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\begin{document}

\title{Wavelets with composite dilations}


\author[K. GUO]{Kanghui Guo}
\address{Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804}
\email{kag026f@smsu.edu}

\author[D. LABATE]{Demetrio Labate}
\address{Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695}
\email{dlabate@math.ncsu.edu}

\author[W. LIM]{Wang-Q Lim}
\address{Department of Mathematics, Washington University, St. Louis, Missouri 63130}
\email{wangQ@math.wustl.edu}

\author[G. WEISS]{Guido Weiss}
\address{Department of Mathematics, Washington University, St. Louis, Missouri 63130}
\email{guido@math.wustl.edu}
\thanks{The fourth author was supported in part by a SW Bell Grant.}

\author[E. WILSON]{Edward Wilson}
\address{Department of Mathematics, Washington University, St. Louis, Missouri 63130}
\email{enwilson@math.wustl.edu}

\date{February 23, 2004 and, in revised form, April 13, 2004}


\subjclass[2000]{Primary 42C15, 42C40}

\keywords{Affine systems, frames, multiresolution analysis (MRA), 
multiwavelets, wavelets}

\commby{Boris Hasselblatt} 

\begin{abstract}
A wavelet with composite dilations is a function generating
an orthonormal basis or a Parseval frame for $L^2({\mathbb R}^n)$ 
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
${\mathbb R}^n$. 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.
\end{abstract}

\maketitle


\section{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. \cite{CD99},
\cite{CM97},  \cite{DH02},  and articles in \cite{Wel03}). 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.~\cite{CD02}).

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 {\bf affine systems with
composite dilations},  have the form
\begin{equation}\label{ab-wav}
\Psi_{AB}= \{D_a \, D_{b} \, T_k \, \psi^\ell: \,\, k \in \Z^n, b
\in B, a \in A, \ell=1, \dots, L\},
\end{equation}
where $\psi^\ell \in L^2(\R^n)$, $T_k$ are the {\bf translations}
defined by $T_k \, f(x) = f(x-k)$, $D_a$ are the {\bf 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 $\psi^\ell$,
$A$, and $B$ appropriately, we can make $\Psi_{AB}$  an
orthonormal (ON) basis or, more generally, a Parseval frame (PF)
for $L^2(\R^n)$. In this case, we call $\Psi = \{\psi^1, \dots,
\psi^L\}$ an {\bf ON $AB$-multiwavelet} or a {\bf PF
$AB$-multiwavelet}, respectively. If the system has only one
generator, that is, $L=1$, then we use the expression {\bf
wavelet} rather than {\bf 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~\ref{ss.ex3} and \ref{ss.ex4}, 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
{\it curvelets} \cite{CD99} and {\it contourlets} \cite{DV03},
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
$\{\psi_i\}_{ i \in \I}$ of elements in a separable Hilbert space
$\mathcal H$ is a {\bf Parseval frame (PF)} for $\mathcal H$, if
\begin{equation*}
\norm{f}^2 = \sum_{i \in \I} |\ip{f}{\psi_i}|^2
\end{equation*}
for each $f \in \mathcal H$. We adopt the convention that an element $x
\in \R^n$ is a column vector, while $\xi \in \widehat \R^n$ is a
row vector. A vector $x$ multiplying a matrix $a \in GL_n(\R)$ on the right,
is understood to be a column vector, while a vector $\xi$ multiplying $a$ on the left
is a row vector. Thus, $a x \in \R^n$ and $\xi a \in \widehat \R^n$.
 The Fourier transform is defined as
$$\hat{f}(\xi) = \int_{\R^n} f(x) \, e^{-2 \pi i \xi  x} \, d x.$$
For any $E \subset \widehat \R^n$, we denote by $L^2(E)^\vee$ the
space $\{f \in L^2(\R^n): \supp \hat f \subset E\}. $






