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The Periodic Problem -- The Fourier Interpolant


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 gif
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

Assume . Then we have

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

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