# How well does a truncated fourier expansion of a stepfunction perform near the expansionpoint

Hey, i'am currently trying to make a stability analysis of a binary fluid at its phase border. As the governing equations in this double-diffusive problem are rather complicated i have to do a series expansion of my density profile. My problem is, that my PDEs contain a hevyside step function right at the border and i want to do a fourier expansion (i can then use symmetries of my problem and do some horrible algebra to actually get a "solution" which can be computed for up to 5 or 6 modes).

Now i know that fourier expansion performs quite horribly at a discontinous jump if only the first few modes are kept. Is it nervertheless possible to obtain a "margin of error"? Numerical precision is not of utmost concern, but the physics behind my problem should not all be thrown out by such an expansion. Do you know of any criteria or "better" expansion where its possible to make heavy use of symmetry/antisymmetry?

all the best, jan

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SFAIK the standard way of handling the "ringing" that comes up in Fourier approximants of discontinuous functions requires multiplication by a sine cardinal (sinc(x)=sin(x)/x), as recommended by Lanczos. Have you tried this already? –  J. M. Aug 12 '10 at 8:02
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## 3 Answers

Isn't it just the Gibbs phenomenom you are asking for? If there is a discontinuity (as for the Heaviside function) the discontinuity in the partial Fourier series does not die out for $n \to \infty$ , but will be about 18% larger than the discontinuity in the original function. This does not give you the precise value for 5 or 6 modes, but tells you that for Heaviside function the overshot will be 0,18 for $n \to \infty$, which will give you the order of magnitude also for 5 or 6 modes. (Please see also http://en.wikipedia.org/wiki/Gibbs_phenomenon.)

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Is this an expansion of Nate Eldridge's answer? –  S. Carnahan Jun 18 '10 at 19:16
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Too lazy to think about the precise answer, but using something like a Cesaro sum should improve matters. There are some useful references in the Wikipedia pages for Gibbs phenomenon, Fejer's theorem, etc.

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As Nate pointed out, Fejer's summation should be better than the standard Fourier sum. The standard Fourier sum will converge pointwise like 1/n near the discontinuity.

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