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As far as I have seen, almost all theoretical filter design occurs in Laplace or Z-space. Also, there is a pervasive connection to real life analog filters in the design. If one is just thinking in a mathematical theoretical thing (or something that could be implemented digitally), why wouldn't one filter signals in Fourier Space?

Why is, say, multiplying the Fourier Transform of a certain function by a unit step up - step down function, and then making the Inverse Transform of the resulting signal a "band pass filter"? Why should one use Butterworth or similar things to make a digital filter?

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    dsp.stackexchange.com might be a better place for this question. – audionuma Sep 16 at 5:06

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