\section{$AB$-MRA}   \label{s.2}
Let $B$ be a countable subset of $\widetilde{SL}_n(\Z) = \{b \in
GL_n(\R): \, |\det b|=1\}$ and $ A = \{a_i: \, i \in \Z\}$, where
$a_i \in GL_n(\R)$. We  say that a sequence $\{V_i\}$ of closed
subspaces of $L^2(\R^n)$ is an {\bf $AB$-multiresolution analysis
($AB$-MRA)} if the following holds:
\begin{enumerate}
    \item[(i)] {\it $ D_b \, T_k \, V_0 = V_0$, for any $b \in B$, $k \in \Z^n$.}
    \item[(ii)] {\it For each  $i \in \Z$,  $V_{i} \subset V_{i+1},$, where $V_i = D_{a_i}^{-1} \, V_0.$}
    \item[(iii)] {\it  $\bigcap V_i = \{0\}$ and $\overline{\bigcup V_i} = L^2(\R^n)$.}
    \item[(iv)]  {\it There exists  $\phi \in L^2(\R^n)$ such that $\Phi_B = \{D_b \, T_k \, \phi: b \in B, \, k \in \Z^n\}$
     is a semi-orthogonal PF for $V_0$, that is,
    $\Phi_B$ is a PF for $V_0$ and, in addition, $D_b \, T_k \, \phi \, \bot \, D_{b'} \, T_{k'} \, \phi$
    for any $b \ne b'$, $b, b' \in B$, $k, k' \in \Z^n$.}
\end{enumerate}
The space $V_0$ is called an {\bf $AB$ scaling space} and the
function $\phi$ is an {\bf $AB$ scaling function} for $V_0$. If,
in addition, $\Phi_B$  is an orthonormal basis for $V_0$, then
$\phi$ is an {\bf ON $AB$ scaling function}. Also, let $W_0$ be
the orthogonal complement of $V_0$ in $V_1$, that is, $V_1 = V_0
\oplus W_0$. We cite the following elementary result
\cite{GLLWW}:

\begin{theorem}
\begin{itemize}
    \item [(i)] Let $\Psi = \{\psi^1, \dots, \psi^L\} \subset L^2(\R^n)$ be such that
    $\{D_b \, T_k \, \psi^\ell: b \in B, \ell =1, \dots, L, k \in \Z^n\}$ is a PF for $W_0$.
 Then $\Psi$ is a PF $AB$-multiwavelet.
    \item [(ii)] Let  $\Psi = \{\psi^1, \dots, \psi^L\} \subset L^2(\R^n)$ be
such that $\{ D_b \, T_k \, \psi^\ell: b \in B, k \in \Z^n, \ell=1,
\dots,L \}$ is an orthonormal basis for $W_0$. Then $\Psi$ is an {ON $AB$-multiwavelet}.
\end{itemize}
\end{theorem}

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


In most cases of interest, the set $A \subset GL_n(\R)$ has the
form $A=\{a_i =a^{i}: \, i \in \Z\}$, where $a \in GL_n(\R)$, and
$B$ is a subgroup of  $\widetilde{SL}_n(\Z)$ which satisfies $a B
a^{-1} \subseteq B$. For the remainder of this section, we will
make this assumption.

We say that the PF MRA $AB$-wavelet $\psi$ is
of {\bf finite filter (FF) type} if there exists an $AB$
scaling function $\phi$ for $V_0$ and a finite set $\{b_1, \dots, b_k\} \subset B$
such that
$$ \hat \phi (\xi a) = \sum_{j=1}^k m_0^{(j)}(\xi) \, \hat \phi(\xi \, b_j), \quad
\hat \psi ( \xi a) = \sum_{j=1}^k m_1^{(j)}(\xi) \, \hat \phi(\xi \, b_j),$$
where $m_0^{(j)}, m_1^{(j)}$, $ 1 \le j \le k$, are periodic functions.
Similarly, the  ON MRA $AB$-multiwavelet
$\Psi$ is of  finite filter (FF) type if there exists an $AB$
scaling function $\phi$ for $V_0$  and a finite set $\{b_1, \dots, b_k\} \subset B$ such that
$$ \hat \phi (\xi a) = \sum_{j=1}^k m_0^{(j)}(\xi) \, \hat \phi(\xi \, b_j), \quad
\hat \psi^\ell (\xi a) = \sum_{j=1}^k m_{1,\ell}^{(j)}(\xi) \, \hat \phi(\xi \, b_j),
\, \ell=1, \dots , L,$$
where $m_0^{(j)}, m_{1,\ell}^{(j)}$, $ 1 \le j \le k$, are periodic functions.


It turns out that, while it is possible to construct a PF
$AB$-wavelet using a single generator, that is, $\Psi=\{\psi\}$,
in the case of {\it orthonormal} MRA $AB$-multiwavelets, multiple
generators are needed, that is, $\Psi = \{\psi^1, \dots,
\psi^L\}$, 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,
\cite{WW}). We refer to~\cite{GLLWW} for a  proof of the following
theorems.

