (ξ)dξ =
∑^ L
`=
∑^ ∞
j=
RN
|̂ψ`((Aᵀ)j^ ξ)|^2 dξ
∑^ L
`=
∑^ ∞
j=
||ψ`||^2 | det A|j^ ≤ L/(| det A| − 1) < ∞.
Hence, (4.3) holds. By (4.1) and Theorem 2.6(h) we have σV 0 (ξ) + σW 0 (ξ) = σV 1 (ξ) = σD(V 0 )(ξ) = σV 0 ((Aᵀ)−^1 ξ).
In particular,
(4.7)
k∈ZN
σV 0 (ξ + k) +
k∈ZN
σW 0 (ξ + k) =
k∈ZN
σV 0 ((Aᵀ)−^1 (ξ + k)).
It remains to describe the quantities appearing in (4.7) in terms of D(ξ) = dimΓ V 0 (ξ). By the isomorphism ZN^ ' (ZN^ /Γ∗) × Γ∗,
(4.8)
k∈ZN
σV 0 (ξ + k) =
ω′∈[ZN^ /Γ∗]
γ∗∈Γ∗
σV 0 (ξ + γ∗^ + ω′) =
ω′∈[ZN^ /Γ∗]
D(ξ + ω′).
Since EZ N (Ψ) forms a Parseval frame for W 0 , and Ψ has L generators, we have
(4.9)
k∈ZN
σW 0 (ξ + k) = dimZ
N W 0 (ξ)^ ≤^ L.
Finally, the isomorphism (Aᵀ)−^1 ZN^ ' ((Aᵀ)−^1 ZN^ /Γ∗) × Γ∗^ yields
k∈ZN
σV 0 ((Aᵀ)−^1 (ξ + k)) =
ω∈[(Aᵀ)−^1 ZN^ /Γ∗]
γ∗∈Γ∗
σV 0 ((Aᵀ)−^1 ξ + γ∗^ + ω)
ω∈[(Aᵀ)−^1 ZN^ /Γ∗]
D((Aᵀ)−^1 ξ + ω).
Combining (4.7)–(4.10) yields ∑
ω′∈[ZN^ /Γ∗]
D(ξ + ω′) + L ≥
ω∈[(Aᵀ)−^1 ZN^ /Γ∗]
D((Aᵀ)−^1 ξ + ω),
which is equivalent with (4.4). In addition, if Ψ is a wavelet, then E(Ψ) is a orthonormal basis for W 0. Since Ψ has L generators, we have
(4.11)
k∈ZN
σW 0 (ξ + k) = dimZ
N W 0 (ξ) =^ L.
Consequently, we obtain (4.5). Conversely, let {Vj }j∈Z be a GMRA satisfying (4.2), (4.3), and (4.4). Define the space W 0 = V 1 V 0. Since V 0 is Γ-SI, V 1 = D(V 0 ) is A−^1 ZN^ -SI, and hence both spaces are ZN^ -SI.
DIMENSION FUNCTIONS OF RATIONALLY DILATED GMRAS AND WAVELETS 13
Proof. Suppose that (4.13) holds. Then we can go one step further and say that for all h < 0, j ≥ 0, k 1 , k 2 , m, n ∈ ZN^ , ,′^ = 1,... , L, we have
〈Tk 1 ψh,m, Tk 2 ψ
′ j,n〉^ =^ 〈Tk 1 −k 2 ψ
` h,m, ψ
`′ j,n〉^ = 0.
Hence, the SI spaces
W 1 =span {Tkψj,n: j < 0 , k, n ∈ ZN^ , = 1,... , L} W 2 =span {Tkψj,n: j ≥ 0 , k, n ∈ ZN^ , = 1,... , L}.
are orthogonal W 1 ⊥ W 2. Recall that V (Ψ) = span {ψj,n: j < 0 , n ∈ ZN^ , = 1,... , L} and, hence, (V (Ψ))⊥^ = span {ψj,n : j ≥ 0 , n ∈ ZN^ , = 1,... , L}. Clearly, we have V (Ψ) ⊂ W 1 and (V (Ψ))⊥^ ⊂ W 2.
Furthermore, since V (Ψ) ⊕ (V (Ψ))⊥^ = L^2 (RN^ ) and W 1 ⊥ W 2 , we must have W 1 = V (Ψ) and W 2 = (V (Ψ))⊥. Since W 1 is, by definition, shift invariant, then so is V (Ψ). Conversely, if V (Ψ) is SI, then so is (V (Ψ))⊥. Thus, W 1 = V (Ψ) and W 2 = (V (Ψ))⊥, and W 1 ⊥ W 2. This implies (4.13).
