ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Pages 78–87 (August 3, 2004) S 1079-6762(04)00132-5

WAVELETS WITH COMPOSITE DILATIONS

KANGHUI GUO, DEMETRIO LABATE, WANG-Q LIM, GUIDO WEISS, AND EDWARD WILSON

(Communicated by Boris Hasselblatt)

Abstract. A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L^{2}(ℝ^{n}) under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets A and B. Typically, the members of B are shear matrices (all eigenvalues are one), while the members of A are matrices expanding or contracting on a proper subspace of ℝ^{n}. These wavelets are of interest in applications because of their tendency to produce “long, narrow” window functions well suited to edge detection. In this paper, we discuss the remarkable extent to which the theory of wavelets with composite dilations parallels the theory of classical wavelets, and present several examples of such systems.

1. Introduction

There is considerable interest, both in mathematics and its applications, in the study of efficient representations of multidimensional functions. The motivation comes partly from signal processing, including applications in image compression and feature extraction, and from the investigation of certain classes of singular integral operators. In particular, it was pointed out in several recent research papers that oriented oscillatory waveforms play a fundamental role in the construction of representations for multidimensional functions and signals (cf. [1], [3], [5], and articles in [11]). For example, it was shown that, in order to be optimally sparse in a certain sense, such representations must contain basis elements with many locations, scales, shapes and directions, unlike the trigonometric bases or even the “classical” wavelets, which are made of essentially isotropic oscillatory bumps at various scales and locations (cf. [2]).

In this paper, we introduce a new class of representation systems which have exactly the features we have described, as well as several other properties which have a great potential in applications. These systems, that we call affine systems with composite dilations, have the form

(1.1)  Ψ_{AB} = {D_{a} D_{b} T_{k} ψ^{ℓ} : k ∈ ℤ^{n}, b ∈ B, a ∈ A, ℓ = 1, …, L},

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

Reverts to public domain 28 years from publication

Received by the editors February 23, 2004 and, in revised form, April 13, 2004.

2000 Mathematics Subject Classification. Primary 42C15, 42C40.

Key words and phrases. Affine systems, frames, multiresolution analysis (MRA), multiwavelets, wavelets.

The fourth author was supported in part by a SW Bell Grant.

where ψ^{ℓ} ∈ L^{2}(ℝ^{n}), T_{k} are the translations defined by T_{k} f(x) = f(x−k), D_{a} are the dilations defined by D_{a} f(x) = | det a|^{−1/2} f(a^{−1}x), and A, B are countable subsets of GL_{n}(ℝ). By choosing ψ^{ℓ}, A, and B appropriately, we can make Ψ_{AB} an orthonormal (ON) basis or, more generally, a Parseval frame (PF) for L^{2}(ℝ^{n}). In this case, we call Ψ = {ψ^{1}, …, ψ^{L}} an ON AB-multiwavelet or a PF ABmultiwavelet, respectively. If the system has only one generator, that is, L = 1, then we use the expression wavelet rather than multiwavelet in this definition.

As we will show, the mathematical theory of these systems provides a simple and flexible framework for the construction of several classes of bases and Parseval frames. For example, in Sections 3.3 and 3.4, we construct composite wavelets with good time-frequency decay properties, whose elements contain “long and narrow” waveforms with many locations, scales, shapes and directions. These examples have similarities to the curvelets [1] and contourlets [4], which have been recently introduced in order to obtain efficient representations of natural images. Our approach is more general and presents a simple method for obtaining orthonormal bases and Parseval frames that exhibit these and other geometric features. In particular, our approach extends naturally to higher dimensions and allows a multiresolution construction which is well suited to a fast numerical implementation.

Before embarking on our presentation, it is useful to establish some notation and definitions. Recall that a countable family {ψ_{i}}_{i∈I} of elements in a separable Hilbert space H is a Parseval frame (PF) for H, if

∥f∥^{2} = ∑_{i∈I} |⟨f, ψ_{i}⟩|^{2}

for each f ∈ H. We adopt the convention that an element x ∈ ℝ^{n} is a column vector, while ξ ∈ ℝ̂^{n} is a row vector. A vector x multiplying a matrix a ∈ GL_{n}(ℝ) on the right, is understood to be a column vector, while a vector ξ multiplying a on the left is a row vector. Thus, ax ∈ ℝ^{n} and ξa ∈ ℝ̂^{n}. The Fourier transform is defined as

f̂(ξ) = ∫_{ℝ^{n}} f(x) e^{−2πiξx} dx.

