We now turn to consider the intriguing case where *a*(*x*) may change
sign.
In this section we take a rather detailed look at
the prototype case of :

We shall show that the solution operator
associated with (weighted.6) is also
similar to a unitary matrix -- consult (weighted.20) below for the precise statement.
This in turn leads to the announced weighted -stability.
It should be noted, however, that the similarity transformation in
this case involves the ill-conditioned Jordan blocks; as the
condition number of the latter may grow linearly with *N*, this in turn
implies *weak* -instability.

We begin by noting that the Fourier approximation (weighted.6)
admits a rather simple representation
in the Fourier
space, using the (2*N*+1)-vector of its Fourier coefficients, . With the periodic extension of
in mind we are able
to express the interpolant of as

so that the Fourier approximation (weighted.6) then reads

augmented by the aliasing boundary conditions,

Thus, in the Fourier space, our approximation is converted into the system of
ODE's

We shall study the stability of (weighted.6) in terms of its
unitarily equivalent Fourier
representation in (weighted.8), which is
decoupled into its real and imaginary parts, .
According to (weighted.7a)-(weighted.7b), the real part of the Fourier coefficients,
, satisfies

augmented with the boundary conditions

The imaginary part of the Fourier coefficients,
, satisfy the same recurrence relations as before

the only difference lies in the augmenting boundary conditions which now read

The weighted stability of the ODE systems (weighted.9a) and
(weighted.10a) is revealed
upon change of variables. For the real part in (weighted.9a) we introduce
the local differences,

for the imaginary
part in (weighted.10a) we consider the local averages,

Differencing consecutive terms in (weighted.9a) while adding consecutive
terms in (weighted.10a) we find

The motivation for considering this specific change of variables
steams from the
side conditions in (weighted.9b) and (weighted.10b),
which are now translated into zero boundary values

Observe that (weighted.11a),(weighted.11b) amount to a fixed translation of
*antisymmetric* ODE systems for
and
, that is, we have

where denotes the antisymmetric matrix

The solution of these systems is expressed in terms of the *unitary* matrix
,

The explicit solution given in (weighted.13) shows that our problem --
when expressed in terms of the new variables , is clearly -stable,

__. We note that this -type argument carries
over for higher derivatives, that is, the -norms
of remain bounded,
__

__
We want to interpret these -type stability statements for the
-variables in term of the original variables -- the real
and imaginary parts of the system (weighted.8). This will be achieved in term of
simple linear transformations involving the Jordan blocks
__

__
To this end, let us assume temporarily that the initial
conditions have zero average, i.e., that
According to (weighted.9a),
remains zero , and so will be temporarily ignored.
Then, if we let
denote
the 'punctured' 2__

__
The equalities (weighted.18) and (weighted.19) confirm our assertion in the
beginning of this section, namely,
__

__
__

__
Assertion. The solution operator
associated with the Fourier approximation, (weighted.6),(weighted.17),
is similar to the unitary matrix ,
in the sense that
__

__
We are now in a position to translate this similarity into an appropriate
weighted -stability.
On the left of (weighted.18) we have a
weighted -norm of .
Also, U(t) being a unitary matrix has an -norm = 1,
hence the right
hand side of (weighted.18) does not exceed,
,
and therefore
satisfies
__

Expanding the last inequality by augmenting it with the zero value of we find the weighted -stability of the real part

Similarly, (weighted.19) gives us the weighted stability of the imaginary part

Summarizing (weighted.21a) and (weighted.21b) we have shown

__
__

__
We close this section by noting three possible extensions of
the last weighted stability result.
Duhammel's principle gives us
__

__
1. .
Let
denote the solution of the inhomogeneous Fourier method
__

Then there exists a constant, , such that the following weighted -stability estimate holds

__
Our second corollary
shows that the weighted -stability of the Fourier method
is invariant under low order perturbations.
__

__
2. .
Let
denotes the solution of the Fourier method
__

Then there exists a constant, , such that the following weighted -stability estimate holds

__
In our third corollary we note that the
last two weighted -stability results
apply equally well to higher order derivatives, which brings
us to
__

__
3. .
Let
denote the solution of the Fourier method
__

Then there exist positive definite matrices, , and a constant , such that the following weighted -stability estimate holds

Here denotes the weighted -norm

__
The last results enable to put forward a complete weighted -stability
theory. The following assertion
contains the typical ingredients.
__

__
__

__
Assertion. The Fourier method
__

satisfies the following weighted -stability estimate

This last assertion confirms the weighted stability of the Fourier method in its non-conservative transport form.

__
. We rewrite (weighted.31) in the 'conservative form'
__

where denotes the usual commutator between interpolation and differentiation. The weighted -stability stated in Theorem 2.1 tells us that this commutator is bounded in the corresponding weighted operator norm. Therefore, we may treat the right hand side of (weighted.31) as a low order term and weighted -stability () follows in view of the second corollary above. The case of general follows with the help of the third corollary..

__
__

Thu Jan 22 19:07:34 PST 1998