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% Author Package file for use with AMS-LaTeX 1.2
\controldates{2-APR-2003,2-APR-2003,2-APR-2003,2-APR-2003}
 
\RequirePackage[warning,log]{snapshot}
\documentclass{era-l}
\issueinfo{9}{05}{}{2003}
\dateposted{April 4, 2003}
\pagespan{32}{39}
\PII{S 1079-6762(03)00109-4}

\copyrightinfo{2003}{American Mathematical Society}

\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}

\theoremstyle{definition}
\newtheorem*{notation}{Notation}
\newtheorem{definition}{Definition}
\newtheorem*{condition}{Basic condition B}
\newtheorem{remark}{\it Remark}

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\newcommand{\R}{\mathbb R}
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\begin{document}


\title[Nonseparable wavelet bases and multiwavelet matrix filters]
{A note on the construction of  nonseparable wavelet bases and 
multiwavelet 
matrix filters  of $L^2(\R^n)$, where $n\geq 2$}
\author{Abderrazek Karoui}
\address{Universit\'e du 7 Novembre \`a  Carthage,  
Institut Sup\'erieur des  Sciences Appliqu\'ees et 
de la Technologie de Mateur, 7030, Tunisia}
\email{abkaroui@yahoo.com}

\date{December 14, 2001}
\commby{Guido Weiss}


\subjclass[2000]{Primary 39B42, 42C05; Secondary 42C15}
\keywords{Multidimensional wavelet bases, multiwavelet bases, 
refinement equation, stability}

\begin{abstract}
 In this note, we announce a general method for the construction of 
nonseparable orthogonal wavelet   bases of  $L^2(\R^n),$ where $n\geq 
2.$ Hence, we prove the existence of such type of wavelet bases for any 
integer $n\geq 2.$ Moreover, we show that this construction method can 
be extended to the construction of $n$-D multiwavelet matrix filters. 
\end{abstract}

\maketitle

\section{Introduction}
In this note, we are interested in the construction of dyadic 
nonseparable compactly supported wavelet bases of $L^2(\R^n).$ This 
type of wavelet bases is defined as follows.

\begin{definition}
Consider a discrete set of functions 
\begin{equation*}
\{\Psi^i_{j,\bk}(\ex)=2^{-j/2}\Psi^i(2^{-j}\ex-\bk),\ j\in 
\Z,\ \bk\in \Z^n,\ i=1,\dots,2^n-1\},
\end{equation*}
 obtained by translations and dilations of $2^n-1$ mother wavelets 
$\Psi^i,\ i=1,\dots,2^n-1$. If  the set $\{\Psi^i_{j,\bk}\}_{i,j,\bk}$ 
satisfies the following three conditions:
\begin{itemize}
\item[($c_1$)] $\{\Psi^i_{j,\bk}\}_{i,j,\bk}$  is orthogonal with 
respect to the usual
$L^2$ inner product  
\begin{equation*}
\langle f,g\rangle=\int_{\R^n} f(\ex)\overline{g}(\ex)\, d\ex,
\end{equation*}
\item[($c_2$)] $\forall f\in L^2(\R^n),$ $ 
f(\ex)=\sum_{i=1}^{2^n-1}\sum_{j\in \Z,\bk\in
\Z^n}\langle f,\Psi^i_{j,\bk}\rangle\Psi^i_{j,\bk}(\ex)$, where the equality  holds 
a.e., and
\item[($c_3$)] $\{\Psi^i_{j,\bk}\}_{i,j,\bk}$ satisfies the stability 
condition, that is, $\exists c_2 > c_1 >0$ such that 
for any $f\in L^2(\R^n), $ we have
\begin{equation*}
c_1\|f\|_2^2 \leq 
\sum_{i=1}^{2^n-1}\sum_{j\in \Z,\bk\in \Z^n}|\langle f,\Psi^i_{j,\bk}\rangle|^2
\leq c_2 \|f\|_2^2,
\end{equation*}
\end{itemize}
then $\{\Psi^i_{j,\bk}\}_{i,j,\bk}$ is called a dyadic orthogonal 
wavelet basis of $L^2(\R^n).$
\end{definition}