For any E ⊂ ℝ̂^{n}, we denote by L^{2}(E)^{∨} the space {f ∈ L^{2}(ℝ^{n}) : supp f̂ ⊂ E}.

2. AB-MRA

Let B be a countable subset of S̃L_{n}(ℤ) = {b ∈ GL_{n}(ℝ) : | det b| = 1} and A = {a_{i} : i ∈ ℤ}, where a_{i} ∈ GL_{n}(ℝ). We say that a sequence {V_{i}} of closed subspaces of L^{2}(ℝ^{n}) is an AB-multiresolution analysis (AB-MRA) if the following holds:

(i) D_{b} T_{k} V_{0} = V_{0}, for any b ∈ B, k ∈ ℤ^{n}.

(ii) For each i ∈ ℤ, V_{i} ⊂ V_{i+1},, where V_{i} = D_{a_{i}}^{−1} V_{0}.

(iii) ⋂ V_{i} = \{0\} and \overline{⋃ V_{i}} = L^{2}(ℝ^{n}).

(iv) There exists ϕ ∈ L^{2}(ℝ^{n}) such that Φ_{B} = {D_{b} T_{k} ϕ : b ∈ B, k ∈ ℤ^{n}} is a semi-orthogonal PF for V_{0}, that is, Φ_{B} is a PF for V_{0} and, in addition, D_{b} T_{k} ϕ ⊥ D_{b′} T_{k′} ϕ for any b ≠ b′, b, b′ ∈ B, k, k′ ∈ ℤ^{n}.

The space V_{0} is called an AB scaling space and the function ϕ is an AB scaling function for V_{0}. If, in addition, Φ_{B} is an orthonormal basis for V_{0}, then ϕ is an ON AB scaling function. Also, let W_{0} be the orthogonal complement of V_{0} in V_{1}, that is, V_{1} = V_{0} ⊕ W_{0}. We cite the following elementary result [6]:

Theorem 2.1. (i) Let Ψ = {ψ^{1}, …, ψ^{L}} ⊂ L^{2}(ℝ^{n}) be such that {D_{b} T_{k} ψ^{ℓ} : b ∈ B, ℓ = 1, …, L, k ∈ ℤ^{n}} is a PF for W_{0}. Then Ψ is a PF ABmultiwavelet.

(ii) Let Ψ = {ψ^{1}, …, ψ^{L}} ⊂ L^{2}(ℝ^{n}) be such that {D_{b} T_{k} ψ^{ℓ} : b ∈ B, k ∈ ℤ^{n}, ℓ = 1, …, L} is an orthonormal basis for W_{0}. Then Ψ is an ON ABmultiwavelet.

In the situation described by the hypotheses of this theorem (where Ψ is not only a PF for L^{2}(ℝ^{n}), but it is also derived from an AB-MRA), we say that Ψ is a PF MRA AB-multiwavelet or an ON MRA AB-multiwavelet, respectively.

In most cases of interest, the set A ⊂ GL_{n}(ℝ) has the form A = {a_{i} = a^{i} : i ∈ ℤ}, where a ∈ GL_{n}(ℝ), and B is a subgroup of S̃L_{n}(ℤ) which satisfies aBa^{−1} ⊆ B. For the remainder of this section, we will make this assumption.

We say that the PF MRA AB-wavelet ψ is of finite filter (FF) type if there exists an AB scaling function ϕ for V_{0} and a finite set {b_{1}, …, b_{k}} ⊂ B such that

ϕ̂(ξa) = ∑_{j=1}^{k} m_{0}^{(j)}(ξ) ϕ̂(ξ b_{j}), ψ̂(ξa) = ∑_{j=1}^{k} m_{1}^{(j)}(ξ) ϕ̂(ξ b_{j}),

where m_{0}^{(j)}, m_{1}^{(j)}, 1 ≤ j ≤ k, are periodic functions. Similarly, the ON MRA ABmultiwavelet Ψ is of finite filter (FF) type if there exists an AB scaling function ϕ for V_{0} and a finite set {b_{1}, …, b_{k}} ⊂ B such that

ϕ̂(ξa) = ∑_{j=1}^{k} m_{0}^{(j)}(ξ) ϕ̂(ξ b_{j}), ψ̂^{ℓ}(ξa) = ∑_{j=1}^{k} m_{1,ℓ}^{(j)}(ξ) ϕ̂(ξ b_{j}), ℓ = 1, …, L,

where m_{0}^{(j)}, m_{1,ℓ}^{(j)}, 1 ≤ j ≤ k, are periodic functions.

