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% Author Package file for use with AMS-LaTeX 1.2
\controldates{18-JAN-1999,18-JAN-1999,18-JAN-1999,20-JAN-1999}
 
\documentclass{era-l}
\usepackage{graphicx}

\newtheorem{theorem}{Theorem}

\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}

\newcommand\supp{\mathrm{supp}}
\newcommand\meas{\mathrm{meas}}
\newcommand\T{\mathrm{T}}
\newcommand\Cd{\mathcal{C}^*}
\newcommand\Q{\mathcal{Q}}
\newcommand\g{\gamma}
\newcommand\gd{{\gamma}^*}
\newcommand\gz{{\gamma}^*_0}
\newcommand\G{\Gamma}
\newcommand\Gz{{\Gamma}^*_0}
\newcommand\Gj{{\Gamma}^*_j}
\newcommand\Gd{{\Gamma}^*}
\newcommand\Ltwo{L^2({\mathbf R}^n)}
\newcommand\Lstwo{L^2({\mathbf R}^n/{\Gamma^*})}
\newcommand\Rtwo{{\mathbf R}^2}
\newcommand\Rn{{\mathbf R}^n}
\newcommand\Z{\mathbf Z}
\newcommand\Zn{{\mathbf Z}^n}
\newcommand\C{{\mathcal C}}
\newcommand\eix{e^{-i\langle x,\xi\rangle}}
\newcommand\jx{\tilde{\jmath}}
\newcommand\js{\mu}
\newcommand\jp{\mu_{+}}
\newcommand\Mj{{M^*}^j}
\newcommand\Msd{{M^*}^{-s} {\mathcal C}^*}
\newcommand\fs{\widehat{f_s}}
\newcommand\gs{\widehat{g_s}}
\newcommand\rda{\sqrt{\det A}}
\newcommand\suhd{\sum_{h=1}^d}
\newcommand\sukd{\sum_{k=1}^d}
\newcommand\suhk{\sum_{h,k=1}^d}
\newcommand\M{{\mathcal M}_0}

\begin{document}
\title[WAVELETS ON GENERAL LATTICES]
{Wavelets on general lattices, 
associated with general expanding maps of $\mathbf R^n$} %.}

\author{A. Calogero}
\address{Dipartimento di Matematica,
        Universit\'a di Milano,
        via Saldini 50,
        20133 Milano, Italy}

\email{Calogero@vmimat.mat.unimi.it}

\issueinfo{5}{01}{}{1999}
\dateposted{January 25, 1999}
\pagespan{1}{10}
\PII{S 1079-6762(99)00054-2}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}

\subjclass{Primary 42C15} %.}
\date{July 13, 1998}

\commby{Stuart Antman}

\keywords{Wavelets, multiresolution analysis (MRA), general lattices in 
${\mathbf R}^n,$
MSF wavelets, multiwavelets} %.}

\begin{abstract}
In the context of a general lattice $\Gamma$ in $\Rn$ and a strictly 
expanding
map $M$ which preserves the lattice, we characterize all the wavelet 
families,
all the MSF wavelets, all the multiwavelets associated with a 
Multiresolution
Analysis (MRA) of multiplicity $d\ge1,$ and all the scaling functions.
Moreover, we give several examples: in particular, we construct a single,
MRA and $C^\infty (\Rn)$ wavelet, which is nonseparable and with 
compactly
supported Fourier transform.
\end{abstract}

\maketitle

\section{Introduction} %.}
%\noindent
An orthonormal wavelet is a function
$\psi\in L^2({\mathbf R})$ such that the set
\begin{equation}\label{eq:1}
\bigl\{\psi_{j,k}\equiv
2^{j/2} \psi(2^j x-k)
:\ j,\ k\in\mathbf{Z}\}
%\eqno(1)\]
\end{equation}
is an orthonormal basis for
$L^2({\mathbf R}).$ A complete characterization of these wavelets
is given by the equations
\begin{equation*}
\begin{array}{l}\displaystyle{
a)\quad
\sum_{j\in \mathbf{Z}}
|\widehat\psi(2^j\xi)|^2=1 \qquad\qquad\qquad\text{ a.e.\ 
}\xi\in{\mathbf R},
}\\
\displaystyle{b)\quad\sum_{j=0}^{\infty}
\widehat\psi(2^j\xi)
\overline{\widehat\psi(2^j(\xi+2k\pi))}=0
\qquad\text{ a.e.\ }\xi\in{\mathbf R},\ k\in 2\mathbf{Z}+1,}
\end{array} %\eqno(I)\]
\tag{$I$}\end{equation*}
together with the assumption $\|\psi\|_2\ge1.$
These two equations have been known since the beginning of the theory
of wavelets (see \cite{L1} and \cite{L2}), but the proof of this 
characterization 
did not
appear in the literature until recently (see \cite{Gr}, \cite{W},
\cite{H--W--W3}, \cite{Ga}).
A much more direct argument gives us two equations that
characterize
the orthonormality of the system $\{\psi_{j,k}\},\ j,\ k\in\Z:$
\begin{equation*}
\begin{array}{l}\displaystyle{
a)\quad\sum_{k\in\mathbf{Z}}
|\widehat\psi(\xi+2k\pi)|^2=1 \qquad\qquad\qquad\text{ a.e.\ 
}\xi\in{\mathbf R},
}\\
\displaystyle{b)\quad\sum_{k\in\mathbf{Z}}
\widehat\psi(2^j(\xi+2k\pi))
\overline{\widehat\psi(\xi+2k\pi)}=0
\qquad\text{a.e.\ }\xi\in{\mathbf R},\ \text{if}\ j>0.}
\end{array}
%\eqno(II)\]
\tag{$II$}
\end{equation*}
These two equations clearly must be a consequence of $(I)$ (see 
\cite{H--W}, Chapter 7).