Note that in general, the set of mother wavelets $\Psi^i,\,  
i=1,\ldots,2^n-1$ is related to a  mother scaling function $\Phi$ which 
is an appropriate solution of  the following refinement equation:
\begin{equation}\label{eq0.1}
\Phi(\ex)= 2^n \sum_{\bk\in \Z^n}\alpha_k\Phi(2\ex-k),
\end{equation}
with $\int \Phi(\ex)\, d\ex =1$.
Note that if in (\ref{eq0.1}),  $(\alpha_{\bk})_{\bk}$ is a finite 
sequence satisfying $\sum_{\bk}\alpha_{\bk}=1$ and if ${\mathcal H}_0$ 
denotes the $n$-variate trigonometric polynomial given by
\begin{equation*}
{\mathcal H}_0(\omega_1,\ldots
\omega_n)=\sum_{\bk} \alpha_{\bk} e^{-i \bk\cdot 
(\omega_1,\ldots,\omega_n)},
\end{equation*}
then 
\begin{equation*}
\widehat{\Phi}(\omega_1,\ldots,\omega_n)
=\prod_{j=1}^{\infty}{\mathcal H}_0
\left(\frac{\omega_1}{2^j},\ldots,\frac{\omega_n}{2^j}\right).
\end{equation*}
 In this case, the different mother wavelets
$\Psi^i, i=1,\ldots,2^n-1$  are given by 
\begin{equation*}
\widehat \Psi^i(\omega_1,\ldots,\omega_n)
={\mathcal H}_i\left(\frac{\omega_1}{2},
\ldots,\frac{\omega_n}{2}\right)\widehat{\Phi}
\left(\frac{\omega_1}{2},\ldots,
\frac{\omega_n}{2}\right),\quad\forall i=1,\ldots,2^n-1,
\end{equation*} 
for some appropriate $n$-variate trigonometric polynomials, also called 
high-pass wavelet filters.

Finally, we should mention that unlike the 1-D case where an extensive 
work has been done,  the construction of nonseparable orthogonal 
wavelet bases is still  a challenging problem. To this date, only very 
few references have dealt with the construction of  such multivariate 
orthogonal wavelet bases in some special cases; see \cite{Ayache}.  
Nonetheless,  many references have dealt with the construction of 
multidimensional biorthogonal wavelet bases. 

Finally, this note  is organized as follows. In section~2, we construct 
the different  orthogonal wavelet filters, candidates for generating 
nonseparable wavelet bases of $L^2(\R^n)$, and study the stability of 
the associated wavelets.  In section~3, we show how to extend the 
previous construction to the design of multidimensional multiwavelet 
filters.

\begin{notation} In this note, we denote
 $\om_k=(\omega_1,\ldots,\omega_k)\in \R^k, 
\et_i=(\eta_i^1,\ldots,\eta_i^n)\in E_n=\{0,\pi\}^n, 
\pi_k=(\pi,\ldots,\pi)\in \R^k$.
\end{notation}

\section{Construction of  $n$-D compactly supported and orthogonal 
wavelet bases}