It turns out that, while it is possible to construct a PF AB-wavelet using a single generator, that is, Ψ = {ψ}, in the case of orthonormal MRA AB-multiwavelets, multiple generators are needed, that is, Ψ = {ψ^{1}, …, ψ^{L}}, where L > 1. This situation is similar to the classical MRA and, as in that case, such restriction is not needed if the system does not come from an MRA (cf., for example, [12]). We refer to [6] for a proof of the following theorems.

Theorem 2.2. Let Ψ = {ψ^{1}, …, ψ^{L}} be an ON MRA AB-multiwavelet for L^{2}(ℝ^{n}), and let N = |B/aBa^{−1}| (= the order of the quotient group B/aBa^{−1}). Assume that | det a| ∈ ℕ. Then L = N | det a| − 1.

Theorem 2.3. Let B ⊂ S̃L_{n}(ℤ) and a ∈ GL_{n}(ℤ) with aBa^{−1} ⊆ B. Let L = N | det a| − 1, where N = |B/aBa^{−1}|. Assume that ϕ ∈ L^{2}(ℝ^{n}) is an ON AB scaling function for V_{0} = \overline{span \{D_{b} T_{k} ϕ : k ∈ ℤ^{n}, b ∈ B\}}. Then:

(i) There exist ON MRA AB-multiwavelets Ψ = {ψ^{1}, …, ψ^{L}} with scaling space V_{0}.

(ii) If ϕ̂ = χ_{U}, where U ⊂ ℝ^{n} is a measurable set, then there are sets T_{ℓ} ⊂ ℝ^{n}, ℓ = 1, …, L, for which Ψ = {ψ^{ℓ} = (χ_{T_{ℓ}})^{∨} : ℓ = 1, …, L} is an ON MRA AB-multiwavelet and Ψ is of FF type.

Remark 2.4. Let a ∈ GL_{n}(ℤ). Under additional assumptions on B (which are satisfied, for example, by the examples in Section 3), there exist sets S ⊂ ℝ̂^{n} such that ψ is an ON AB-wavelet for L^{2}(ℝ^{n}), where ψ̂ = χ_{S}. It is clear, by Theorem 2.2, that these AB-wavelets are not of MRA type.

3. Examples

This section shows that there are several examples of affine systems with composite dilations forming ON bases or PFs for L^{2}(ℝ^{n}). In particular, Sections 3.1 and 3.2 contain examples of PF MRA AB-multiwavelets and ON MRA AB-multiwavelets for L^{2}(ℝ^{2}), respectively; Sections 3.3 and 3.4 describe how to construct PF ABwavelets which are well-localized both in ℝ^{2} and ℝ̂^{2}; Section 3.5 describes generalizations of these examples for dimensions n > 2; finally, Section 3.6 describes a class of singly generated (non-MRA) ON AB-wavelets. More details about these constructions are found in [6].

3.1. Example 1. Let B = \{b_{j} = ( \begin{matrix}1 & j \\ 0 & 1 \end{matrix} ) : j ∈ ℤ\}. For 0 ≤ α < β, let S(α, β) = {ξ = (ξ_{1}, ξ_{2}) ∈ ℝ̂^{2} : α ≤ |ξ_{1}| < β} and let T (α, β) = T^{+}(α, β) ∪ T^{−}(α, β), where T^{+}(α, β) is the trapezoid with vertices (α, 0), (α, α), (0, β) and (β, β), and T^{−}(α, β) = {ξ ∈ ℝ̂^{2} : −ξ ∈ T^{+}(α, β)}. Observe that T (0, β) is the union of two triangles. A simple computation shows that T (α, β) is a B-tiling region for S(α, β), that is, S(α, β) is the disjoint union, modulo null sets, of the sets T (α, β) b, b ∈ B.