These four equations are most useful for the construction of very
general examples of wavelets. An immediate consequence is that
$|\widehat\psi(\xi)|\le1$ a.e. when $\psi$ satisfy $(I)$ a), or $(II)$ b).
Integrating the series in $(II)$ a) on $[0,2\pi)$ it then follows that
$\meas(\supp\ \widehat\psi)\ge 2\pi$ and we have equality
if and only if $|\widehat\psi|=\chi_K,$ with $K$ a measurable subset of
${\mathbf R}$ of measure $|K|=2\pi.$
Wavelets of this last described form are known as Minimally Supported
Frequency (MSF) wavelets.
It is an easy exercise that the MSF wavelets are characterized by the
equations $(I)$ a) and $(II)$ a) (see \cite{H--W}).

Another important way for constructing wavelets is the use of the
Multiresolution Analysis (MRA) method which yields what is known as the
MRA wavelets.
This type of wavelets has been completely characterized by Gripenberg 
and Wang (\cite{Gr}, \cite{W})
in the following way:
the wavelet $\psi$ is
an MRA wavelet if and only if
\begin{equation}\label{eq:2}
D_{\psi}(\xi)=\sum_{j=1}^\infty\sum_{k\in\Z}
|\widehat\psi\bigl(2^j(\xi+2k\pi)\bigr)|^2=1 \qquad\qquad\qquad 
\mbox{a.e.}\ \xi\in{\mathbf R}.
%\eqno(2)\]
\end{equation}

These notions extend to $\Rn,\ n>1.$
Certain needed changes are obvious
(we translate by lattice points and the preservation of the norm
is affected by multiplying $\psi$ in \eqref{eq:1} by $2^{jn/2}$ instead of 
$2^{j/2}).$
Other changes, however, are more crucial and of a different character.
There is a natural extension of the notion of an MRA involving one 
scaling function.
This leads to the same characterization \eqref{eq:2} in the $n$-dimensional
case.
However, an easy computation shows the necessity of introducing
$2^n-1$ functions
$\psi^1,\  \psi^2,\dots,\psi^L,\ L=2^n-1,$
that generate the wavelets basis.
Moreover, another way in which more than one generator appears is by 
considering
an MRA given by more than one scaling function.
Taking into account these features one can, again, characterize
all these wavelets by two equations \cite{F--G--W--W} and the basic 
structure
of the 1-dimensional proof applies.
This general approach also includes singly generated wavelets
(which, of course, cannot be MRA wavelets).


The principal purpose of our work is to show that all this can be 
extended
to the case where the ``dilations'' are affected by using more general
matrices $M$ (not just $M=2I_n)$ and the translations involve a
general lattice.
In fact, we obtain a theorem that characterizes these more general 
wavelets
and, also, one that characterizes MRA wavelets and scaling functions.
Let us be explicit: in the paper \cite{Ca1} that is to appear in the
Journal of Geometric Analysis we obtain a generalization of $(I)$ a), b).
The characterization of MSF wavelets offers no surprises.
We do this explicitly in the second section. Others
have considered these dilations and lattices in the general MRA case:
Gr\"ochenig, Madych and Strichartz (we will refer to them below).
In the third section we give a characterization of the appropriate 
Multiresolution
Analyses and associated scaling functions (that extend the
1-dimensional result in \cite{H--W}, Chapter 7). In this connection 
see also \cite{Ma}.

An interesting feature of these general wavelets is that for certain
dilations we can construct singly generated MRA wavelets.
These and other observations will be made in the fourth section where
we list several examples of these more general wavelets.
We include a specific method for ``smoothing'' an MSF wavelet of the plane
that, almost certainly, is applicable in much more general situations.