\subsection{Construction of multivariate orthogonal wavelet filters}

We first construct a family of 2-bands orthogonal and compactly 
supported low-pass wavelet filters ($n$-variate trigonometric 
polynomials) 
$H_n(\omega_1,\ldots,\omega_n).$  This family is given by the following 
proposition.
\begin{proposition} 
Let $H_1(\omega)$ and ${\displaystyle G_1(\omega)=e^{-i\omega}\overline
H_1(\omega+\pi)}$ be any  1-D low-pass and the corresponding  high-pass 
orthogonal wavelet filters of  $L^2(\R),$ respectively.
Define an $n$-variate low-pass wavelet filter 
$H_n(\omega_1,\ldots,\omega_n)$ by the following
iterative process: 
\begin{eqnarray*}
\forall \,\,\, 2\leq k\leq n & &\mbox{ choose an integer\ } 1\leq 
l_k\leq k-1,\\
P_{k-1}(\omega_1,\ldots,\omega_k) &=& 
H_{k-1}(2\omega_1,\ldots,2\omega_{k-1}),\\
Q_{k-1}(\omega_1,\ldots,\omega_k) &=& H_{k-1}(2\omega_1+
\pi,\ldots,2\omega_{k-1}+\pi),\\
H_k(\omega_1,\ldots,\omega_k)&=&
P_{k-1}(\omega_1,\ldots,\omega_{k-1})H_{l_k}(\omega_{k-l_k+
1},\ldots,\omega_k)\\
&& +\,Q_{k-1}(\omega_1,\ldots,\omega_{k-1})G_{l_k}(\omega_{k-l_k+
1},\ldots,\omega_k),
\end{eqnarray*}
where 
$G_l(\omega_1,\ldots,\omega_l)=e^{-i\omega_l}\overline{H_l}(\omega_1+
\pi,
\ldots, \omega_l+\pi).$ 
Then  $H_n(0,\ldots,0)=1.$  
Moreover, $H_n$ satisfies the following orthogonality condition:
\begin{equation}\label{eq2.4}
|H_n(\omega_1,\ldots,\omega_n)|^2+|H_n(\omega_1+\pi,\ldots,\omega_n+
\pi)|^2=1,\quad
\forall (\omega_1,\ldots,\omega_n)\in {\bf \R}^n.
\end{equation}
\end{proposition}

\begin{proof} See \cite{karoui2}.
\end{proof}

It is well known that the construction  of dyadic, orthogonal and 
compactly supported wavelet bases of 
$L^2(\R^n)$  requires the construction of one low-pass filter 
${\mathcal H}_0$ and  $2^n-1$ high-pass
filters  ${\mathcal H}_i, i=1,\ldots,2^n-1$. These different wavelet 
filters must satisfy the following 
matrix equation (see \cite{kv,ym}):\\[0.1in]
$
\left[\begin{array}{cccc}
\overline{{\mathcal H}_0(\mbox{\boldmath $\om$})}& 
\overline{ {\mathcal H}_0(\mbox{\boldmath $\om$}+
\mbox{\boldmath $\eta$}_1)}&\cdots &
\overline{{\mathcal H}_0(\mbox{\boldmath $\om$}+\mbox{\boldmath 
$\eta$}_{2^n-1})}\\
\overline{ {\mathcal H}_1(\mbox{\boldmath $\om$})} & 
\overline{ {\mathcal H}_1(\mbox{\boldmath $\om$}+\mbox{\boldmath 
$\eta$}_1)}&\cdots &
\overline{ {\mathcal H}_1(\mbox{\boldmath $\om$}+\mbox{\boldmath 
$\eta$}_{2^n-1})}\\
\vdots &\vdots&&\vdots\\
\overline{ {\mathcal H}_{2^n-1}(\mbox{\boldmath $\om$})} & 
\overline{ {\mathcal H}_{2^n-1}(\mbox{\boldmath $\om$}+
\mbox{\boldmath $\eta$}_1)}&
\cdots &\overline{{\mathcal H}_{2^n-1}(\mbox{\boldmath $\om$}+
\mbox{\boldmath $\eta$}_{2^n-1})}\\
\end{array}
\right]\\[0.1in]
\hspace*{1.2cm}\times\left[\begin{array}{cccc}
{\mathcal H}_0(\mbox{\boldmath $\om$}) & {\mathcal H}_1(\mbox{\boldmath 
$\om$})& 
\cdots & {\mathcal H}_{2^n-1}(\mbox{\boldmath $\om$})\\
{\mathcal H}_0(\mbox{\boldmath $\om$}+\mbox{\boldmath $\eta$}_1) & 
{\mathcal H}_1(\mbox{\boldmath $\om$}+\mbox{\boldmath $\eta$}_1) & 
\cdots & {\mathcal H}_{2^n-1}(\mbox{\boldmath $\om$}+\mbox{\boldmath 
$\eta$}_{1})\\
\vdots &\vdots&&\vdots\\
{\mathcal H}_0(\mbox{\boldmath $\om$}+\mbox{\boldmath $\eta$}_{2^n-1}) & 
{\mathcal H}_{1}(\mbox{\boldmath $\om$}+\mbox{\boldmath 
$\eta$}_{2^n-1})& 
\cdots &{\mathcal H}_{2^n-1}(\mbox{\boldmath $\om$}+\mbox{\boldmath 
$\eta$}_{2^n-1})
\end{array}  
\right] ={I}_{2^n}, $\\[0.1in]
where the  $\et_i$ are the different points of the set  
$E_n=\{0,\pi\}^n.$ A solution to the above
matrix equation is obtained by the use of a family of  $n-1$
matrices $D_i\in \Z^{n\times n}$ satisfying the following three 
properties:
\begin{itemize}
\item[($P_1$)] $\forall \et_j\in E_n,$  
$D_i(\et_j)=D_i(\et_j^s) \bmod (2\pi \Z^n),$ where $\et_j^s=\pi_n-\et_j.$
\item[($P_2$)] For all $\et_{j'}\neq\et_j$ and all  
$\et_{j'}\neq\et_j^s,$ we have
$D_i(\et_j)\neq D_i(\et_{j'}) \bmod (2\pi {\bf \Z}^n).$
\item[($P_3$)] If  $F_i=D_iD_{i-1}\cdots D_1(E_n)/_{2\pi \Z^n}$, 
then 
$ |F_i|=\frac{|E_n|}{2^i}$ and $F_i$ is a symmetric 
subset of  $E_n,$ i.e. $\forall \et \in F_i,\,\,\et^s\in F_i.$
\end{itemize}