Theorem 4.4. Suppose Ψ = {ψ^1 ,... , ψL} ⊂ L^2 (RN^ ) is a wavelet associated with a dilation A ∈ MN (R). Then, V (Ψ) is shift invariant if for all h < 0 and j > 0 we have
(4.14) ZN^ ⊂ AhZN^ + Aj^ ZN^.
Proof. Take any h < 0, j ≥ 0, k, m, n ∈ ZN^ , ,′^ = 1,... , L. If j = 0, then
〈Tkψh,m, ψ ′ j,n〉^ =^ 〈Tkψ
` h,m, Tnψ
′ 〉 = 〈ψ h,m, Tn−kψ
′ 〉 = 〈ψ h,m, ψ
`′ j,n−k〉^ = 0.
If, on the other hand, j > 0 then we choose γ, β ∈ ZN^ such that k = A−hγ + A−j^ β. We are guaranteed the existence of such γ, β ∈ ZN^ by (4.14). Then,
〈Tkψh,m, ψ ′ j,n〉^ =^ 〈TA−hγ+A−j^ β D
hT mψ `, Dj (^) T nψ
`′ 〉
= 〈DhTm+γ ψ, Dj^ Tn−β ψ
′ 〉 = 〈ψh,m+γ , ψ
′ j,n−β 〉^ = 0.
Therefore, V (Ψ) is shift invariant by Lemma 4.3.
As an illustration we demonstrate how Theorem 4.4 can provide for the shift invariance of wavelets associated with certain classes of dilations. For instance, it is well known that the space of negative dilates of any wavelet associated with an integer dilation is shift invariant. Theorem 4.4 provides a very quick proof of this fact by simply noting that if A ∈ MN (Z), then ZN^ ⊂ AhZN^ for all h < 0. Slightly more interesting, however, is the case of diagonal rational dilations as we see below.
Proposition 4.5. Suppose Ψ = {ψ^1 ,... , ψL} ⊂ L^2 (RN^ ) is a wavelet associated with a dila- tion A ∈ MN (Q). If A is diagonal, then V (Ψ) is shift invariant.
Proof. Suppose A = diag
(p 1 q 1 ,... ,^
pN qN
where pi, qi ∈ Z with gcd(pi, qi) = 1 for each i =
1 ,... , N. Given any k = (k 1 ,... , kN ) ∈ ZN^ and any h < 0 and j > 0, choose (ω 1 ,... , ωN ) ∈ ZN^ satisfying the two following conditions for each i = 1,... , N ,
ωi ≡ − ki (mod p−i h) ωi ≡0 (mod qji ).
14 MARCIN BOWNIK AND KENNETH HOOVER
Then for each i = 1,... , N , we have
(pi qi
)h (ωi + ki) ∈ Z and
(pi qi
)j ωi ∈ Z. Hence, Ah(ω + k) ∈
ZN^ and Aj^ ω ∈ ZN^. Thus, for each m, n ∈ ZN^ and ,′^ = 1,... , L we have
〈Tkψh,m, ψ ′ j,n〉^ =^ 〈TkDhTmψ
`, Dj (^) T nψ
`′ 〉 = 〈T
ω+kDhTmψ
`, T
ωD j (^) T nψ
`′ 〉
= 〈DhTAh(ω+k)+mψ, Dj^ TAj^ ω+nψ
′ 〉 = 〈ψh,Ah(ω+k)+m, ψ
′ j,Aj^ ω+n〉^ = 0.
Therefore, V (Ψ) is shift invariant by Lemma 4.3.
It is important to understand, however, that Theorem 4.4 does not guarantee the shift invariance of V (Ψ) for every rationally dilated wavelet Ψ. The following simple example was communicated to us by Daniel Chan of the University of New South Wales, Australia.
Example 4.6. If A =
, then A is an expansive matrix that does not satisfy
(4.14) for h = − 1 and j = 1.
Proof. Note that A is expansive, since A^2 = 4Id. We claim that (4.14) fails for h = −1 and j = 1. Indeed, A−^1 = 14 A implies that
A−^1 Z^2 + AZ^2 =
AZ^2 + AZ^2 =
AZ^2.
Assuming that Z^2 ⊂ 14 AZ^2 , we have AZ^2 ⊂ 14 A^2 Z^2 = Z^2 , which is a contradiction.
- Dimension function of GMRAs The goal of this section is to derive the properties satisfied by a dimension function of any rationally dilated GMRA.