Finally, we wish to thank Guido Weiss for his invaluable advice and help.
\section{Characterization of wavelet families and MSF wavelets} %.}
In the mathematical literature there are several papers devoted to the 
study of
wavelets and multiresolution analyses from the point of view of general 
lattices.
The papers \cite{Ma} and \cite{S1} contain detailed illustrations of 
multiresolution analyses,
generated by lattice translates of the scaling
function and their dilations. Moreover, they show how to construct 
orthonormal wavelets bases in
this context. In particular, Gr\"ochenig and Madych \cite{G--M}, and 
Strichartz \cite{S2}, starting from self-affine and self-similar 
tilings of $\Rn,$ construct
multiresolution analyses and wavelet bases with the translation lattice 
and scaling naturally associated
with the tiling.

We consider a
%{\sl 
\textit{lattice} $\G$ of $\Rn$ as a discrete additive subgroup of $\Rn$ 
given by the integral linear
combinations of $n$ independent vectors. In other words, $\G=A\Zn$
where $A$ is a $n\times n$ real matrix with
$\det A> 0.$
Moreover we denote by $\C$ the
%{\sl 
\textit{fundamental domain  of $\G$}
which is the set $\C=
A[-1/2,+1/2)^n.$

We define the %{\sl 
\textit{dual lattice $\Gd$} as the set of points $\gd\in\Rn$
such that
$\langle\g,\gd\rangle\in\ 2\pi\mathbf{Z}$ for all $\g\in\G.$
It is immediate to see that
$\Gd=
2\pi(A^*)^{-1}\Zn.$
Note that for $n=1$ and $\G=\Z,$ we simply have $\Gd=2\pi\Z.$

We say that a linear trasformation
$M:\Rn\rightarrow \Rn$ is an %{\sl 
\textit{acceptable dilation for $\G$}
\cite{Ma} if it satisfies the following conditions:

\begin{itemize}
\item[(1)] $M$ is %{\sl 
\textit{strictly expanding,} i.e. all the eigenvalues 
$\lambda$ of $M$ satisfy
$|\lambda|>1,$
%\smallskip

\item[(2)] $M$ %{\sl
\textit{ preserves the lattice $\G,$} i.e. $M\G\subseteq \G.$
%\smallskip
\end{itemize}
%\noindent
It is easy to see that
$m=|\det M|$ is an integer $\ge2.$
We denote by $M^*$ the dual map of $M.$

A family of functions $\Psi=\{\psi^1,\  \psi^2,\dots,\psi^L\}
\subset\Ltwo$
is a %{\sl
\textit{ family of orthonormal wavelets} when
\begin{equation*}
\bigl\{\psi^l_{j,\g}:\ \g\in\G,\ j\in\mathbf{Z},\ l=1,\dots,L\bigr\}
%\eqno(1')\]
\tag{1$'$}\end{equation*}
is an orthonormal basis for $\Ltwo,$ where
$
\psi^l_{j,\g}=
m^{j/2} \psi^l(M^j x-\g).
$

In \cite{Ca1} we have characterized these wavelet families as follows:

%\te{1.}%{\sl
\begin{theorem}\label{thm:1}
Suppose
$\Psi=\{\psi^1,\  \psi^2,\dots,\psi^L\}
\subset\Ltwo,$ with $\|\psi^l\|_2\ge1$ for $1\le l\le L.$
Then
$\Psi$ is a family of orthonormal wavelets
if and only if the functions $\psi^1,\  \psi^2,\dots,\psi^L$ satisfy 
the following two equations:
\begin{equation*}
\begin{array}{l}\displaystyle{
a)\ \sum_{l=1}^L\sum_{j\in\mathbf{Z}}
|\widehat\psi^l({M^*}^j\xi)|^2=\det A \quad\qquad\qquad \mbox{a.e.}\ 
\xi\in\Rn,
}\\
\displaystyle{b)\
\sum_{l=1}^L\sum_{j=0}^{\infty}
\widehat\psi^l({M^*}^j\xi)
\overline{\widehat\psi^l({M^*}^j(\xi+\gd))}=0
\quad \mbox{a.e.}\ \xi\in\Rn,\ \gd\in\Gd\setminus M^*\Gd.}
\end{array}
%\eqno(I')\]
\tag{$I'$}
\end{equation*}
\end{theorem}

This characterization follows the strategy of the proof of Frazier, 
Garrig\'os, Wang and Weiss
in the classical case. However,
there are several differences. For example, in the general case
$M$ will not be isotropic, since there may be different eigenvalues.
Moreover, $M$ may involve a rotation, so that
a neighborhood $U$ of the origin in general will not be contained in 
$MU.$
Thus, we have to take into account several features of lattices which 
do not show up
in the classical case.
In some parts of the proof this results in more technical complications;
but, in other parts, we have to prove additional results in order to 
obtain
finer estimates.