Now let a = ( \begin{matrix}c & d \\ 0 & e \end{matrix}) ∈ GL_{2}(ℝ), with |c| > 1, S_{0} = S(0, 1/2|c|), S_{i} = S_{0} a^{i}, i ∈ ℤ,

and define a sequence {V_{i}}_{i∈ℤ} of nested subspaces of L^{2}(ℝ^{2}) by V_{i} = L^{2}(S_{i})^{∨}, i ∈ ℤ. Also, let U = T (0, (2|c|)^{−1}) and ϕ = (χ_{U})^{∨}. Since U is a B-tiling region for S_{0}, it follows that \overline{span \{D_{b} T_{k} ϕ : k ∈ ℤ^{2}, b ∈ B\}} = L^{2}(S_{0})^{∨} = V_{0}. This shows that the spaces {V_{i}} form an AB-MRA and ϕ is the AB scaling function for V_{0}. In addition, for R = T ((2|c|)^{−1}, 2^{−1}), the function ψ = (χ_{R})^{∨} is a PF MRA AB-wavelet, since {D_{b} T_{k} ψ : b ∈ B, k ∈ ℤ^{2}} is a PF for W_{0} = L^{2}(S_{1} \ S_{0})^{∨}. Observe that W_{0} is the orthogonal complement of V_{0} in V_{1}. Also, ψ is of FF type, since U a^{−1} and R a^{−1} are contained in ⋃_{j=0}^{k−1} Ub_{j} for k ≥ |c|.

3.2. Example 2. Let B, S(α, β), and T (α, β) be defined as in Example 1. Let

a = ( \begin{matrix}k_{1} & k_{2} \\ 0 & k_{3} \end{matrix} ) ∈ GL_{2}(ℤ), with k_{1} > 1 and N = |k_{1}/k_{3}| ∈ ℕ, and let L = | det a|N − 1

= k_{1}^{2} − 1. Similarly to Example 1, let S_{0} = S(0, 1), S_{i} = S_{0} a^{i}, i ∈ ℤ, and define a sequence of spaces {V_{i}}_{i∈ℤ} by V_{i} = L^{2}(S_{i})^{∨}, i ∈ ℤ. Let U = T (0, 1) and ϕ = (χ_{U})^{∨}. Then ϕ is an ON basis for \overline{span \{D_{b} T_{k} ϕ : k ∈ ℤ^{n}, b ∈ B\}} = L^{2}(S_{0})^{∨} = V_{0}, and the spaces {V_{i}}_{i∈ℤ} form an AB-MRA, where ϕ is the corresponding ON AB scaling function.

Observe that the set R = T (1, k_{1}) is a B-tiling region for S_{1} \ S_{0}. We can pick disjoint subsets R_{ℓ}, ℓ = 1, …, L of R such that R = ⋃_{ℓ=1}^{L} R_{ℓ} and, for each ℓ, R_{ℓ} is a fundamental domain for ℤ^{2}. Figure 1 illustrates this construction in the special case where k_{1} = 1 and, thus, L = 3. Now let Ψ = {ψ^{1}, …, ψ^{L}}, where ψ^{ℓ} = (χ_{R_{ℓ}})^{∨}, ℓ = 1, …, L. Since {D_{b} T_{k} ψ^{ℓ} : b ∈ B, ℓ = 1, …, L, k ∈ ℤ^{2}} is an ON basis for W_{0} = L^{2}(S_{1} \ S_{0})^{∨}, and W_{0} is the orthogonal complement of V_{0} in V_{1}, it follows that Ψ is an ON MRA AB-multiwavelet. In addition, Ψ is of FF type, since the sets U a^{−1} and R_{ℓ} a^{−1}, ℓ = 1, …, L, are contained in ⋃_{j=0}^{k_{1}−1} U b_{j}.

The examples described in Sections 3.1 and 3.2 are well-localized in ℝ̂^{n}, since the AB-multiwavelets that we constructed are band-limited, but they do not have good localization in ℝ^{n}, since they decay only as fast as |x|^{−1} when |x| → ∞. The examples described in the next two sections, on the contrary, are well-localized both in ℝ^{n} and ℝ̂^{n}.

[figure: ξ_{1}; ξ_{2}; U^{+}; R_{1}^{+}; R_{2}^{+}; R_{3}^{+}; R_{1}^{−}; R_{3}^{−}; R_{2}^{−}; U^{−}; 1; 2; −1; −2; 1; 2; −1; −2; S_{0}; S_{0}; S_{1} \ S_{0}; S_{1} \ S_{0}; U = U^{+} ∪ U^{−}; R = ⋃_{ℓ=1}^{3} R_{ℓ}; R_{ℓ} = R_{ℓ}^{+} ∪ R_{ℓ}^{−}]

Figure 1. The sets U, R and R_{ℓ}, ℓ = 1, 2, 3, in Example 2, where L = 3.

