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We have seen that given the ``moments''

we can recover smooth functions *w*(*x*) within spectral accuracy. Now,
suppose we are given discrete data of *w*(*x*): specifically, assume *w*(*x*)
is known at equidistant collocation points

Without loss of generality we can assume that *r* -- which measures a fixed
shift from the origin, satisfies

Given the equidistant values , we can approximate the above
``moments,'' , by the trapezoidal rule

Using instead of in (app_fourier.7),
we consider now the
pseudospectral approximation

The error, , consists of two parts:

The first contribution on the right is the *truncation error*

We have seen that it is spectrally small provided *w*(*x*) is sufficiently
smooth. The second contribution on the right is the *aliasing
error*

This is pure discretization error; to estimate its size we need the

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The proof of (app_ps.7) is based on the pointwise representation of
by its Fourier
expansion (app_fourier.31),
Since w(x) is assumed to be in , the summation on the right is
absolutely convergent
and hence we can interchange the order of summation
Straightforward calculation yields
and we end up with the asserted equality
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- Aliasing and spectral accuracy
- Fourier differentiation matrix
- Fourier interpolant revisited on an even number of gridpoints

Thu Jan 22 19:07:34 PST 1998