Theorem 5.1. Suppose that {Vj }j∈Z is a GMRA associated with the dilation A ∈ MN (Q).
Then, its dimension function D(ξ) = dimZ
N V 0 (ξ)^ satisfies the following four conditions: (D1) D : RN^ → N ∪ { 0 , ∞} is a measurable ZN^ -periodic function, (D2) D satisfies the consistency inequality
(5.1)
ω∈[(Aᵀ)−^1 Γ˜/ZN^ ]
D(ξ + ω) ≥
ω′∈[Γ˜/ZN^ ]
D(Aᵀξ + ω′) for a.e. ξ ∈ RN
where ˜Γ = ZN^ + AᵀZN^ , (D3)
k∈ZN^1 ∆(ξ^ +^ k)^ ≥^ D(ξ) for a.e.^ ξ^ ∈^ R
N (^) , where
∆ =
ξ ∈ RN^ : D
(Aᵀ)−j^ ξ
≥ 1 for all j ∈ N ∪ { 0 }
(D4) lim infn→∞ D((Aᵀ)−nξ) ≥ 1 for a.e. ξ ∈ RN^.
Proof. Clearly, the dimension function of any ZN^ -SI space V 0 must satisfy (D1). To show (D2) observe that the space V 1 is A−^1 ZN^ -SI. Hence, the spaces V 0 and V 1 are SI with respect to the common lattice Γ = ZN^ ∩ (A−^1 ZN^ ). Note that Γ∗^ = ZN^ + Aᵀ(ZN^ ) = Γ. The inclusion,˜ V 0 ⊂ V 1 implies that
dimΓ V 1 (ξ) ≥ dimΓ V 0 (ξ) for a.e. ξ.
16 MARCIN BOWNIK AND KENNETH HOOVER
τ : RN^ → [− 1 / 2 , 1 /2)N^ be the translation projection, τ (ξ) = ξ + k, where k ∈ ZN^ is the unique element such that ξ+k ∈ [− 1 / 2 , 1 /2)N^. Finally, given any measurable subset E˜ ⊂ RN^ , let E ⊂ E˜ be any measurable set such that τ (E) = τ ( E˜) and τ |E is injective.
Algorithm 1. Assume that D satisfies the conditions (D1)–(D4). For m ∈ N, let
Am = {ξ ∈ [− 1 / 2 , 1 /2)N^ : D(ξ) ≥ m}. (1) Let Q be any measurable subset of RN^ satisfying the following four properties: (i) Q ⊂ AᵀQ, (ii) limn→∞ 1 Q
(Aᵀ)−nξ
= 1 for a.e. ξ ∈ RN^ , (iii) τ
Q is injective (iv) D(ξ) ≥ 1 for all ξ ∈ Q. (2) Suppose that Si’s are already defined for all i = 1,... , m − 1 and some m ∈ N. In the case when m = 1 (meaning that none of Si’s were defined yet) let F˜m, 1 = Q. Otherwise, let F˜m, 1 = (AᵀPm− 1 \ Pm− 1 ) ∩ APm, where Pm− 1 =
⋃m− 1 i=1 Si. (3) For each n ∈ N, define iteratively F˜m,n+1 =
AᵀFm,n \
⋃n i=1 F^
P m,i
∩ APm. Then, let Sm =
n=1 Fm,n. (4) Finally, let S =
m∈N Sm. Then, we have the following result due to [19, 22].
Theorem 5.2. Assume that A ∈ MN (Z), and a function D, which is not ∞ constantly a.e., satisfies the conditions (D1)–(D4). Let S be the result of the above Algorithm. Define the spaces Vj = Lˇ^2 ((Aᵀ)j^ S) for j ∈ Z. Then, {Vj }j∈Z is a GMRA such that its dimension
function dimZ
N V 0 ≡^ D. In particular, if D satisfies the consistency equation (1.2), then the Algorithm produces a generalized scaling set S. Thus, defining W to be AᵀS \ S gives a wavelet set associated with A. Nevertheless, it turns out that Theorem 5.2 is false for rational dilations A. For a counterexample see Section 8.
- Dimension function of wavelets The goal of this section is to establish the formula for the dimension function of rationally dilated wavelets. Moreover, we derive the necessary properties which every wavelet dimension function must satisfy. Theorem 4.2 suggests the following definition.