\begin{theorem} \label{th2}
Let  $\Psi = \{\psi^1, \dots, \psi^L\}$ 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 $|\det a| \in \N$.
 Then $L= N \, |\det a| -1$.
\end{theorem}

\begin{theorem}
Let $B \subset\widetilde{SL}_n(\Z)$ and $a \in GL_n(\Z)$ with $a B a^{-1} \subseteq B$.
Let  $L= N \, |\det a| -1$, where $N= |B / aBa^{-1}|$. Assume that $\phi \in
L^2(\R^n)$ is an ON $AB$ scaling function for $V_0= \overline{\span\{D_b \, T_k \, \phi: k \in \Z^n, b \in B\}}$. Then:
\begin{itemize}
    \item[(i)] There exist ON MRA $AB$-multiwavelets $\Psi = \{\psi^1, \dots, \psi^L\}$ with scaling space $V_0$.
    \item[(ii)] If $\hat \phi = \chi_U$, where $U \subset \R^n$ is a measurable set, then there are sets $T_\ell \subset \R^n$,
    $\ell=1, \dots, L$, for which $\Psi = \{\psi^\ell = (\chi_{T_\ell})^\vee: \, \ell= 1, \dots, L\}$ is an ON MRA $AB$-multiwavelet
    and $\Psi$ is of FF type.
\end{itemize}
\end{theorem}

\begin{remark} \label{th.ws}
{\rm Let  $a \in GL_n(\Z)$.
Under additional assumptions on $B$ (which are satisfied, for example, by the examples in Section~\ref{s.ex}),
there exist sets $S \subset \widehat \R^n$ such that $\psi$ is an ON $AB$-wavelet for $L^2(\R^n)$, where $\hat \psi = \chi_S$.
It is clear, by Theorem~\ref{th2}, that these $AB$-wavelets  are {\bf not} of MRA type}.
\end{remark}



\section{Examples} \label{s.ex}

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~\ref{ss.ex1}
and~\ref{ss.ex2} contain examples of PF MRA $AB$-multiwavelets and
ON MRA $AB$-multiwavelets for $L^2(\R^2)$, respectively;
Sections~\ref{ss.ex3} and ~\ref{ss.ex4} describe how to construct
PF $AB$-wavelets which are well-localized both in $\R^2$ and
$\widehat \R^2$; Section~\ref{ss.ex5} describes  generalizations
of these examples for dimensions $n >2$; finally,
Section~\ref{ss.ex6} describes a class of singly generated 
(non-MRA) ON $AB$-wavelets.  More details about these constructions are
found in \cite{GLLWW}.



\subsection{Example 1.} \label{ss.ex1}
Let $B = \{b_j= \left(\begin{smallmatrix}
     1 & j \\ 0 & 1
 \end{smallmatrix}\right): j \in \Z\}$.
For $0 \le \mya < \myb$, let $S({\mya, \myb}) = \{\xi=(\xi_1, \xi_2) \in
\widehat{\R}^2: \, \mya \le |\xi_1| < \myb \}$ and let $T({\mya, \myb})=
T^+({\mya, \myb}) \cup T^-({\mya, \myb})$, where $T^+({\mya, \myb})$ is the
trapezoid with vertices $(\mya, 0)$, $(\mya,\mya)$, $(0, \myb)$ and
$(\myb,\myb)$, and $T^-({\mya, \myb})=\{\xi \in \widehat{\R}^2: -\xi \in
T^+({\mya, \myb})\}$. Observe that $T({0, \myb})$ is the union of two
triangles. A simple computation shows that $T({\mya, \myb})$ is a {\bf
$B$-tiling region} for $S({\mya, \myb})$, that is, $S({\mya, \myb})$ is
the disjoint union, modulo null sets, of the sets $T({\mya, \myb}) \,
b$, $b \in B$.

Now let $a= \left(\begin{smallmatrix}
     c & d \\ 0 & e
 \end{smallmatrix}\right) \in GL_2(\R)$, with $|c| > 1$,
$S_0 =S({0, 1/2|c|})$, $S_i = S_0 \, a^i$, $i\in \Z$, and define a
sequence $\{V_i\}_{i \in \Z}$ of nested subspaces of $L^2(\R^2)$
by $V_i = L^2(S_i)^\vee$, $i \in \Z$. Also, let $U =  T({0,
(2|c|)^{-1}})$ and $\phi = (\chi_{U})^\vee$. Since $U$ is a
$B$-tiling region for $S_0$, it follows that $\overline{\span\{D_b \, T_k \,
\phi: k \in \Z^2, b \in B\}}=L^2(S_0)^\vee = V_0$. This shows that
the spaces $\{V_i\}$ form an $AB$-MRA and $\phi$ is the $AB$
scaling function for $V_0$. In addition, for $R =
T((2|c|)^{-1},2^{-1})$, the function $\psi = (\chi_{R})^\vee$ is a
PF MRA  $AB$-wavelet, since $\{D_b \, T_k \, \psi: b \in B, k \in
\Z^2\}$ is a PF for $W_0 =L^2(S_1 \setminus S_0)^\vee$. Observe
that $W_0$ is the orthogonal complement of $V_0$ in $V_1$. Also,
$\psi$ is of FF type, since $U \, a^{-1} $ and  $R \, a^{-1} $ are
contained in $\bigcup_{j=0}^{k-1}  U b_j$ for $k \ge |c|$.