\begin{remark} If  $\{e_k\}_{k=1,\ldots,n}$ denotes the usual basis of  
$\R^n$, and if  $\forall 1\leq i\leq n,$ we denote  $D_i$
the matrix defined by
\begin{equation*}
D_i(e_k)=e_k \mbox{\ if \ } k\neq i,i+1,\quad
D_i(e_i)=e_i+e_{i+1},\quad D_i(e_{i+1})=\sum_{k=1, k\neq i}^{n}e_k-e_i,
\end{equation*}
then the family  $\{ D_i;\ i=1,\dots,n-1\}$ satisfies the above three  
properties.
\end{remark}

The construction of 
${\mathcal H}_0$  is then given by the following proposition.

\begin{proposition}
Let  $H_0(\omega_1,\ldots,\omega_n)$ be the $n$-D filter of  
Proposition~1.
Define  ${\mathcal H}_0$ by
\begin{equation*}
{\mathcal H}_0(\omega_1,\ldots,\omega_n)=\prod_{k=0}^{n-1} 
H_0(D_k\cdots D_0 
(\omega_1,\ldots,\omega_n)).
\end{equation*}
 Then 
${\mathcal H}_0(0,\ldots,0)=1.$  Moreover,
 ${\mathcal H}_0$ satisfies the following orthogonality condition:
\begin{equation}\label{eq2.6}
\sum_{i=0}^{2^n-1} \left|{\mathcal H}_0(\om_n+
\et_i)\right|^2=1,\quad\forall
\om_n\in {\bf \R}^n.
\end{equation}
\end{proposition}