Definition 6.1. Suppose that Ψ = {ψ^1 ,... , ψL} ⊂ L^2 (RN^ ) is a semi-orthogonal wavelet associated with a GMRA and the dilation A ∈ MN (Q). Define the wavelet dimension of Ψ as
(6.1) DΨ(ξ) :=
∑^ L
`=
∑^ ∞
j=
k∈Γ∗
|̂ψ`((Aᵀ)j^ (ξ + k))|^2 ,
where Γ∗^ = (Aᵀ)−^1 ZN^ ∩ ZN^ is a dual lattice to Γ = AZN^ + ZN^.
Theorem 6.2. Suppose that Ψ = {ψ^1 ,... , ψL} ⊂ L^2 (RN^ ) is a semi-orthogonal wavelet as- sociated with a GMRA {Vj }j∈Z. Then, its dimension function DΨ satisfies the following five conditions:
DIMENSION FUNCTIONS OF RATIONALLY DILATED GMRAS AND WAVELETS 17
(W1) DΨ : RN^ → N ∪ { 0 } is a measurable Γ∗-periodic function, (W2) DΨ satisfies the consistency inequality (equality if Ψ is a wavelet)
(6.2)
ω∈[(Aᵀ)−^1 ZN^ /Γ∗]
DΨ(ξ + ω) ≤ L +
ω′∈[ZN^ /Γ∗]
DΨ(Aᵀξ + ω′) for a.e. ξ.
(W3)
k∈Γ∗^1 ∆(ξ^ +^ k)^ ≥^ DΨ(ξ) for a.e.^ ξ^ ∈^ R
N (^) , where
∆ =
ξ ∈ RN^ : DΨ
(Aᵀ)−j^ ξ
≥ 1 for all j ∈ N ∪ { 0 }
(W4) lim infn→∞ DΨ
(Aᵀ)−nξ
≥ 1 for a.e. ξ ∈ RN^ , (W5)
RN^ /Γ∗^ DΨ(ξ)dξ^ ≤^
L | det A|− 1 (with equality if^ Ψ^ is a wavelet).
Proof. By Theorem 4.2, the core space V 0 is Γ-SI. Moreover, Theorem 3.1 and (4.6) yields
(6.3) dimΓ V 0 (ξ) =
k∈Γ∗
σ VΓ 0 (ξ + k) =
∑^ L
`=
∑^ ∞
j=
k∈Γ∗
|̂ψ`((Aᵀ)j^ (ξ + k))|^2 = DΨ(ξ).
Thus, (4.3) implies that DΨ(ξ) takes values in N ∪ { 0 } for a.e. ξ. Hence, (W1) holds. The consistency inequality (W2) can be verified directly from the definition (6.1). However, it follows immediately from (4.4), (4.5), and (6.3). To verify (W3), we simply repeat the proof of (D3) in Theorem 5.1. Indeed, (5.3) implies that σV 0 (ξ) ≤ (^1) ∆(ξ) for a.e. ξ ∈ RN^. Therefore, by Theorem 2.6(i), ∑
k∈Γ∗
(^1) ∆(ξ + k) ≥
k∈Γ∗
σ VΓ 0 (ξ + k) = DΨ(ξ),
which shows (W3). Condition (W4) follows from the fact that for each n ∈ N and ξ ∈ RN^ we have
DΨ
(Aᵀ)−nξ
∑^ L
`=
∑^ ∞
j=
∣̂ψ`((Aᵀ)j−nξ)∣∣^2 =
∑^ L
`=
∑^ ∞
j=1−n
∣̂ψ`((Aᵀ)j^ ξ)∣∣^2 ,
and by the Calder´on condition, see [15],
lim inf n→∞
DΨ
(Aᵀ)−nξ
∑^ L
`=
j∈Z
∣̂ψ`
(Aᵀ)j^ ξ
∣^2 = 1 for a.e. ξ.
Finally, (W5) is verified by the argument ∫
RN^ /Γ∗
DΨ(ξ)dξ =
∑^ L
`=
∑^ ∞
j=
k∈ZN
RN^ /Γ∗
∣̂ψ`((Aᵀ)j^ (ξ + k))∣∣^2 dξ
∑^ L
`=
∑^ ∞
j=
RN
∣̂ψ`
(Aᵀ)j^ ξ
∣^2 dξ =
∑^ ∞
j=
| det A|−j^ ·
∑^ L
`=
‖̂ψ`‖^2 ≤
L
| det A| − 1
In the case when Ψ is a wavelet, the last step is an equality which proves (W5).
DIMENSION FUNCTIONS OF RATIONALLY DILATED GMRAS AND WAVELETS 19
- Characterization of 3 interval wavelet sets In this section we characterize all possible wavelet sets consisting of 3 intervals for all dilation factors a > 1. While such characterization is of interest by itself, it also leads to a large class of examples of dimension functions. This direction will be explored in the next section.
