Next: Solving for the scaling
Up: Representation in a Helmholtz
Previous: Linear function space at
Because the scaling eigenfunctions
defined above all have
the same , and are normalizable, they fall into the above linear space.
They can be represented by
coefficient vectors
of a
Helmholtz basis set with a single wavenumber ,

(6.17) 
where
is some small error function for each state.
(The error function is a Helmholtz solution and is orthogonal to the basis
set space).
The functions will not be orthogonal over , and there will
be some (numerical) nullspace of vectors that has negligible effect
on wavefunctions inside (see Section 5.3.1).
The boundary tension is a quadratic form
in the linear space.
Applying this form to the error
gives the
achievable tension minimum for each state.
In the dimensional above basis, the quadratic form is written
where is the tension matrix.
In the methods of the previous chapter the only requirement on the
dependence of this basis set was that at a given all the functions
are Helmholtz solutions at that .
In this chapter we must specialize to a scaling basis,

(6.18) 
Note that
implies a scaling function whereas
implies
some more general dependence.
The expansion of any scaling function, as in (6.17), now has a
coefficient vector which is constant as changes.
This will be necessary for our particular case of the
scaling eigenfunctions.
Note that certain basis sets, most notably that of the BIM, cannot be
used because their basis functions
(Green's functions) have origins at different points
`pinned' to the boundary .
Next: Solving for the scaling
Up: Representation in a Helmholtz
Previous: Linear function space at
Alex Barnett
20011003