We recall (see \cite{H--W--W1} and \cite{H--W--W2}) that a Minimally 
Supported Frequency (MSF) wavelet is
a wavelet in $\Ltwo$ such that the measure of $\supp(\widehat\psi)$ is 
minimal.
Here, if $\psi$ is an MSF wavelet, then $|\widehat\psi|=\sqrt{\det 
A}\cdot\chi_T$ for some measurable set $T$ in $\Rn$
with $\meas(T)=(2\pi)^n (\det A)^{-1}.$
Fang and Wang present a thorough study of MSF wavelets in
$L^2({\mathbf R})$ in the classical context.
Dai, Larson and Speegle \cite{D--L--S1} proved that for $\G=\Zn$ and 
every acceptable map there exists an MSF wavelet.
In other words, there always exists a measurable set $K\subset\Rn$ such 
that
the single function $\psi,$ defined by $\widehat\psi= 
\mathrm{const}\cdot\chi_K,$
is an orthonormal wavelet for $\Ltwo.$
This type of set $K$ is called a wavelet set.
Soardi and Weiland \cite{S--W} provided explicit examples of such $K$, 
for $\G=\Zn$ and $M=2I.$

The characterization of all the MSF wavelets can be given in terms of the
following definition.

%\de{1.}
\begin{definition}
Let us define a %{\sl
\textit{ partition} of a measurable set $K\subseteq\Rn$ as a
countable collection $\{K_n\}$ of measurable subsets of $K,$ such that
$K_n\cap K_m$ has measure zero if $n\not= m$ and such that
$\bigcup_{n} K_n=K$ up to a set of measure zero.


The characterization of all the MSF follows directly from the equations 
$(I')$.
\end{definition}
%\te{2.}{\sl 
\begin{theorem}\label{thm:2}
Let $\psi$ be a function such that
$|\widehat\psi|=\alpha\cdot\chi_T,$ for some measurable set $T$ and 
some positive constant $\alpha.$
Then $\psi$ is an MSF wavelet if and only if the following three 
conditions are
satisfied:
\begin{itemize}
\item[(a)] \quad$\alpha^2=\det A$;

\item[(b)] \quad$\{ {M^*}^j T:\ j\in\mathbf{Z}\}$ is a partition of $\Rn$;

\item[(c)] \quad$\{ T+\gd:\ \gd\in\Gd\}$ is a partition of $\Rn.$


\end{itemize} %}
\end{theorem}
%\medskip

\noindent
Another characterization of MSF wavelets can be found in \cite{B--M--M}.
The problem which naturaly arises is how to construct
the support $T$ of the Fourier transform of an MSF wavelet. In 
\cite{Ca1} we give a general
and constructive method to do this and apply this method to produce 
some examples which, in particular, involve the quincunx
lattice.



\section{Characterization
of multiwavelet families and scaling functions} %.}
In the wavelet literature and, in particular, in the study of digital 
signal processing
there is an increasing interest in
wavelets associated with an MRA of multiplicity $d>1,$
the so called multiwavelets (see for example \cite{H}, \cite{Ma}, 
\cite{D--S}). Recent papers show how to convert a single input
stream to multi-inputs which give samples of the data at 
half-intervals. An
example is the Geronimo--Hardin--Massopust multiwavelets \cite{G--H--M} 
which give
experimental results in compression.

Let us recall the defition of Multiresolution Analyses (MRA) of 
multiplicity $d.$
We say that a sequence
$\{V_j\},\ j\in\mathbf{Z},$
of closed subspaces of $\Ltwo$
is a %{\sl
\textit{ multiresolution analysis of multiplicity} $d$ if the 
following four conditions are satisfied:
%\smallskip
\begin{itemize}
\item[(i)]
$\bigcup_{j=-\infty}^{+\infty} V_j\ \ \mbox{is dense in}\ \Ltwo;$
%\smallskip

\item[(ii)]
$V_{j-1}\subseteq V_j,$
for every $j\in\mathbf{Z};$
%\smallskip

\item[(iii)]
for every $f\in \Ltwo$ and $j\in\mathbf{Z}$ we have
$f(x)\in V_j\ \ \Longleftrightarrow\ \  f(M^{-j}x)\in V_0;$
%\smallskip

\item[(iv)]
there exist $d$ functions $\varphi_1, \varphi_2,\dots, \varphi_d$ in 
$V_0$ such that
$\{\varphi_h(x-\g);\ \g\in\G,\ h=1,\dots,d\}$
is an orthonormal basis for $V_0.$
%\smallskip
\end{itemize}

The functions $\varphi_1, \varphi_2,\dots, \varphi_d$ are called the 
%{\sl
\textit{ scaling functions } of the MRA.
Furthermore
the wavelets associated with an MRA of multiplicity $d$ are
called multiwavelets.

