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The Periodic Problem -- The Fourier Interpolant

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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

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