7.1. Two interval wavelet sets. As a preliminary, we first characterize 2 interval wavelet sets. We say that a measurable set W ⊂ R is a wavelet set associated with the dilation a > 1 if and only if ∑
k∈Z
(7.1) (^1) W (ξ + k) = 1 for a.e. ξ ∈ R
∑
j∈Z
(7.2) (^1) W (aj^ ξ) = 1 for a.e. ξ ∈ R
Notice that (7.2) implies that W cannot contain any intervals of positive length containing 0, even as an endpoint. Indeed, if I ⊂ W is an interval with 0 ∈ I, then for every ξ > 0, or ξ < 0, we would have ξ ∈ aj^ I for infinitely many j ∈ Z and, hence,
j∈Z 1 W^ (a
j (^) ξ) =
∞. Furthermore, it also implies that W must have at least one negative component of positive measure and at least one positive component of positive measure. If, for instance,∣ ∣W ∩ (−∞, 0)
∣ (^) = 0, then we would have ∑ j∈Z 1 W (aj^ ξ) = 0 for all ξ < 0. Thus, if we
wish to construct a wavelet set W that is composed of exactly two intervals, we must have W = [b, c] ∪ [d, e] with b < c < 0 < d < e. Clearly, (7.2) is equivalent to [b, c] and [d, e] partitioning (−∞, 0) and (0, ∞), respectively, by dilations (modulo null sets) as shown in Figure 1. Note that in Figures 1 & 2 we have I 1 = [b, c] and I 2 = [d, e].
oo 0 //
[ (^) aI 1
][ I
1
][
a−^1 I 1
] [
a−^1 I 2
][ I
2 ][ (^) aI 2
]
Figure 1. Dilation partition condition for two intervals.
On the other hand, (7.1) is equivalent to the fact that d is an integer shift of c and the lengths of the two intervals sum to 1, as shown in Figure 2 In other words, (7.2) is equivalent to b = ac and e = ad, while (7.1) is equivalent to (a − 1)(d − c) = 1 and d = c + n for some n ∈ N with c + n > 0. Solving for a yields a = n+1 n. Hence, we obtain the following theorem.
Theorem 7.1. There exist two interval wavelet sets corresponding to the dilation a if and only if a = n+1 n for some n ∈ N. Furthermore, if a = n+1 n for some n ∈ N, then W is a two interval wavelet set corresponding to a if and only if:
W =
[
ax, x
]
[
x +
a − 1
, ax +
a a − 1
]
for some x ∈
a− 1 ,^0
20 MARCIN BOWNIK AND KENNETH HOOVER
oo 0 //
[ I
1
] [ I
1 +n^
][ I
2
]
% %
Figure 2. Translation partition condition for two intervals.
7.2. Construction Of Three Interval Wavelet Sets. In [16] one can find an example of Speegle which provides a formula for a family of wavelet sets in R consisting of three intervals and depending on the dilation a > 1 (see [16], Remark 3). Our goal is to extend this example to a more general form, characterizing all wavelet sets in R consisting of three intervals. We consider W = [b, c] ∪ [d, e] ∪ [f, g] where b < c < 0 < d < e < f < g. This is sufficient since W satisfies (7.1) and (7.2) if and only if −W does. Notice that in the case of three intervals (7.2) implies a slightly more complicated relationship. This condition is satisfied if and only if [b, c] partitions (−∞, 0) by dilations (modulo null sets) and [d, e] and [f, g] partition (0, ∞) by dilations (modulo null sets) in an interlacing pattern as shown in Figure
- Note that in Figures 3–5 we have I 1 = [b, c], I 2 = [d, e], and I 3 = [f, g]. The number p ∈ N is called an interlacing parameter.
oo 0 //
[ (^) aI 1
][ I
1
][
a−^1 I 1
] [
a−p−^1 I 3
][ I
2
][
a−pI 3 ] [ (^) apI 2
][ I
3
][
ap+1I 2
]
# # $ $ d d c c
Figure 3. Dilation partition condition for three intervals.
On the other hand, the relationship implied by (7.1) is similar to the two interval case, with the exception that there are now two ways in which it can be satisfied. These are shown in Figures 4 and 5. Initially we will concern ourselves only with three interval wavelet sets that satisfy the translation partition condition as shown in Figure 4. Let us construct all such wavelet sets. Notice that each one represents a solution to the following system of equations for some set of values m, n, p ∈ N.