\subsection{Example 2.} \label{ss.ex2}



Let $B$, $S(\mya,\myb)$, and $T(\mya,\myb)$ be defined as in Example~1.
Let $a= \left(\begin{smallmatrix}
     k_1 & k_2 \\ 0 & k_3
 \end{smallmatrix}\right) \in GL_2(\Z)$, with $k_1>1$ and $N = |k_1/k_3| \in
 \N$, and let $L= |\det a| N-1\break = k_1^2-1$. Similarly to Example~1, let
$S_0 = S(0,1)$, $S_i = S_0 \, a^i$, $i \in \Z$, and define a
sequence of spaces $\{V_i\}_{i \in \Z}$ by $V_i = L^2(S_i)^\vee$,
$i \in \Z$.
 Let $U = T({0, 1})$  and $\phi = (\chi_U)^\vee$. Then $\phi$ is an ON basis
for  $\overline{\span\{D_b \, T_k \, \phi: k \in \Z^n, b \in B\}} = L^2(S_0)^\vee = V_0$, and the spaces $\{V_i\}_{i \in \Z}$
form an $AB$-MRA, where $\phi$ is the corresponding ON $AB$ scaling function.

 Observe that the set $R = T({1,k_1})$ is a $B$-tiling region for  $S_1 \setminus S_0$. We can pick disjoint subsets
 $R_\ell$, $\ell = 1, \dots, L$ of $R$ such that $R = \bigcup_{\ell =1}^L R_\ell$ and, for each $\ell$, $R_\ell$ 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 $\Psi = \{ \psi^1, \dots, \psi^L\}$, where
 $\psi^\ell = (\chi_{R_\ell})^\vee$, $ \ell =1, \dots, L$. Since $\{D_b \, T_k \, \psi^\ell: \, b \in B, \ell = 1, \dots, L, k \in \Z^2\}$
 is an ON basis for $W_0= L^2(S_1\setminus S_0)^\vee$, and $W_0$ is the orthogonal complement of $V_0$ in $V_1$, it follows that
 $\Psi$ is an  ON MRA $AB$-multiwavelet.
 In addition, $\Psi$ is of FF type, since the sets $U \, a^{-1} $ and  $R_\ell \, a^{-1} $, $\ell = 1, \dots, L$,
 are contained in $\bigcup_{j=0}^{k_1-1} U \, b_j $.

The examples described in Sections \ref{ss.ex1} and \ref{ss.ex2} are well-localized in $\widehat \R^n$, 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 $|x| \rightarrow \infty$.
The examples described in the next two sections, on the contrary, are  well-localized both in $\R^n$  and $\widehat \R^n$.

{\setlength{\unitlength}{0.2mm}
\begin{figure}
\begin{center}
  \begin{picture}(590,450)(0,0)
    \put(-25,200){\vector(1,0){450}}
    \put(460,200){$\xi_1$}
    \put(200,-20){\vector(0,1){450}}
    \put(212,438){$\xi_2$}
    \put(300,-5){\line(0,1){415}}
    \put(400,-5){\line(0,1){415}}
    \put(100,-5){\line(0,1){415}}
    \put(0,-5){\line(0,1){415}}
    \put(300,300){\line(1,0){100}}
     \put(300,200){\line(1,1){100}}
    \put(0,100){\line(1,0){100}}
    \put(0,100){\line(1,1){100}}
    \put(0,0){\line(1,1){400}}
    \put(250,218){\small $U^+$}
    \put(348,220){\small $R_{1}^+$}
    \put(317,260){\small $R_{2}^+$}
    \put(348,320){\small $R_{3}^+$}
    \put(25,165){\small $R_{1}^-$}
    \put(25,64){\small $R_{3}^-$}
    \put(53,125){\small $R_{2}^-$}
    \put(125,168){\small $U^-$}
    \put(290,203){\small $1$}
     \put(405,202){\small $2$}
     \put(76,202){\small $-1$}
      \put(-24,202){\small $-2$}
    \put(200,300){\circle*{3}}
    \put(205,296){\small $1$}
    \put(200,400){\circle*{3}}
    \put(205,396){\small $2$}
    \put(200,100){\circle*{3}}
    \put(205,96){\small $-1$}
    \put(200,0){\circle*{3}}
    \put(205,-4){\small $-2$}
    \put(244,12){\small $S_0$}
    \put(144,12){\small $S_0$}
      \put(324,-2){\small $S_1 \setminus S_0$}
      \put(32,-2){\small $S_1 \setminus S_0$}
  \put(470,350){\small $U = U^+ \, \cup \, U^-$}
      \put(470,310){\small $R = \bigcup_{\ell=1}^3 R_\ell$}
   \put(470,270){\small $R_\ell = R_\ell^+ \, \cup \, R_\ell^-$}
  \end{picture}
\end{center}
\smallskip
\caption{ The sets $U$, $R$ and $R_\ell$, $\ell=1,2,3$, in Example~2, where $L=3$.}
\end{figure}}



