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As I understand it, one typically performs Fourier deconvolution for the data produced by some instrument of a linear system by:

  1. Taking the observation of some unit impulse signal $m$, with the property that every other input can be represented as the sum of scaled versions of $m$ - let's call the data of this observation $s$.

  2. Taking the observation of the phenomenon we care about, let's call this $o$.

Then, we say that to go from $m$ to $s$ the instrument has convolved with some other function and then say, via certain remarkable properties of Fourier transforms and convolutions, that $O = IS$, where $i$ is the ideal data we wish to obtain from $o$ via deconvolution and capital letters denote the Fourier transform of a function.

The other day, a friend suggested the following in a situation related to measuring the spectra of certain light sources:

Because we couldn't measure a unit impulse signal, we should instead measure a known blackbody radiation curve, $bbr$, as the data $s$. Then, he said that "via a parallel" we see that for a general input $i$ to our system and a general output $o$ associated with that input, we have $O = SI$ and then you solve for $I$ using inverse fourier transforms.

I pointed out that it was definitely not known that $bbr$ could represent every other signal in the form of the superposition of scaled versions of itself and so that there was no reason I knew of for this to work. Given that they said "it was a form of genuine physical insight" and the fact that they are generally correct, I believe them - but why is what they said true?

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