We note that once *w*(*x*) is assumed to be smooth, it is completely
determined ( - in the pointwise sense)
by its Fourier coefficients ; so
are its equidistant values and so are its
discrete Fourier coefficients . The aliasing formula shows that
are determined in terms of , by folding back high
modes on the lowest ones, due to the discrete resolution of the moments of
*w*(*x*): all modes are aliased to the same place
since they are equal on the gridpoints

Let us rewrite (app_ps.7) in the form

Returning to the aliasing error in (app_ps.6), we now have

We note that the truncation error lies outside ,
while the aliasing error lies
in , hence by -orthogonality

Both contributions involve only the *high
amplitudes* - higher than
*N* in absolute value; in fact they involve precisely all of these high
amplitudes. This leads us to aliasing estimate

We conclude that the aliasing error is dominated by
the truncation error (at least for any ),

Augmenting this with our previous estimates on the truncation error
we end up with spectral accuracy
as before, namely

Thu Jan 22 19:07:34 PST 1998