\subsection{Example 3.} \label{ss.ex3}

Let $\psi_1 \in L^2(\R)$ be a 
(one-dimensional) dyadic band-limited wavelet with $\supp 
\hat \psi_1 \subset [-\myO,\myO]$, $\myO>0$,
and $\psi_2 \in L^2(\R)$ be another 
band-limited function with $\supp \hat \psi_2 \subset [-1,1]$ and satisfying
\begin{equation}\label{eq.psi2}
\sum_{j \in \Z} |\hat \psi_2(\xi +j)|^2 =1 \quad \text{a.e. } \xi \in \R.
\end{equation}
As we will show later on, there are several choices of functions $\psi_1$ and $\psi_2$ satisfying these properties.

For any $\myo =(\myo_1, \myo_2) \in \R^2$, 
$\myo_1 \ne 0$, define $\psi \in L^2(\R^2)$ by
\begin{equation}\label{eq.psi}
    \hat \psi (\myo) = \hat \psi_1(2^s \, \myo_1) \,
\hat \psi_2 \Bigl(\frac{\myo_2}{\myo_1}\Bigr),
\end{equation}
where $s \in \Z$ satisfies $2^s \ge  2 \, \myO.$ It turns out that
$\psi$ is a PF $AB$-wavelet, where $A = \{a_i= \left(\begin{smallmatrix}
     2^i & 0 \\ 0 & 1
 \end{smallmatrix}\right): i \in \Z\}$ and  $B = \{b_j= \left(\begin{smallmatrix}
     1 & j \\ 0 & 1
 \end{smallmatrix}\right): j \in \Z\}$.
The proof of this fact is based on an application of a result from~\cite{HLW02}, 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 $\psi_1$ and $\psi_2$ that satisfy the assumptions we have described above.
For example, we can choose $\psi_1$ to be the Lemari\`e-Meyer wavelet (see~\cite[
Sec. 1.4]{HW96}) defined by $\hat \psi_1 (\xi) = e^{i \pi \xi} \, b(\xi)$, where
$$ b(\xi) = \begin{cases}  & \sin(\frac{\pi}2 \, (3 |\xi| -1) ), \quad 
\frac13 \le |\xi| \le \frac23, \\
& \sin(\frac{3 \pi}4 \, (\frac43 -|\xi|) ), \quad \frac23 < |\xi| \le \frac43, \\
& 0 \quad \hskip6em\text{ otherwise.}
\end{cases} $$
In order to construct $\psi_2$, let $\phi$ be a compactly supported $C^\infty$ bump function, with $\supp \phi \subset [-1,1]$
 (examples can be found in \cite[Sec.~3.3]{SW70} or \cite[Sec.~1.4]{Hor}),
and define $\psi_2$ by
$$\hat \psi_2(\xi) = \frac{\phi(\xi)}{\sqrt{\sum_{k \in \Z} |\phi(\xi+k)|^2}} \,. $$
It is clear that $\psi_2 \in C^\infty(\R)$  and satisfies (\ref{eq.psi2}).
It follows that $\hat \psi$, given by (\ref{eq.psi}), is in $C^{\infty}(\R^2)$, 
and this implies that $|\psi(x)| \le K_N \, ({1+|x|})^{-N}$, $K_N > 0$,  for any $N \in \N$.



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

\subsection{Example 4.} \label{ss.ex4}
Let $\psi_1 \in L^2(\R)$ be a (one-dimensional) dyadic
band-limited MRA wavelet with $\supp \hat \psi_1 \subset
[-\myO,\myO]$, $\myO>0$, and $\phi_1$ be its associated scaling
function. Let $m_0$ and $m_1$ be the low pass and high pass
filters, respectively, associated with $\phi_1$ and $\psi_1$, that
is, $m_0$, $m_1$ are the periodic functions satisfying the
equations $$ \hat \phi_1(\myo_1) = m_0(\frac{\myo_1}2) \, \hat
\phi_1(\frac{\myo_1}2) \quad \text{and} \quad \hat \psi_1(\myo_1) =
m_1(\frac{\myo_1}2) \, \hat \phi_1(\frac{\myo_1}2).$$ Let $\psi_2 \in
L^2(\R)$ be defined by $$ \psi_2(x) = e^{i (N+1) \pi x} \, \Bigl(
\frac{\sin \pi x}{\pi x} \Bigr)^{N+1},$$ where $N \in \N$. That
is, $\hat \psi_2$ is a basic spline of order $N$
(cf.~\cite{HW96}).