\begin{proof} Since  $H_0(0,\ldots,0)=1$, we have
\begin{equation*}
{\mathcal  H}_0(0,\ldots,0)=\prod_{k=0}^{n-1}H_0(D_k\cdots 
D_0(0,\ldots,0))=1.
\end{equation*}
Moreover, since
\begin{equation*}
D_1(\et_i)=D_1(\et_i^s)\bmod (2\pi {\bf \Z}^n),\quad\forall \et_i\in E_n,
\end{equation*}
it follows that
\begin{eqnarray*}
\lefteqn{\sum_{i=0}^{2^n-1}\left|{\mathcal H}_0(\om+\et_i)\right|^2}\\
&=&\!\!\!\!\sum_{\et_i\in F_1}\left[|{H}_0|^2(\om+\et_i)+|{ H}_0|^2(\om+
\et_i^s)\right]\cdot \prod_{k=1}^{n-1}|H_0|^2(D_k\cdots (D_1
(\om)+\et_i))\\
&=&\!\!\!\!\sum_{\et_i\in F_1} \prod_{k=1}^{n-1}|H_0|^2(D_k\cdots (D_1
(\om)+\et_i))\\
&=&\!\!\!\!\!\sum_{\et_i\in F_2}\!\!\!\left[|{H}_0|^2(D_1(\om)+\et_i)+|{
H}_0|^2(D_1(\om)+\et_i^s)\right]\!\cdot\! 
\prod_{k=2}^{n-1}|H_0|^2(D_k\cdots (D_1
(\om)+\et_i))\\
&=&\!\!\!\!\sum_{\et_i\in F_2} \prod_{k=2}^{n-1}|H_0|^2(D_k\cdots (D_1
(\om)+\et_i))\\
&\vdots&\\
&=&\!\!\!\! \sum_{\et_i\in F_{n-1}} |H_0|^2(D_{n-1}\cdots D_1(\om )+
\et_i)\\
&=&|H_0|^2(D_{n-1}\cdots D_1(\om)+\et_0)+|H_0|^2(D_{n-1}\cdots D_1(\om)+
\et_0^s)=1.\quad 
\end{eqnarray*}
\end{proof}

Once ${\mathcal H}_0$ is constructed, one is faced with the problem of 
finding the remaining 
 $2^n-1$ high-pass wavelet filters  ${\mathcal H}_i, i=1,\ldots,2^n-1$, 
satisfying the following 
equations:
\begin{equation}\label{eq1}
\sum_{\et_i\in E_n}{\mathcal H}_j(\om_n+\et_j)\overline{{\mathcal 
H}_{j'}}(\om_n+\et_i)=\delta_{jj'},\quad\forall \, 0\leq j,j'\leq 
2^n-1.
\end{equation}

In general, a solution of (\ref{eq1}) is hard to obtain. Nevertheless 
and thanks to the particular structure
of ${\mathcal H}_0,$ a solution to the above equations is given by the 
following theorem. Note that the biorthogonal version of this theorem 
is given in \cite{karoui}.

\begin{theorem}
Let  $H_0$  be the $n$-D wavelet filter of  Proposition~1.
Let   $D_1,D_2,\ldots$, $D_{n-1}$ be a set of matrices with integer
coefficients and satisfying
the three properties  ($P_1$), ($P_2$) and ($P_3$). 
Define a filter  $H_1$  by
\begin{equation*}
H_1(\om_n)=e^{-i\omega_1} \overline{H_0}(\om_n+\pi_n).
\end{equation*}
For $i=1,\ldots, 2^n-1$, define ${\mathcal H}_i$ by
\begin{equation}\label{eq2.88}
{\mathcal H}_i(\om_n)=\prod_{j=0}^{n-1}[\ep_j^iH_0(D_jD_{j-1}\cdots  
D_0\om_n)+
(1-\ep_j^i)H_1(D_jD_{j-1}\cdots D_0\om_n)],
\end{equation}
where  $(\ep_0^i,\ep_1^i,\ldots,\ep_{n-1}^i)_{i=1,2^n-1}$ are the 
different  points of
$\{0,1\}^n\setminus (0,0,\ldots,0)$. Then ${\mathcal H}_i,
 i=0,2^n-1,$ is a  solution of  \eqref{eq1}.
\end{theorem}

\begin{proof} See \cite{karoui2}.
\end{proof}

\subsection{Stability of the orthogonal wavelet bases}

It is well known that a solution of  (\ref{eq1})  does not ensure the 
construction of wavelet bases of 
 $L^2(\R^n).$ In fact, the translates of the scaling function  $\Phi$ 
derived from ${\mathcal H}_0$ must satisfy
the stability condition, that is,
\begin{equation}\label{eq2}
0< c_1\leq \sum_{{\bf k}\in \Z^n} |\widehat{\Phi}(\om_n+2\pi {\bf 
k})|^2 \leq c_2 <+\infty,
\end{equation}
for some constants $c_1, c_2.$ 