3.3. Example 3. Let ψ_{1} ∈ L^{2}(ℝ) be a (one-dimensional) dyadic band-limited wavelet with supp ψ̂_{1} ⊂ [−Ω, Ω], Ω > 0, and ψ_{2} ∈ L^{2}(ℝ) be another band-limited function with supp ψ̂_{2} ⊂ [−1, 1] and satisfying

(3.1)  ∑_{j∈ℤ} |ψ̂_{2}(ξ + j)|^{2} = 1 a.e. ξ ∈ ℝ.

As we will show later on, there are several choices of functions ψ_{1} and ψ_{2} satisfying these properties.

For any ω = (ω_{1}, ω_{2}) ∈ ℝ^{2}, ω_{1} ≠ 0, define ψ ∈ L^{2}(ℝ^{2}) by

(3.2)  ψ̂(ω) = ψ̂_{1}(2^{s} ω_{1}) ψ̂_{2}(\frac{ω_{2}}{ω_{1}} ),

where s ∈ ℤ satisfies 2^{s} ≥ 2 Ω. It turns out that ψ is a PF AB-wavelet, where A = \{a_{i} = ( \begin{matrix}2^{i} & 0 \\ 0 & 1 \end{matrix} ) : i ∈ ℤ\} and B = \{b_{j} = ( \begin{matrix}1 & j \\ 0 & 1 \end{matrix} ) : j ∈ ℤ\}. The proof of this fact is based on an application of a result from [7], where the characterization equations for a very general class of Parseval frames are obtained.

As we mentioned before, there are many choices for the functions ψ_{1} and ψ_{2} that satisfy the assumptions we have described above. For example, we can choose ψ_{1} to be the Lemariè-Meyer wavelet (see [8, Sec. 1.4]) defined by ψ̂_{1}(ξ) = e^{iπξ} b(ξ), where

b(ξ) = \{ \begin{matrix}sin(\frac{π}{2} (3|ξ| − 1)), \frac{1}{3} ≤ |ξ| ≤ \frac{2}{3}, \\ sin(\frac{3π}{4} (\frac{4}{3} − |ξ|)), \frac{2}{3} < |ξ| ≤ \frac{4}{3}, \\ 0 otherwise. \end{matrix}

In order to construct ψ_{2}, let ϕ be a compactly supported C^{∞} bump function, with supp ϕ ⊂ [−1, 1] (examples can be found in [10, Sec. 3.3] or [9, Sec. 1.4]), and define ψ_{2} by

ψ̂_{2}(ξ) = \frac{ϕ(ξ)}{\sqrt{∑_{k∈ℤ} |ϕ(ξ + k)|^{2}}} .

It is clear that ψ_{2} ∈ C^{∞}(ℝ) and satisfies (3.1). It follows that ψ̂, given by (3.2), is in C^{∞}(ℝ^{2}), and this implies that |ψ(x)| ≤ K_{N} (1 + |x|)^{−N}, K_{N} > 0, for any N ∈ ℕ.

The following example shows how to construct MRA AB-wavelets for L^{2}(ℝ^{2}) which are well-localized both in ℝ^{n} and ℝ̂^{n}.

3.4. Example 4. Let ψ_{1} ∈ L^{2}(ℝ) be a (one-dimensional) dyadic band-limited MRA wavelet with supp ψ̂_{1} ⊂ [−Ω, Ω], Ω > 0, and ϕ_{1} be its associated scaling function. Let m_{0} and m_{1} be the low pass and high pass filters, respectively, associated with ϕ_{1} and ψ_{1}, that is, m_{0}, m_{1} are the periodic functions satisfying the equations

ϕ̂_{1}(ω_{1}) = m_{0}(\frac{ω_{1}}{2} ) ϕ̂_{1}(\frac{ω_{1}}{2} ) and ψ̂_{1}(ω_{1}) = m_{1}(\frac{ω_{1}}{2} ) ϕ̂_{1}(\frac{ω_{1}}{2} ).

Let ψ_{2} ∈ L^{2}(ℝ) be defined by

ψ_{2}(x) = e^{i(N+1)πx} (\frac{sin πx}{πx})^{N+1},

where N ∈ ℕ. That is, ψ̂_{2} is a basic spline of order N (cf. [8]).