For $\myo = (\myo_1, \myo_2) \in \R^2$, $\myo_1 \ne 0$,  let $\phi \in L^2(\R^2)$ be defined by
\begin{equation*}
 \hat \phi (\myo)= \hat \phi_1(2^s \, \myo_1) \, \frac{\hat \psi_2(\frac{
\myo_2}{\myo_1})}{\sqrt{\sum_{m \in \Z} 
\bigl|\hat \psi_2(\frac{\myo_2}{\myo_1}+m)\bigr|^2}},
\end{equation*}
where $s \in \Z$ satisfies $2^s \ge  4 \, \myO \, (\frac N2 +1).$ This assumption on $s$ ensures that
\begin{equation*}
    \supp \Bigl\{ \hat \phi_1(2^s \, \myo_1) \, \hat \psi_2(\frac{\myo
_2}{\myo_1}) \Bigr\} \subset [-\frac 14, \frac 14]^2.
\end{equation*}
Also, let $\psi \in L^2(\R^2)$ be defined by 
$$ \hat \psi (\myo) =
\sum_{k=0}^{N+1} d_k^{(N)} \, m_1(2^{s-1} \myo_1) \, M_0(a^{-1} \myo)
\, \hat \phi ((b^T)^{-k} a^{-1} \, \myo), $$ where the matrices $a$
and $b$ are as in Section~\ref{ss.ex3}, $d_k^{(N)} = 2^{-N} \,
\binom{N+1}{k}$, and $M_0(\myo)$ is the $\Z^2$-periodic function
which, restricted to the fundamental region
$[-\frac12,\frac12]^2$, is   given by $$ M_0(\myo) =
\Biggl(\frac{\sum_{m \in \Z} \bigl|\hat
\psi_2(\frac{\myo_2}{\myo_1}+m)\bigr|^2}{\sum_{m \in \Z} \bigl|\hat
\psi_2(2^{-1} \, \frac{\myo_2}{\myo_1}+m)\bigr|^2}\Bigg)^{1/2}, \qquad
\myo \in [-\frac12,\frac12]^2.$$ It turns out that $\psi$ is a PF
MRA $AB$-wavelet for $L^2(\R^2)$. Indeed, the spaces $\{V_j:\break j \in
\Z\}$, where $V_j = D_a^{-j} V_0$, $j \in \Z$, and $V_0 =
\overline{\span\{D_b \, T_m \, \phi: b \in B, m \in \Z^2\}}$, form
an $AB$-MRA. Observe, however, that this system is somewhat
different from those described in Section~\ref{s.2}, since the
spaces $V_0$ and $W_0=\overline{\span\{D_b \, T_m \, \psi: b \in
B, m \in \Z^2\}}$ are {\it not} mutually orthogonal. Also, by
construction, $\hat \psi \in C^N(\widehat \R^2)$, so $|\psi(x)|
\le K_N \, (1+|x|)^{1-N}$, for some $K_N >0$.




\subsection{Extensions for $n>2$.} \label{ss.ex5}
For $n >2$, there are several generalizations of the examples described in the previous sections. The general idea is to write
$\widehat R^n = F \oplus E$ and to define matrices $a$, $b$ such that:
\begin{itemize}
    \item[(i)] $Ea \subseteq E$;
    \item[(ii)] the induced action of $a$ on $\widehat R^n /E$ is expanding;
    \item[(iii)] $b$ is the identity map on $E$ and $b-I$ maps $F$ to $E$.
\end{itemize}
This allows us to construct a space $V_0 = L^2(S)^\vee$ as we did in the 
$2$-dimensional examples,
for strip domains $S = c \times E$, where $c \subset F$ is contained in a neighborhood of the origin.