In the study of MRA wavelet families, we note an interesting difference 
between
the dyadic map and some general acceptable maps of $\Rn.$
In the first case one cannot have a single
MRA wavelet if $n>1,$ while for such $M$ it suffices to
consider the case $m=|\det M|=2$ to construct such a single $\psi,$ as 
shown in section 4.
%\te{3.}{\sl
\begin{theorem}\label{thm:3}
Let
$\Psi=\{\psi^1,  \psi^2,\dots,\psi^{(m-1)d}\}
\subset\Ltwo$
be a wavelet family associated with the lattice $\G$ and the acceptable 
map $M.$ $\Psi$ is an
MRA wavelet family of multiplicity $d$
if and only if %}
\begin{equation*}
D_{\Psi}(\xi)=\sum_{l=1}^{(m-1)d}\sum_{j=1}^\infty\sum_{\gd\in\Gd}|%
\widehat
\psi^l\bigl({M^*}^j(\xi+\gd)\bigr)|^2=d\cdot\det A 
\quad\mbox{a.e.}\ 
\xi\in\Rn.
%\eqno(2')\]
\tag{2$'$}\end{equation*}
\end{theorem}

A proof of this result is given in \cite{C--G} as a consequence of a 
characterization of wavelet families arising from biorthogonal MRA's 
of multiplicity $d.$ Given the result of Theorem \ref{thm:3}, it is of some 
interest to 
know which functions can be scaling functions of an MRA. Let us mention 
existing 
results on this subject.
%\vskip .5cm

The characterization of single scaling functions was previously known 
in some
particular cases: for $n=1$ and so called localized scaling functions
(see e.g. \cite{K--L}, \cite{Co}); for scaling functions which are 
characteristic functions of a prototile
of a self-similar tiling of $\Rn$ \cite{G--M};
for certain scaling functions constructed by means of tilings of
$\Rn$ \cite{S1}.
The book \cite{H--W} (see also \cite{H--W--W3}) contains a 
characterization of
all scaling functions in the case $n=1.$
Madych, in this context of general lattices and
acceptable dilations, characterizes the single functions $\varphi$ which
generate an MRA and those whose scaling functions are characteristic 
functions are completely described.

In the case $d>1,$
Geronimo, Hardin and Massopust \cite{G--H--M} give necessary
and sufficient conditions for certain functions, constructed using 
fractal functions,
to form an MRA of $L^2({\mathbf R}).$
Herv\'e \cite{H} studied the scaling filters which yield an MRA of 
multiplicity $d>1,$
while De Michele and Soardi \cite{D--S} studied the basic properties of 
MRA of multiplicity $d.$

We can prove the following general characterization (see \cite{Ca2}).

%\te{4.}{\sl
\begin{theorem}\label{thm:4}
 Let $\varphi_1, \varphi_2,\dots, \varphi_d$ be functions of 
$\Ltwo$ and, for every $j\in\mathbf{Z},$ let
\[
V_j= \overline{span \{m^{j/2}\varphi_h(M^jx-\g):\ \g\in\G,\ 
h=1,\dots,d\}}.\]
Then the scale of spaces $\{V_j\}_{j\in\mathbf{Z}}$ is an MRA of 
multiplicity $d$ with scaling
functions
$\varphi_1,\varphi_2,\dots,\varphi_d$ if 
and only if for a.e. $\xi\in\Rn$
the following conditions hold:
\begin{equation}
\sum_{\gd\in\Gd} |\widehat\varphi_h(\xi+\gd)|^2=\det A\quad\text{ for}\ 
h=1,
\dots,d,
\end{equation}
\begin{equation}
\sum_{\gd\in\Gd} \widehat\varphi_h(\xi+
\gd)\overline{\widehat\varphi_k
(\xi+\gd)}=0\quad \text{for}\ h,k=1,\dots,d\ \text{with}\ 
k\not=h,
\end{equation}
\begin{equation}
\lim_{j\to{+\infty}} 
\suhd|\widehat\varphi_h({M^*}^{-j}\xi)|^2={\det
A},
\end{equation}
\begin{equation}\label{eq:6}
\widehat\varphi_h(M^* \xi)=\sukd m_{hk}(\xi) 
\widehat\varphi_k(\xi)
\quad \text{for}\ h=1,\dots, d,
\end{equation}
for some functions $m_{hk}$ of
$\Lstwo.$ %}
\end{theorem}

