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The following paper is excellent and not totally irrelevant:

http://home.agh.edu.pl/~turcza/sr/Splitting%20the%20Unit%20D...

The effects you're describing (blurring, numerical stability) are readily formalized in DSP jargon. It's a shame this post didn't provoke more discussion here.

Casey did mention that Fabian and cbloom know the math behind this and might elaborate on the topic on their own blogs later.