For ω = (ω_{1}, ω_{2}) ∈ ℝ^{2}, ω_{1} ≠ 0, let ϕ ∈ L^{2}(ℝ^{2}) be defined by

ϕ̂(ω) = ϕ̂_{1}(2^{s} ω_{1}) \frac{ψ̂_{2}(\frac{ω_{2}}{ω_{1}} )}{\sqrt{∑_{m∈ℤ}|ψ̂_{2}(\frac{ω_{2}}{ω_{1}} + m)|^{2}}} ,

where s ∈ ℤ satisfies 2^{s} ≥ 4 Ω (\frac{N}{2} + 1). This assumption on s ensures that

supp \{ϕ̂_{1}(2^{s} ω_{1}) ψ̂_{2}(\frac{ω_{2}}{ω_{1}} )\} ⊂ [−\frac{1}{4}, \frac{1}{4}]^{2}.

Also, let ψ ∈ L^{2}(ℝ^{2}) be defined by

ψ̂(ω) = ∑_{k=0}^{N+1} d_{k}^{(N)} m_{1}(2^{s−1}ω_{1}) M_{0}(a^{−1}ω) ϕ̂((b^{T})^{−k}a^{−1} ω),

where the matrices a and b are as in Section 3.3, d_{k}^{(N)} = 2^{−N} \binom{N+1}{k}, and M_{0}(ω) is the ℤ^{2}-periodic function which, restricted to the fundamental region [−\frac{1}{2}, \frac{1}{2}]^{2}, is given by

M_{0}(ω) = (\frac{∑_{m∈ℤ}|ψ̂_{2}(\frac{ω_{2}}{ω_{1}} + m)|^{2}}{∑_{m∈ℤ}|ψ̂_{2}(2^{−1} \frac{ω_{2}}{ω_{1}} + m)|^{2}} )^{1/2}, ω ∈ [−\frac{1}{2}, \frac{1}{2}]^{2}.

It turns out that ψ is a PF MRA AB-wavelet for L^{2}(ℝ^{2}). Indeed, the spaces {V_{j} : j ∈ ℤ\}, where V_{j} = D_{a}^{−j}V_{0}, j ∈ ℤ, and V_{0} = \overline{span \{D_{b} T_{m} ϕ : b ∈ B, m ∈ ℤ^{2}\}}, form an AB-MRA. Observe, however, that this system is somewhat different from those described in Section 2, since the spaces V_{0} and W_{0} = \overline{span \{D_{b} T_{m} ψ : b ∈ B, m ∈ ℤ^{2}\}} are not mutually orthogonal. Also, by construction, ψ̂ ∈ C^{N}(ℝ̂^{2}), so |ψ(x)| ≤ K_{N} (1 + |x|)^{1−N}, for some K_{N} > 0.

3.5. Extensions for n > 2. For n > 2, there are several generalizations of the examples described in the previous sections. The general idea is to write R̂^{n} = F ⊕E and to define matrices a, b such that:

(i) Ea ⊆ E;

(ii) the induced action of a on R̂^{n}/E is expanding; (iii) b is the identity map on E and b − I maps F to E.

This allows us to construct a space V_{0} = L^{2}(S)^{∨} as we did in the 2-dimensional examples, for strip domains S = c×E, where c ⊂ F is contained in a neighborhood of the origin.

In particular, in order to extend Examples 1 and 2, let n = k + ℓ, and A = {a^{i} : i ∈ ℤ\}, where a = ( \begin{matrix}a_{0} & a_{1} \\ 0 & a_{2} \end{matrix} ) ∈ GL_{n}(ℝ), and a_{0} ∈ GL_{k}(ℝ) is an expanding matrix. The matrices B can be chosen according to one of the following patterns.

(i) Let k = 1, ℓ = n − 1 and B = \{b_{j} = ( \begin{matrix}1 & j \\ 0 & I_{ℓ} \end{matrix} ) : j ∈ ℤ^{ℓ}\}. Then ψ = (χ_{R})^{∨}, where R = \{(ξ_{0}, …, ξ_{ℓ}) ∈ ℝ^{n} : \frac{1}{2|a_{0}|} ≤ |ξ_{0}| < \frac{1}{2}, 0 ≤ ξ_{i}/ξ_{0} < 1, for 1 ≤ i ≤ ℓ}, is a PF MRA AB-wavelet.

(ii) Let k ≥ ℓ > 0 and

B = \{b_{j_{1},...,j_{ℓ}} = (\begin{matrix}I_{k−ℓ} & 0 & 0 \\ 0 & I_{ℓ} & (\begin{matrix}j_{1} & … & 0 \\ . & . & . \\ 0 & … & j_{ℓ} \end{matrix}) \\ 0 & 0 & I_{ℓ} \end{matrix}) : j_{i} ∈ ℤ, for each 1 ≤ i ≤ ℓ\} .