In particular, in order to extend Examples 1 and 2, let $n = k + \ell$,
and $A = \{a^i: i \in \Z\}$, where $a= \left(\begin{smallmatrix}
     a_0 & a_1 \\ 0 & a_2
 \end{smallmatrix}\right) \in GL_n(\R)$, and $a_0 \in GL_k(\R)$ is an expanding matrix.
 The matrices $B$ can be chosen according to one of the following patterns.
 \begin{itemize}
    \item[(i)] Let $k =1$, $\ell = n-1$ and $B = \{b_j= 
\big(\begin{smallmatrix}
     1 & j \\ 0 & I_\ell
 \end{smallmatrix}\big): j \in \Z^\ell\}$. Then $\psi = (\chi_R)^\vee$, where $R= \{(\xi_0, \dots, \xi_\ell) \in \R^n: \, \frac 1{2|a_0|} \le |\xi_0| < \frac 12,
 \, 0 \le \xi_i/\xi_0 < 1, \text{ for } 1 \le \break i \le \ell\}$, 
is a PF MRA  
$AB$-wavelet.
    \item[(ii)] Let $k \ge \ell > 0$  and 
$$
B = \left\{b_{j_1, \dots, j_\ell}= \begin{pmatrix}
     I_{k-\ell} & 0 & 0 \\ 0 & I_\ell & \begin{pmatrix}
     j_1 & \dots & 0 \\ & \ddots & \\ 0 & \dots & j_\ell
 \end{pmatrix}  \\ 0 & 0 & I_\ell
 \end{pmatrix}: j_i \in \Z, \text{ for each } 1 \le i \le 
\ell\right\}.
$$ 
To illustrate this example, let $k=3, \ell=2$, 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 $\psi = (\chi_R)^\vee$, where $R= \{(\xi_0, \dots, \xi_4) \in \R^n: \, \frac 1{2|c_i|} \le |\xi_i| < \frac 12,
i=0,1,2, \, \text{ and } 0 \le \xi_{i+2}/\xi_{i} < 1, \text{ for }
i =1,2\}$, is a PF MRA  $AB$ wavelet. The construction is similar
for any $k \ge \ell$.
    \item[(iii)] Let $k =1$, $\ell = n-1$ and 
$$
B = \left\{b_{j_1, \dots, j_\ell}= 
\begin{pmatrix}
     1 & j_1 & \dots & 0 & 0 \\ 0 & 1 & \ddots& 0 & 0 \\  & \ddots & \ddots & \ddots &  \\ 0 & 0 & \dots & 1 & j_\ell \\0 & 0 & \dots & 0 & 1
 \end{pmatrix}: j_i \in \Z,  1 \le i \le \ell\right\}.
$$
 Then $\psi = (\chi_R)^\vee$, where $R= \{(\xi_0, \dots, \xi_\ell) \in \R^n: \, \frac 1{2|a_0|} \le |\xi_0| < \frac 12,
 \, 0 < \xi_i/\xi_{i-1} < 1, \text{ for } 1 \le i \le \ell\}$, is a PF MRA  $AB$ wavelet. Observe that, unlike the cases
 (i) and (ii), here the matrices $B$ do not form a group.
 \end{itemize}


In order to generalize Examples 3 and 4 to dimensions $n >2$, we can proceed as follows.
Again, let $n = k + \ell$ and $A = \{a^i: i \in \Z\}$,
where  
$$
a= \begin{pmatrix}
     {\begin{pmatrix}
     a_1 & \dots & 0 \\ & \ddots & \\ 0 & \dots & a_k
 \end{pmatrix} } & 0 \\ 0 & I_\ell
 \end{pmatrix} \in GL_n(\R)
$$ 
and $|a_i| > 1$, for each $i = 1, \dots, k$.
 The matrices $B$ can be chosen according to one of the following patterns.
 \begin{itemize}
    \item[(i)] Let $k =1$, $\ell = n-1$ and $B = \{b_j= \big(\begin{smallmatrix}
     1 & j \\ 0 & I_\ell
 \end{smallmatrix}\big): j = (j_1, \dots, j_\ell) \in \Z^\ell\}$.
 Let $\psi_1 \in L^2(\R)$ be a (one-dimensional) band-limited wavelet with
respect to the dilation $a_1$, having $\supp \hat \psi_1 \subset [-\myO,\myO]$,
$\myO>0$,
and let $\psi_2, \dots, \psi_n \in L^2(\R)$ be also band-limited functions with $\supp \hat \psi_i \subset [-1,1]$, $i=2, \dots, n$,
and satisfying
\begin{equation}\label{eq.psi2b}
\sum_{j \in \Z} |\hat \psi_i(\xi +j)|^2 =1 \quad \text{ for a.e. } \xi \in \R, \text{ for any } i=2, \dots, n.
\end{equation}
Now, for any $\myo =(\myo_1, \dots, \myo_n) \in \R^n$, 
$\myo_1 \ne 0$, define $\psi \in L^2(\R^n)$ by
\begin{equation*}
    \hat \psi (\myo) = \hat \psi_1(a_1^s \, \myo_1) \,
\hat \psi_2 \Bigl(\frac{\myo_2}{\myo_1}\Bigr) \times \dots \times \hat \psi_n
\Bigl(\frac{\myo_n}{\myo_1} 
\Bigr),
\end{equation*}
where $s \in \Z$ satisfies $a_1^s \ge  2 \, \myO.$ 
Then one can show that $\psi$ is a PF $AB$-wavelet.

