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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 .

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998