It is known (see \cite{shen1,law,shen2}) that 
if  $T_{{\mathcal H}_0}$ denote the transition operator associated with 
 ${\mathcal H}_0$ and defined by
\begin{equation*}
T_{{\mathcal H}_0}(f)(\om_n)=\sum_{\et_j\in E_n}|{\mathcal 
H}_0(\frac{\om_n}{2}+\et_j)|^2 f(\frac{\om_n}{2}+\et_j)
\end{equation*}
and if  $V$ is a finite-dimensional subspace of $n$-variate 
trigonometric polynomials and $V$ is invariant under the action of 
$T_{{\mathcal H}_0},$ then the stability and the orthogonality of our 
wavelet basis can be deduced
from the spectral radius of   $T_{{\mathcal H}_0}/_V$, the restriction 
of  $T_{{\mathcal H}_0}$ to  
 $V.$  More precisely, if  $1$ is a simple eigenvalue of  
$T_{{\mathcal H}_0}/_V$ and if the other eigenvalues are inside the 
unit circle, then  ${\mathcal H}_0$ 
gives rise to an orthogonal wavelet basis of  $L^2(\R^n).$ The 
following theorem 
provides us with an explicit invariant subspace  $V$ under  
$T_{{\mathcal H}_0}$ as well as the different
coefficients of the matrix representing  $T_{{\mathcal H}_0}/_V.$

\begin{theorem}
Let  ${\mathcal H}_0$ be the filter of  Proposition~2 and  
\begin{equation*}
|{\mathcal H}_0(\omega_1,\ldots,\omega_n)|^2=\sum_{m_1,\ldots,m_n=-M}^M
\beta_{m_1,\ldots,m_n}\cos\left[\sum_{i=1}^n m_i\omega_i\right].
\end{equation*}
Let  $V=\operatorname{span} \Omega,$ where  $ \Omega=\{U_{\bf k}=\cos
\left[\sum_{i=1}^n k_i\omega_i\right],\ {\bf k}=(k_1,\ldots,k_n)\in 
\supp |{\mathcal H}_0|^2\}$. 
Define a matrix  $B_{{\mathcal H}_0}$  by
\begin{equation*}
 B_{{\mathcal H}_0}(U_{{\bf k}})=2^{n-1}\sum_{{\bf l,k,l+2k}\in \supp 
|{\mathcal H}_0|^2} \beta_{{\bf k+2l}}[U_{{\bf l+k}}+U_{{\bf l}}].
\end{equation*}
Then, $ B_{{\mathcal H}_0}$ represents the restricted operator  
$T_{{\mathcal H}_0}/_V.$
In particular, if  1 is a simple eigenvalue of  $B_{{\mathcal H}_0}$ 
and if the other eigenvalues 
are  inside the unit circle, then ${\mathcal H}_0$  generates an 
orthogonal wavelet basis of   $L^2({\bf \R}^n).$
\end{theorem}

\begin{proof} See \cite{karoui2}.
\end{proof}

\section{Construction of $n$-D multiwavelet filters}

In this section we show that the techniques of the previous section 
can be used for the construction of orthogonal multiwavelet bases of 
$L^2(\R^n).$ Note that a 1-D orthogonal multiwavelet basis of 
multiplicity $r\geq 2, r\in \mathbb{N}$ is an orthogonal and  stable set 
generated by translations and dilations of a vector-valued mother 
multiwavelet function $\Psi=(\psi^1,\ldots,\psi^r)^T.$ The associated 
vector-valued multiscaling function $\Phi=(\phi^1,\ldots,\phi^r)^T$ 
satisfies the functional equation
\begin{equation}\label{eq1.1}
\Phi(x)=\sum_{\alpha\in \Z}P_{\alpha}\Phi(2x -\alpha),
\end{equation}
where $P_{\alpha}$ is an $r\times r$ matrix with real coefficients. If 
we define the
trigonometric matrix function $P(\omega)$ by
\begin{equation}\label{eq1.2}
P(\omega) = \frac{1}{2}\sum_{\alpha\in \Z} P_{\alpha}\exp(-i\alpha 
\omega),
\end{equation}
then by applying Fourier transform to (\ref{eq1.1}), the latter can be 
written as
\begin{equation}\label{eq1.3}
\widehat\Phi(\omega)= P\left(\frac{\omega}{2}\right)\cdot\widehat\Phi%
\left(\frac{\omega}{2}\right).
\end{equation}
Unlike the one-dimensional case, where an extensive work has been done 
in the theory and the design of  1-D multiwavelet bases, 
only very few references have dealt with the theory and the design of 
multidimensional multiwavelet bases \cite{lai}. For the sake of 
simplicity, we will be concerned with the construction of $n$-D
multiwavelet matrix filters  in the special case $r=2.$ Moreover, we 
will use very often the following basic condition B on the low-pass 
matrix filter $P(\om)$, which is
a necessary condition for the existence of multiwavelet basis and it is 
 due to \cite{shen2}. This condition is given as follows:

\begin{condition} We say that the matrix mask $P$ satisfies the 
basic condition B, if
the following two conditions are satisfied:
\begin{itemize}
\item[($C_1$)] $P(0)$ has the form $ 
\left[\begin{smallmatrix} 1 & 0\\0& \lambda\end{smallmatrix}\right]$,
where $\lambda$ is an $(r-1)\times (r-1)$ matrix such that 
$\rho(\lambda) <1.$ Here
$\rho(\cdot)$ denotes the spectral radius of a given matrix.
\item[($C_2$)] $\forall \mu\in \{0,1\}^n,$   $e_1^T P(\mu 
\pi)=\delta_{\alpha} e_1^T,$ where $e_1^T=(1,0,\ldots,0)\in \R^r.$
\end{itemize}
\end{condition}

It is shown in \cite{shen2} that if $P(\om)$ satisfies condition B, 
then the infinite product $\prod_{j=1}^{\infty} 
P\left(\frac{\om}{2^j}\right)$ converges to the $r\times r$ matrix 
$\big[\widehat \Phi(\om) A,0,\ldots,0\big]$, where $A$ is an 
eigenvector of $P(0)$ corresponding to the eigenvalue $1.$ Also, it is 
well known that a necessary condition for the orthonormality of the $r$ 
components of $\Phi$ is the following equality:
\begin{equation}\label{eq1.4}
P(\omega) P^*(\omega)+P(\omega+\pi)P^*(\omega+\pi) = I_r,
\end{equation}
where $P^* = \overline{P}^T,$ the transpose of the conjugate of $P.$
Also, it is well known that the construction of $n$-D dyadic 
multiwavelet matrix filters with multiplicity $r=2$ requires the
design of one low-pass $2\times 2$ matrix filter ${\mathcal P}_n(\om)$  
and $2^n-1$ high-pass $2\times 2$ matrix filters ${\mathcal 
Q}_n^i(\om)$ such that the following matrix equation holds:
\begin{equation}\label{eq3.11}
\left[\begin{array}{l} {\mathcal P}_n(\om+\mu  \pi)\\ {\mathcal 
Q}^1_n(\om+\mu\pi)\\ \vdots\\{\mathcal Q}^{2^n-1}_n(\om+\mu\pi)
\end{array}\right]_{\mu\in \{0,1\}^n}\cdot \left[\begin{array}{l} 
{\mathcal P}_n(\om+\mu  \pi)\\ {\mathcal Q}^1_n(\om+\mu\pi)\\ 
\vdots\\{\mathcal Q}^{2^n-1}_n(\om+\mu\pi)
\end{array}\right]^*_{\mu\in \{0,1\}^n}=I_{2^{2n}}.
\end{equation}