To illustrate this example, let k = 3, ℓ = 2, and a_{0} be a diagonal matrix with diagonal entries c_{0}, c_{1}, c_{2} and |c_{i}| > 1, i = 0, 1, 2. Then ψ = (χ_{R})^{∨}, where R = \{(ξ_{0}, …, ξ_{4}) ∈ ℝ^{n} : \frac{1}{2|c_{i}|} ≤ |ξ_{i}| < \frac{1}{2}, i = 0, 1, 2, and 0 ≤ ξ_{i+2}/ξ_{i} < 1, for i = 1, 2}, is a PF MRA AB wavelet. The construction is similar for any k ≥ ℓ.

(iii) Let k = 1, ℓ = n − 1 and

B = \{b_{j_{1},...,j_{ℓ}} = (\begin{matrix}1 & j_{1} & … & 0 & 0 \\ 0 & 1 & ... & 0 & 0 \\ & ... & ... & ... \\ 0 & 0 & … & 1 & j_{ℓ} \\ 0 & 0 & … & 0 & 1 \end{matrix}) : j_{i} ∈ ℤ, 1 ≤ i ≤ ℓ\} .

Then ψ = (χ_{R})^{∨}, where R = \{(ξ_{0}, …, ξ_{ℓ}) ∈ ℝ^{n} : \frac{1}{2|a_{0}|} ≤ |ξ_{0}| < \frac{1}{2}, 0 < ξ_{i}/ξ_{i−1} < 1, for 1 ≤ i ≤ ℓ}, is a PF MRA AB wavelet. Observe that, unlike the cases (i) and (ii), here the matrices B do not form a group.

In order to generalize Examples 3 and 4 to dimensions n > 2, we can proceed as follows. Again, let n = k + ℓ and A = {a^{i} : i ∈ ℤ}, where

a = (\begin{matrix}(\begin{matrix}a_{1} & … & 0 \\ . & . & . \\ 0 & … & a_{k} \end{matrix}) & 0 \\ 0 & I_{ℓ} \end{matrix}) ∈ GL_{n}(ℝ)

and |a_{i}| > 1, for each i = 1, …, k. The matrices B can be chosen according to one of the following patterns.

(i) Let k = 1, ℓ = n − 1 and B = \{b_{j} = ( \begin{matrix}1 & j \\ 0 & I_{ℓ} \end{matrix} ) : j = (j_{1}, …, j_{ℓ}) ∈ ℤ^{ℓ}\}. Let

ψ_{1} ∈ L^{2}(ℝ) be a (one-dimensional) band-limited wavelet with respect to the dilation a_{1}, having supp ψ̂_{1} ⊂ [−Ω, Ω], Ω > 0, and let ψ_{2}, …, ψ_{n} ∈ L^{2}(ℝ) be also band-limited functions with supp ψ̂_{i} ⊂ [−1, 1], i = 2, …, n, and satisfying

(3.3)  ∑_{j∈ℤ} |ψ̂_{i}(ξ + j)|^{2} = 1 for a.e. ξ ∈ ℝ, for any i = 2, …, n.

Now, for any ω = (ω_{1}, …, ω_{n}) ∈ ℝ^{n}, ω_{1} ≠ 0, define ψ ∈ L^{2}(ℝ^{n}) by

ψ̂(ω) = ψ̂_{1}(a_{1}^{s} ω_{1}) ψ̂_{2}(\frac{ω_{2}}{ω_{1}} ) × ⋯ × ψ̂_{n}(\frac{ω_{n}}{ω_{1}} ),

where s ∈ ℤ satisfies a_{1}^{s} ≥ 2 Ω. Then one can show that ψ is a PF ABwavelet.

(ii) Let n = 2k and

B = \{b_{j} = (\begin{matrix}I_{k} & (\begin{matrix}j_{1} & … & 0 \\ . & . & . \\ 0 & … & j_{k} \end{matrix}) \\ 0 & I_{k} \end{matrix}) : j = (j_{1}, …, j_{k}) ∈ ℤ^{k}\} .

