Fourier interpolant revisited on an even number of gridpoints

  We assume w(x) is known at the 2N gridpoints ,
 
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 2N 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 .