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

 
• Aliasing and spectral accuracy
• Fourier differentiation matrix
• Fourier interpolant revisited on an even number of gridpoints