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Aliasing and spectral accuracy

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 tex2html_wrap_inline11151, while the aliasing error lies in tex2html_wrap_inline11151, 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

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