Weighted -stability

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 (2N+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' 2N-vector of real part associated with (weighted.8), it is related to the 2N-vector of local differences, , through

This enables us to rewrite the solution given in as
 
Similarly, since in the real case, it will be temporarily ignored. Then, the 'punctured' 2N-vector of imaginary part associated with (weighted.8),

is related to the 2N-vector of local averages, , through

which enables us to rewrite the solution given in as
 

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..