 \item[(ii)] Let $n = 2k$ and 
$$
B = \left\{b_j= \begin{pmatrix}
     I_k & {\begin{pmatrix}
     j_1 & \dots & 0 \\ & \ddots & \\ 0 & \dots & j_k
 \end{pmatrix} } \\ 0 & I_k
 \end{pmatrix}: j = (j_1, \dots, j_k) \in \Z^k\right\}.
$$
  Let $\psi_1, \dots, \psi_k \in L^2(\R)$ be band-limited wavelets with 
respect to the dilations $a_1, \dots, a_k$, having
  $\supp \hat \psi_i \subset [-\myO_i,\myO_i]$, $\myO_i>0$,
and let $\psi_{k+1}, \dots, \psi_n \in L^2(\R)$ be also band-limited functions with $\supp \hat \psi_i \subset [-1,1]$, $i=k+1, \dots, n$,
and satisfying (\ref{eq.psi2b}) for any $i=k+1, \dots, n$.
Now, for any $\myo =(\myo_1, \dots, \myo_n) \in \R^n$, $\myo_1,
 \dots, \myo_k  \ne 0$, define $\psi \in L^2(\R^n)$ by
\begin{equation*}
    \hat \psi (\myo) = \hat \psi_1(a_1^{s_1} \, \myo_1) 
\times \dots \times \hat \psi_k(a_k^{s_k} \, \myo_k) \,
\hat \psi_{k+1} \Bigl(\frac{\myo_{k+1}}{\myo_1}\Bigr) \times \dots \times \hat \psi_n 
\Bigl(\frac{\myo_{2k}}{\myo_k} \Bigr),
\end{equation*}
where $s_k \in \Z$ satisfies $a_k^{s_k} \ge  2 \, \myO_k.$ 
Then one can show that $\psi$ is a PF $AB$-wavelet.
 \end{itemize}

\subsection{Composite wavelet sets} \label{ss.ex6}
In this section, we consider $AB$-wavelets where $\psi \in
L^2(\R^n)$ is given by $\hat \psi = \chi_S$, with $S \subset
\widehat \R^n$. These systems are the analog of the so-called MSF
wavelets in the classical wavelet theory.

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  \end{picture}
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\medskip
\caption{A singly generated ON $AB$-wavelet (Section~\ref{ss.ex6}).}
\end{figure}}



As a specific example of such systems, consider, for simplicity, the case $n =2$.
 Let $B = \{b_j= \left(\begin{smallmatrix}
     1 & j \\ 0 & 1
 \end{smallmatrix}\right): j \in \Z\}$, and $A= \{a^i: i \in \Z\}$, where $a= \left(\begin{smallmatrix}
     a_1 & 0 \\ 0 & a_2
 \end{smallmatrix}\right)$, with $|a_1| >1$ and $a_2 \ne 0$. Given any $0 < \delta \le 1$, let $S = \{(\xi_1, \xi_2) \in \R^2:
\myd \le |\xi_1| \le \myd \, 
|a_1| , \text{ and } 0 \le \xi_2 \le 
|\xi_1|\}$, and define $\psi$ by $\hat \psi = \chi_S$.
A direct computation shows that
\begin{equation}\label{eq.omega}
\Omega = \bigcup_{i,j \in \Z} S \, b^j \, a^i = \{(\xi_1, \xi_2) \in \R^2: \, \xi \ne 0\},
\end{equation}
where the union is disjoint, and this implies that $\psi$ is a PF $AB$-wavelet.

If, in addition, we take $a \in GL_n(\Z)$, 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 $T(0,\mya)$ be as in
Example~1, and let $T(0,\mya) + (\myb,\g)$ denote the triangle
obtained by translating $T(0,\mya)$ by $(\myb,\g) \in \R^2$. Let
 $a= \left(\begin{smallmatrix}
     2 & 0 \\ 0 & 2
 \end{smallmatrix}\right)$, $B$ as above, and $\psi$ 
be given by $\hat\psi=\chi_S$, where $S=S^+\cup S^-$, $S^-=-S^+$
and $S^+$ is the subset of $\mathbb{R}^2$ equal to
\[\Bigl[T(0,1)\setminus\bigcup_{k=1}^{\infty}
\bigl\{T(0,2^{-k})+(1-2^{-k+1})(1,1)\bigl\}\Bigr]\cup
\Bigl[\,\bigcup_{k=1}^\infty
\bigl\{T(0,2^{-k})+(2-2^{-k+1})(1,1)\bigr\}\Bigr]\,.\]
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~(\ref{eq.omega})
is satisfied. This shows that $\psi$ is an ON $AB$-wavelet.


Observe that, since this orthonormal system is generated by a single function, it follows 
from Theorem~\ref{th2} that
$\psi$ is {\bf not} an ON MRA $AB$-wavelet.

For $a= \left(\begin{smallmatrix}
     2 & 0 \\ 0 & 1
 \end{smallmatrix}\right)$, one can construct an analogous unbounded set $S$.
 Again, $\psi$, where  $\hat \psi = \chi_S$, is  not an ON MRA $AB$-wavelet  by Theorem~\ref{th2}.
 Higher-dimensional generalizations of these examples can also be
obtained.

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\end{document}