Why the spatial/mathematician's Fourier Transform?

Question

I was wondering why the sign-change in the exponential of the spatial/mathematician's Fourier Transform

and

why is it called

• mathematician's
• spatial

in either case?

Answer 1

Either the professor is making a joke about mathematicians (the same way that we mathematicians often joke about engineers), or that he is mis-remembering/misinterpreting. First some evidence:

A quick survey of the available textbooks on my shelf behind me indicates that among mathematicians such as ...
Richard Courant and David Hilbert (Methods of mathematical physics), Elias Stein (Introduction to Fourier Analysis on Euclidean Spaces), Michael Taylor (Partial Differential Equations, Vol 1), Lars Hormander (The Analysis of Linear Partial Differential Operators Vol 1), Kosaku Yoshida (Functional Analysis), Walter Rudin (Functional Analysis and Real and Complex Analysis), Anthony Knapp (Advanced Real Analysis), LC Evans (Partial differential equations)
... not a single one defines the Fourier transform using the so-called "Mathematicians style".

(And if the slides are any indication of your institutional affiliation, I remark that down the street in the Math Department at Georgia Tech, both Christopher Heil and Michael Lacey (I'm pretty sure) would agree with me on the matter of convention.)

When mathematicians do disagree with each other about the definition of the Fourier transform, it is always about where to put factors of $2\pi$: should the transform be taken against $e^{-i\omega x}$ or $e^{-2\pi i \omega x}$, or whether there should be factors of $\frac{1}{2\pi}$ or $\frac{1}{\sqrt{2\pi}}$ in front one or both of the forward and inverse Fourier transforms. But never about the sign in front of the imaginary unit!

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Now, why do I suspect that the professor is "misremembering"? It turns out that in the software package Mathematica the default normalization of the Fourier transform is the version that in the slides is called "Mathematician's". So it is possible that the professor at one point knew/meant that to be the "Mathematica convention", instead of the "Mathematician convention", and got corrupted over the ages, or that the professor mistakenly thought that the convention choice of Mathematica represents that of most mathematicians.

As to why Mathematica use that strange convention, it can be conveniently explained by this page published by the software company itself. One should remember that the originator of Mathematica is Stephen Wolfram, who was trained in theoretical physics at CalTech. And it turns out that in some parts of the physics literature, the convention $$ \hat{f}(\xi) = \int f(x) \exp(i x\xi) \mathrm{d}x $$ is used, at least according to the Wolfram Reference page cited above. So it would be likely that because Wolfram is trained in the tradition with that particular convention (which I again emphasize is not the commonly used one in both mathematics and engineering, a fact acknowledged on the same Reference page), he aligned the software he designed to do it in the way he preferred.

Incidentally, MATLAB uses the math/engineering convention; both MATLAB and Mathematica have options to change the convention to whatever you want.

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That said, there really is no different in the definitions: $\pm i$ are the two square roots of $-1$, and which one you choose to call $i$ and which $-i$ is rather irrelevant.