\section{General $W$-Matrices}

In this section we give some properties of general $W$-matrices.  
Detailed 
proofs are omitted.  As mentioned in the preceding
section, there is more than one way of constructing  the 
beginning and trailing rows of a $W$-matrix and all valid
construction gives rise to multiresolution analysis with discrepancies that
affect only a few boundary values.  We therefore consider two
$W$-matrices equivalent if they differ only in some boundary rows,
in other words, if they are generated by the same
$ g $- and $ h $-vectors.  

\begin{THEOREM}
If $ W_1 $ and $ W_2 $ are $W$-matrices of equal size, then their product
$ W_1W_2 $ is again a $W$-matrix.
\end{THEOREM}

One can use this theorem to generate new $W$-matrices from known ones, especially
ones of higher order from those of lower order.

\begin{THEOREM}\Label{eq} 
Let $ W $ be generated by a pair of vectors 
$ (\mbox{\bf g},\mbox{\bf h}) $, and let $ a,b $ be two
nonzero real numbers.  Then the $W$-matrix generated by
$ (a\mbox{\bf g},b\mbox{\bf h}) $ is functionally the
same as $ W $ in the sense that they correspond to the same
multiresolution analysis.  In a similar vein, the $W$-matrix generated by 
$ (\mbox{\bf h},\mbox{\bf g}) $ differs from the first one
only by a permutation, and we can regard them as being
functionally equivalent.
\end{THEOREM}

$W$-matrices of order 4 can be completely characterized by the
following theorem.
The inverse of the general $W$-matrix of order 4 have been computed
explicitly with the help of  Maple.  We omit the 
formula.  

\begin{THEOREM}\Label{wm} 
Let $ A,B,C,D,a $, and $ b $ be any six given real numbers such that
\begin{equation}    AD - BC \neq  0  \quad  \mbox{ and } \quad  a \neq  b. \end{equation}
Then the vectors
\begin{equation}     \left[ \begin{array}{cccccccccc}    aA, &  aB, &  bC, &  bD  \end{array} \right ] \end{equation}
and
\begin{equation}     \left[ \begin{array}{cccccccccc}    A, &  B, &  C, &  D  \end{array} \right ] \end{equation}
generate a $W$-matrix.  Conversely, any $W$-matrix of order 4 can be 
generated this way.
\end{THEOREM}

\begin{THEOREM}
In general, given any vector $ \mbox{\bf h}=[h_1,h_2,\cdots ,h_{2n}] $
of even length,
one can supplement it
with a vector $ \mbox{\bf g} $ to form a $W$-matrix if the 
$ (2n-1)\times 2n $ matrix
\begin{equation}    \mbox{\bf A} =  \left( \begin{array}{cccccccccc}    h_2 &  -h_1  \\  h_4 &  -h_3 &  h_2 &  -h_1  \\   &   &   &   & \ddots  \\   &   &   &   &   &  -h_{2n-1} &  h_{2n}     \end{array} \right ) \end{equation}
has full rank.  More precisely, let {\bf B} be the matrix obtained
from {\bf A} by deleting the middle row of {\bf A}.  The solution
space of the matrix equation 
\begin{equation}    \mbox{\bf B} \, [ z_1, z_2, \cdots , z_{2n} ]' = 0 \end{equation}
is a two-dimensional linear space that contains $ \mbox{\bf h} $.
Any nonzero vector in the solution space other than 
a multiple of $ \mbox{\bf h} $ can be used as {\bf g}.

If the vector {\bf h} is of odd length, 
it can be considered to be a vector of even length with 
the last component being zero, and the above result can be
applied.
\end{THEOREM}

A simple corollary of Theorem~4 is the existence of 
$W$-matrices generated by symmetric basic vectors.

\begin{THEOREM}
For any given $ n $ numbers $ h_1,h_2,\cdots ,h_n $, one forms the
vector $ \mbox{\bf h}=[h_1,h_2,\cdots ,h_n,-h_n,\cdots ,-h_2,-h_1] $.
If \,{\bf h} satisfies the conditions in Theorem~4, then there exists
a symmetric {\bf g} such that $ ({h},{\bf g}) $ and {\bf g} generates a
$W$-matrix.
\end{THEOREM}

\begin{THEOREM}
All orthogonal $W$-matrices of order 4 are generated by the pair
of basic vectors (after being normalized to be of unit length)
\begin{equation}    \mbox{\bf g} = [ 1, \alpha , \alpha \beta , -\beta  ] \end{equation}
\begin{equation}    \mbox{\bf h} = [ 1, \alpha , -\alpha /\beta , 1/\beta  ] \end{equation}
for arbitrary real numbers $ \alpha  $ and $ \beta  $.  If, in addition, we
require that {\bf h} has vanishing zero-th moment, then
\begin{equation}    \beta  = \frac{\alpha -1}{\alpha +1} . \end{equation}
The particular choice of $ \alpha =\sqrt{3} $ leads to the $ D_4 $  matrix.
\end{THEOREM}
