fourier transform normalization constant

Question

I'm confused about the two Fourier transform formulas that crop up. One where the constant of $ \frac{1}{\sqrt{2\pi}}$ is used in both the forward and reverse transform and the other where just $ \frac{1}{{2\pi}}$ is used for the inverse. I understand that it's a normalization constant or something and so both are equivalent somehow, but if the Fourier transform of a signal tells you what the amplitude of the signal is at various frequencies how is one of these not wrong. If the first formula tells you that there's a peak and 5Hz and the other says something else then surely only one of them is right. Or will they both give the same result, and if so how is that possible if they are multiplied by different constants. I hope that makes sense, thanks in advance

Answer 1

You are right that the purely formulaic presentation of Fourier transform and Fourier inversion give the impression that Plancherel and/or other aspects of the situation depend on the normalizations. And, of course, in a way, they do, because one can choose partly-correct normalizations and then make incompatible choices in the rest. Also, coordinate-dependent interpretations of formulas are doomed... Rather, I'd recommend thinking that "Fourier (-Plancherel) transform" is an isometry (or "isometric isomorphism") from one Hilbert space to another, and Fourier inversion is the inverse isometry.

The several differing choices of locations of the $2\pi$ can easily be accommodated by altering the measure for the $L^2$ space that is the target of Fourier (-Plancherel), and similarly for the inverse.

That is, the way to think about this (in my opinion) is that there cannot possibly be any genuine issue about mathematical facts, only one of correct interpretation of formulaic things in terms of the underlying reality.