# Approximation using a Fourier transform with low pass filter

I need to approximate a function f, but I cannot do so with frequencies that exceed 1kHz

What is the best approximation I can get? Is taking the Fourier transform then zeroing any term above 1kHz the best approximation? Or can I fiddle with the lower order terms and get a better fit?

This is a made up scenario, but I have to prove the same concept with Walsh transforms. I am fairly certain that the lower order terms form the best approximation from random twiddling and hill climbing searches, but I need proof.

I believe the proof is something very similar to a least squares regression proof, but I can't get it. Has this problem been solved before? At least in the Fourier domain?

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I think this belongs on digital signal processing...if someone can migrate it there? – Fixed Point Mar 20 '13 at 7:06

The Fourier transform is unitary. Therefore the best fit in the $L^2$ norm in the frequency domain is also the best fit in the time domain. This means that suppressing all frequencies outside the allowed band indeed gives the best approximation.
Is there a proof or a deeper explanation of this? I do not understand it. I read about Unitary. That means the matrix has orthogonal rows/columns. I don't understand what $L^2$ norm is and how that relates across the domains. I can see intuitively how it is an orthogonal decomposition and therefore probably is the best approximation. – SwimBikeRun Mar 21 '13 at 4:00
Ah $L^2$ is just the norm of the squares. So the key factor is that it is unitary? – SwimBikeRun Mar 21 '13 at 6:41