One major difficulty in the design of multidimensional multiwavelet 
matrix filters is the noncommutativity of the matrix product.
As we will see, this extra difficulty does not affect very much the 
construction method of the previous section.  
To solve (\ref{eq3.11}), we  need the result of the following 
proposition, whose proof is given in \cite{karoui2}.
\begin{proposition}
Let $P_1, Q_1$ be a 1-D matrix filters generating 1-D multiscaling 
function and multiwavelet with multiplicity 2. Let 
$H_k$ be the $p$-variate wavelet filter of Proposition~1.
For $2\leq k\leq n,$ define two sequences of matrix filters $P_k, \, 
Q_k$ by the following iterative process:
\begin{eqnarray*}
\forall \,\,\, 2\leq p\leq k, & &\mbox{ let } \\
L_p(\om_p) &=& H_p(2\om_p),\quad M_p(\om_p)=H_p(2\om_p+\pi_p),\\
P_p(\om_p)&=&L_p(\om_p)P_{p-1}(\om_{p-1})+
\overline{M_p}(\om_p)Q_{p-1}(\om_{p-1}),\\
Q_p(\om_p)&=&M_p(\om_p)P_{p-1}(\om_{p-1})-\overline{L_p}(\om_p)Q_{p-1}(%
\om_{p-1}).
\end{eqnarray*}
Then $P_n(\cdot), Q_n(\cdot)$ is a solution of the following equation:
\begin{equation}\label{eq3.7}
\left[\begin{array}{ll}P_n(\om_n)&P_n(\om_n+
\pi_n)\\Q_n(\om_n)&Q_n(\om_n+\pi_n)\end{array}\right]\cdot
\left[\begin{array}{ll}P^*_n(\om_n)&Q^*_n(\om_n)\\P^*_n(\om_n+
\pi_n)&Q^*_n(\om_n+\pi_n)\end{array}\right]=I_4.
\end{equation}
\end{proposition}

Next, the following proposition gives us an $n$-variate matrix filter 
candidate for
generating an $n$-D multiscaling function.

\begin{proposition}
Let $P_n$ be as given by Proposition~3 and let $D_0=I_n$ and $D_i, 
i=1,\dots,n-1$ be a set of matrices satisfying the three properties ($P_1$), 
($P_2$) and ($P_3$).  Define an $n$-variate matrix filter ${\mathcal 
P}_n(\om_n)$ by
\begin{equation}\label{eq3.9}
{\mathcal P}_n(\om_n)=\prod_{k=0}^{n-1} P_n(D_{n-k-1}\cdots D_0 \om_n);
\end{equation}
then ${\mathcal P}_n$ satisfies the basic condition B. Moreover, 
${\mathcal P}_n$ is a solution of the following matrix equation:
\begin{equation}\label{eq3.8}
\sum_{\et_k\in E_n=\{0,\pi\}^n}{\mathcal P}_n(\om_n+\et_k){\mathcal 
P}^*_n(\om_n+\et_k)=I_2,\quad \forall \om_n\in \R^n.
\end{equation}
\end{proposition}

\begin{proof} See \cite{karoui2}.
\end{proof}

Finally, the different $2^n-1$ matrix filters candidates for generating 
the different $n$-D mother multiwavelet functions 
are given by the following theorem, whose proof is given in 
\cite{karoui2}.

\begin{theorem}
Let $P_n,$ $Q_n$ be as given by Proposition~3 and let ${\mathcal P}_n$ 
be as given by the previous proposition.
Let $D_0=I_n$ and let $D_1,\ldots,D_{n-1}$ be a set of matrices satisfying the 
three properties $P_1,P_2,P_3.$
Define $2^n-1$ matrix filters ${\mathcal Q}_n^i, i=1,\dots,2^n-1$ by
\begin{equation}\label{eq3.12}
{\mathcal 
Q}^i_n(\om)=\prod_{k=0}^{n-1}\left[\epsilon_k^iP_n(D_{n-k-1}\cdots 
D_0\om)+(1-\epsilon_k^i)Q_n( D_{n-k-1}\cdots D_0\om)\right],
\end{equation}
where $(\ep_0^i,\ldots,\ep_{n-1}^i)_{i=1,2^n-1}$ are the different 
points of $\{0,1\}^n\setminus \{\mbox{\boldmath $0_n$}\}.$
The set \begin{equation*}
\{{\mathcal P}_n,{\mathcal Q}^1_n,\ldots,{\mathcal Q}^{2^n-1}_n\}
\end{equation*}
 is a solution of \eqref{eq3.11}.
\end{theorem}

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\end{document}