Etranger wrote:Am I correct in assuming that by that, you mean "pick up the math as you need it and proofs while nice, aren't a necessity"? If yes, when taking grad courses or doing your research in condensed matter, did you ever feel that this somehow held you back? Is studying math from the perspective of a pure mathematician useful in your work?
I'm just slow when it comes to pure math...
We know it works for many functions, since we've tried it many times and get plots that look exactly the same. Let's thing about the general case. Large wavelengths capture large-scale fluctuations. Small wavelengths capture small-scale fluctuations. If we first fit the general shape of a function using large wavelengths, we should be able to correct for local inconguencies by using small wavelengths, regardless of the shape of the function (we know from example it works for functions that don't look even close to sine waves, so there's no reason to suspect this wouldn't be true). There might be an issue if the function changes its value too quickly--that is, has a large slope. Large slopes mean a large derivative, so we probably need a condition on the derivative. Since we can go as small as we want on the wavelengths, that condition is probably only that the derivative doesn't become infinite--in other words, that the derivative exists at every point. If the derivative doesn't exist somewhere, we're probably ok unless we get very close, since everywhere else around that point the function behaves totally fine. Since again we can go as small as we want on wavelengths, it seems reasonable to suppose that we can trust the fourier transform except exactly at the point where the derivative becomes singular--that is, if we cut off our function before the singular derivative, we should be totally fine.
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