%\medskip
We recall that in the case $d=1,$ the function $m$ which satisfies 
relation \eqref{eq:6}
is called a low-pass filter.
%\bigskip



\section{Examples of MSF and MRA wavelets, of multiwavelets and a
$C^\infty(\Rtwo)$, nonseparable wavelet with compactly supported 
Fourier transform} %.}
In this last section we give some examples of wavelets constructed
from the previous characterizations.
In particular, we concentrate our attention on the case
$\G=\Z^2$ and the acceptable map
$M(x,y)=(x-y,x+y)$
which was studied in some detail by Cohen and Daubechies \cite{C--D}.
For this map $m=|\det M|=2.$
Clearly $M^*(x,y)=(x+y,y-x)$ and ${M^*}^{-1}(x,y)=\bigl( (x-y)/2,(y+
x)/2\bigr).$

In the first example we produce an MRA, MSF wavelet $\psi_0.$
In the second example, we smooth the wavelet $\psi_0.$ So we obtain a 
single
$C^\infty(\Rtwo),$ nonseparable wavelet with compactly supported 
Fourier transform.
Using the ideas of the construction of $\psi_0,$ in the third example, 
we present
two wavelets $\psi_1$ and $\psi_2$ that are multiwavelets associated
with an MRA of multiplicity 2.
In the fourth example, we consider a different map and give examples of 
singly
generated wavelts in $\Rn.$

%\bigskip\noindent EXAMPLE 1.
\begin{example}
Let us define $T=M^*\Cd\setminus\Cd.$ We recall that 
$\Cd=[-\pi,\pi)^2.$ We claim that $\widehat\psi_0=\chi_T$
defines a wavelet $\psi_0$ associated with a multiresolution analysis of
multiplicity 1. Moreover, $\psi_0$ is also an MSF wavelet.
%\vskip 2.5cm

\begin{figure}[ht]\label{fig:1}
%\quad
%\vskip-1in
%\quad
%\centerline{
%\epsfysize=3in
%\epsfxsize=2.5in
%\epsfbox{figuone.eps}}
\includegraphics{era54el-fig-1}
\caption{
The support $T$ of the Fourier transform of the wavelet $\psi_0.$}
\end{figure}
%\quad 
%\vskip .5cm

The proof is an easy application of Theorems \ref{thm:2} and \ref{thm:3}.
We do not give this proof here, but we indicate how it is possible
to construct the wavelet $\psi_0$ starting from the scaling function
$\varphi_0$ of the MRA.
We define the scale of spaces $\{V_j\}_{j\in\Z}$ of $\Ltwo$, 
$V_j= \overline{span \{2^{j/2}\varphi_0(M^jx-\g):\g\in\Z^2\}}$, with 
the scaling
function $\widehat\varphi_0=\chi_{\Cd}$ (see
Theorem \ref{thm:4}).
Moreover an easy computation shows that
the low-pass filter is
$m_0(\xi)=\sum_{\gd\in 2\pi\Z^2}\chi_{({M^*}^{-1}\Cd)}(\xi+\gd).$

With this fixed map and lattice, we use that a generic scaling function
$\varphi$ and the associated low-pass filter $m$ produce the wavelet 
$\psi$ by
\begin{equation}\label{eq:7}
\widehat\psi(M^*(x,y))=\overline{m(x+\pi,y+\pi)}\widehat\varphi(x,y)
e^{i(x+y)/2}. %\eqno(7)\]
\end{equation}
We recall that
$e^{i(x+y)/2}$ is called the phase and
$p(x,y)=\overline{m(x+\pi,y+ \pi)}
e^{i(x+y)/2}$ is called the high-pass filter.
Moreover, we remember (see \cite{D--S}) that $p$ is such that the matrix
\begin{displaymath}
\left(
\begin{array}{cc}
m(x,y)& m(x+\pi,y+\pi)\\
p(x,y)& p(x+\pi,y+\pi)
\end{array}
\right)
\end{displaymath}
is unitary, a.e. This implies the so called filter relation,
\[
|m(x,y)|^2+|m(x+\pi,y+\pi)|^2=1.
\]
Returning to our example, since the wavelet $\psi_0$ is MSF, we can 
choose an arbitrary
phase in \eqref{eq:7} and it is an easy exercise to obtain $\psi_0$ from 
$\varphi_0.$
Now the question is the following: in order to construct a 
$C^\infty(\Rtwo)$ wavelet, is it possible to smooth the
previous wavelet $\psi_0$?
The next example gives us a positive answer.
\end{example}
%\bigskip\noindentEXAMPLE 2.
\begin{example}
In order to answer the previous question, we do not smooth directly the 
wavelet $\psi_0,$
but we smooth the filter $m_0.$ For every fixed $\epsilon,$ with 
$0<\epsilon\le\pi,$
we construct two low-pass filters,
$\widetilde{m_\epsilon}$ and $m_\epsilon.$ We will see that the 
construction of $\widetilde{m_\epsilon}$ is very easy but
also that this function belongs only to $\mathrm{Lip}_1(\Rtwo).$
However, the idea underlying the construction of 
$\widetilde{m_\epsilon}$ allows us to produce
a more complicated function $m_\epsilon\in C^\infty({\Rtwo})$.

