with , and is fixed. We use the trapezoidal rule to approximate the Fourier coefficients in (2.2.1)

to obtain the pseudospectral approximation

__: We now have only 2 N pieces of discrete data at the
different 2N grid points __

and they correspond to 2N waves, as we have a ``silent'' last mode, i.e., with . Thus is well-defined; in view of (Fourier_even.3) it is the unique interpolant of w(x) at the 2N gridpoints :

The aliasing relation in this case reads - compare (app_ps.7)

and spectral convergence follows - compare with (app_ps.16)

In the usual sin-cos formulation it takes the form

Noting that we have 2N free parameters to match our data at .

__
__

Thu Jan 22 19:07:34 PST 1998