Let ψ_{1}, …, ψ_{k} ∈ L^{2}(ℝ) be band-limited wavelets with respect to the dilations a_{1}, …, a_{k}, having supp ψ̂_{i} ⊂ [−Ω_{i}, Ω_{i}], Ω_{i} > 0, and let ψ_{k+1}, …, ψ_{n} ∈ L^{2}(ℝ) be also band-limited functions with supp ψ̂_{i} ⊂ [−1, 1], i = k + 1, …, n, and satisfying (3.3) for any i = k + 1, …, n. Now, for any ω = (ω_{1}, …, ω_{n}) ∈ ℝ^{n}, ω_{1}, …, ω_{k} ≠ 0, define ψ ∈ L^{2}(ℝ^{n}) by

ψ̂(ω) = ψ̂_{1}(a_{1}^{s_{1}} ω_{1}) × ⋯ × ψ̂_{k}(a_{k}^{s_{k}} ω_{k}) ψ̂_{k+1}(\frac{ω_{k+1}}{ω_{1}} ) × ⋯ × ψ̂_{n}(\frac{ω_{2k}}{ω_{k}} ),

where s_{k} ∈ ℤ satisfies a_{k}^{s_{k}} ≥ 2 Ω_{k}. Then one can show that ψ is a PF AB-wavelet.

3.6. Composite wavelet sets. In this section, we consider AB-wavelets where ψ ∈ L^{2}(ℝ^{n}) is given by ψ̂ = χ_{S}, with S ⊂ ℝ̂^{n}. These systems are the analog of the so-called MSF wavelets in the classical wavelet theory.

As a specific example of such systems, consider, for simplicity, the case n = 2.

Let B = \{b_{j} = ( \begin{matrix}1 & j \\ 0 & 1 \end{matrix} ) : j ∈ ℤ\}, and A = \{a^{i} : i ∈ ℤ\}, where a = ( \begin{matrix}a_{1} & 0 \\ 0 & a_{2} \end{matrix} ), with

|a_{1}| > 1 and a_{2} ≠ 0. Given any 0 < δ ≤ 1, let S = {(ξ_{1}, ξ_{2}) ∈ ℝ^{2} : δ ≤ |ξ_{1}| ≤ δ |a_{1}|, and 0 ≤ ξ_{2} ≤ |ξ_{1}|}, and define ψ by ψ̂ = χ_{S}. A direct computation shows that

(3.4)  Ω = ⋃_{i,j∈ℤ} S b^{j} a^{i} = {(ξ_{1}, ξ_{2}) ∈ ℝ^{2} : ξ ≠ 0},

where the union is disjoint, and this implies that ψ is a PF AB-wavelet.

If, in addition, we take a ∈ GL_{n}(ℤ), then, by choosing S appropriately, we can construct systems which are not only PF but even ON bases for L^{2}(ℝ^{n}). For example, let T (0, α) be as in Example 1, and let T (0, α)+(β, γ) denote the triangle obtained by translating T (0, α) by (β, γ) ∈ ℝ^{2}. Let a = ( \begin{matrix}2 & 0 \\ 0 & 2 \end{matrix} ), B as above, and ψ

[figure: ξ_{1}; ξ_{2}; 1; 2; −1; −2; 1; 2; −1; −2]

Figure 2. A singly generated ON AB-wavelet (Section 3.6).

be given by ψ̂ = χ_{S}, where S = S^{+} ∪ S^{−}, S^{−} = −S^{+} and S^{+} is the subset of ℝ^{2} equal to

[T (0, 1)\ ⋃_{k=1}^{∞}{T (0, 2^{−k})+(1−2^{−k+1})(1, 1)}]∪[ ⋃_{k=1}^{∞}{T (0, 2^{−k})+(2−2^{−k+1})(1, 1)}] .

This construction is illustrated in Figure 2. It is easy to see that S is a fundamental domain for the ℤ^{2}-translations and equation (3.4) is satisfied. This shows that ψ is an ON AB-wavelet.

Observe that, since this orthonormal system is generated by a single function, it follows from Theorem 2.2 that ψ is not an ON MRA AB-wavelet.

For a = ( \begin{matrix}2 & 0 \\ 0 & 1 \end{matrix} ), one can construct an analogous unbounded set S. Again, ψ, where ψ̂ = χ_{S}, is not an ON MRA AB-wavelet by Theorem 2.2. Higher-dimensional generalizations of these examples can also be obtained.

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Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804

E-mail address: kag026f@smsu.edu

Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695

E-mail address: dlabate@math.ncsu.edu

Department of Mathematics, Washington University, St. Louis, Missouri 63130

E-mail address: wangQ@math.wustl.edu

Department of Mathematics, Washington University, St. Louis, Missouri 63130

E-mail address: guido@math.wustl.edu

Department of Mathematics, Washington University, St. Louis, Missouri 63130

E-mail address: enwilson@math.wustl.edu