These constructions are similar to the one that produces the 
Lemari\'e--Meyer wavelet from the Shannon wavelet.
We begin by defining a $C^\infty({\mathbf R})$ function $\theta$ with 
the following properties:
%\[\displaylines{\hfill 
\begin{gather*}
0\le\theta(t)\le 1 \quad\text{and}\quad
\theta(t)=1-\theta(1-t)\qquad \text{for every}\ t\in{\mathbf 
R},\\ %\hfill\cr\hfill 
\theta(t)=0\qquad \text{if}\ t\le0,\qquad\qquad
\theta(t)=1\qquad \text{if}\ t\ge1. %\hfill\cr}\]
\end{gather*}
Moreover let us define the function $\nu(t)=
\cos(\frac{\pi}{2}\theta(t))$. Clearly
$\nu\in C^\infty({\mathbf R})$ and $\nu^2(t)+\nu^2(1-t)=1.$
Starting from this function $\nu,$ we construct the low-pass filters.

Let us fix $\epsilon,$ with $0<\epsilon\le\pi.$
For every $(x,y)\in[-\pi,\pi]^2$ we set
\[
\widetilde{m_\epsilon}(x,y)=
\nu\bigl[(|x|+|y|-\pi+\epsilon)/(2\epsilon)\bigr].\]
Next, we extend periodically the definition of $\widetilde{m_\epsilon}$ 
so that, for every $\gd\in 2\pi\Z^2\setminus\{0\}$ and
$\xi\in[-\pi,\pi]^2,$ we have $\widetilde{m_\epsilon}(\xi+
\gd)=\widetilde{m_\epsilon}(\xi).$
Clearly, $\widetilde{m_\epsilon}\in L^2({\mathbf R}^2/{\Gamma^*})$ and 
belongs to $\mathrm{Lip}_1(\Rtwo).$
It is easy to verify that $\widetilde{m_\epsilon}$ is a low-pass filter.

Let us construct the second low-pass filter $m_\epsilon$.
For every $(x,y)\in\Rtwo,$ we set
\[
\begin{align*}
n_\epsilon(x,y)&=
\nu\bigl[(x+y-\pi+\epsilon)/(2\epsilon)\bigr]
\nu\bigl[(x-y-\pi+\epsilon)/(2\epsilon)\bigr]\\
&\quad\cdot
\nu\bigl[(-x+y-\pi+\epsilon)/(2\epsilon)\bigr]
\nu\bigl[(-x-y-\pi+\epsilon)/(2\epsilon)\bigr].\\
\end{align*}
\]
Since $\supp(n_\epsilon)\subseteq[-\pi-\epsilon,\pi+\epsilon]^2,$
we define the function $N_\epsilon$ by periodizing $n_\epsilon:$
\[
N_\epsilon(\xi)=\sum_{
\gd\in 2\pi\Z^2} n_\epsilon(\xi+\gd).
\]
Clearly, $N_\epsilon\in L^2({\mathbf R}^2/{\Gamma^*})\cap
C^\infty(\Rtwo),$ but
the filter relation is not satisfied.
Since $N^2_\epsilon(x,y)+N^2_\epsilon(x+\pi,y+\pi)>0,$ we can prove 
that the function
\[
m_\epsilon(x,y)=\frac{N_\epsilon(x,y)}
{\sqrt{N^2_\epsilon(x,y)+N^2_\epsilon(x+\pi,y+\pi)}},\qquad
\text{for\ every}\ (x,y)\in\Rtwo,\]
is a low-pass filter
belonging to
$C^\infty(\Rtwo).$ We remark that 
$\supp(\widetilde{m_\epsilon})=\supp(m_\epsilon).$

%\vskip 2cm\bigskip\quad
\begin{figure}[ht]\label{fig:2}
%\quad
%\vskip-1in
%\centerline{
%\epsfysize=4in
%\epsfxsize=4.5in
%\epsfbox{figutwo.eps}}
\includegraphics{era54el-fig-2}
\caption{
The  function $|\widehat{\psi}_\epsilon|,$ with $\epsilon=2/5.$}
\end{figure}
%\centerline{\epsfbox{figutwo.eps}}
%\vskip 5cm
%\quad
%\centerline{Figure 2: The  function $|\widehat{\psi}_\epsilon|,$ with 
%$\epsilon=2/5.$}
%\bigskip
\end{example}
%\te{5.}{\sl
\begin{theorem}\label{thm:5}
For every fixed $\epsilon,$ with $0<\epsilon\le\pi/5,$ the function
$\psi_\epsilon$ defined by
\[
\widehat{\psi_\epsilon}(x,y)=
\prod_{j=2}^5
m_\epsilon\bigl({M^*}^{-j}(x,y)\bigr)
m_\epsilon\bigl({M^*}^{-1}(x,y)+(\pi,\pi)\bigr)
e^{ix/2},
\]
is a wavelet of
$L^2(\Rtwo)$ with lattice $\G=\Z^2$ and acceptable map $M$. This
wavelet is MRA and belongs to
$C^\infty(\Rtwo)$. 
Moreover, 
$\psi_\epsilon$ is nonseparable and
 with compactly supported Fourier transform. %}
\end{theorem}
%\noindent\vskip .5cm

The proof of Theorem \ref{thm:5} uses \eqref{eq:7} and
the relation
\begin{equation}\label{eq:8}
\widehat{\varphi}_\epsilon(\xi)=
\prod_{j=1}^\infty
m_\epsilon\bigl({M^*}^{-j}\xi\bigr), %\eqno(8)\]
\end{equation}
where $\varphi_\epsilon$ is the scaling function of the associated MRA.


We remark that the assumption on $\epsilon$ guarantees that
the set 
\[
\supp(\widehat{\varphi}_\epsilon)=
\bigcap_{j\ge1} (\supp(m_\epsilon({M^*}^{-j}\cdot)))
\]
is a compact and connected subset of $[-2\pi,2\pi]^2.$
Hence, $\supp(\widehat{\psi}_\epsilon)$ is also compact and connected.
These properties together with \eqref{eq:8}
show that $\widehat{\varphi}_\epsilon$
is the product of only four factors.

It is possible to show that if we assume $\pi/5<\epsilon\le\pi,$ then
we can construct again an MRA wavelet $\psi_\epsilon$ but the scaling 
function
$\varphi_\epsilon$ is not a finite product of functions
and $\supp(\widehat\varphi_\epsilon)$ is unbounded.

Finally, similar results hold with $\widetilde{m_\epsilon}$ in place of 
$m_\epsilon;$
the corresponding functions $\widehat{\varphi_\epsilon}$ and 
$\widehat{\psi_\epsilon}$
are simpler to describe but less regular.

%\bigskip\noindentEXAMPLE 3.
\begin{example}
Let us define $\Psi=\{\psi^1,\ \psi^2\}$ such that
$\widehat\psi^1=\chi_{T_1}$ and
$\widehat\psi^2=\chi_{T_2}.$

%\bigskip
\begin{figure}[ht]\label{fig:3}
%\centerline{
%\epsfysize=4.5in
%\epsfxsize=2.5in
%\epsfbox{figuthr.eps}}
%\caption{
%\end{figure}
%\vskip -2.5in
%\noindent
%\quad
%\bigskip
\includegraphics{era54el-fig-3}
\caption{The supports $T_1$ and $T_2$ 
of $\widehat\psi^1$ and $\widehat\psi^2$
respectively.}
\end{figure}
%\bigskip

We claim that $\Psi$
is a wavelet family associated with a multiresolution analyses of
multiplicity 2. The proof uses Theorems \ref{thm:1} and \ref{thm:3}.
\end{example}

%\bigskip\noindentEXAMPLE 4.
\begin{example}
In this last example,
we want to show that for every integer $n\ge 2$, there exists an easy MRA 
wavelet
on the lattice
$\Gamma=\Z^n,$ associated with an acceptable map $M_n$ with $\det M_n=2$.
Let  $\widetilde\psi\in
L^2({\mathbf R})$ be an MRA wavelet on the usual lattice $\G=\Z$, 
associated
with the usual dyadic map. Moreover, let $\widetilde\varphi$ be the 
scaling function
of the MRA.
Let us fix $n\ge2$ and let us consider the map
$
M_n^*(x_1,x_2,\dots,x_n)=
(2x_n,x_1,x_2,\dots,x_{n-1})$.
We claim that the function $\psi$
such that
\[
\widehat\psi(x_1,x_2,\dots,x_n)=
\widehat{\widetilde\psi}(x_1)
\widehat{\widetilde\varphi}(x_2)
\widehat{\widetilde\varphi}(x_3)\cdots
\widehat{\widetilde\varphi}(x_n)
\]
is an MRA wavelet on the lattice $\Z^n$, associated with the map $M_n.$
The proof is an easy exercise.
\end{example}
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\end{document}

<\PRE>