There are several conventions for the definition of the Fourier transform on the real line.
1 . No $2\pi$. Fourier (with cosine/sine), Hörmander, Katznelson, Folland.
$ \int_{\bf R} f(x) e^{-ix\xi} \, dx$
2 . $2\pi$ in the exponent. L. Schwartz, Trèves
$\int_{\bf R} f(x) e^{-2i\pi x\xi} \, dx$
3 . $2\pi$ square-rooted in front. Rudin.
${1\over \sqrt{2\pi}} \int_{\bf R} f(x) e^{-ix\xi} \, dx$
I would like to know what are the mathematical reasons to use one convention over the others?
Any historical comment on the genesis of these conventions is welcome.
Who introduced conventions 2 and 3? Are they specific to a given context?
From the book of L. Schwartz, I can see that the second convention allows for a perfect parallel in formulas concerning Fourier transforms and Fourier series. The first convention does not make the Fourier transform an isometry, but in Fourier's memoir the key formula is the inversion formula, I don't think that he discussed what is now known as the Plancherel formula. Regarding the second convention, Katznelson warns about the possibility of increased confusion between the domains of definition of a fonction and its transform.
Best Answer
The version number 2 is the only one that makes the Fourier transform both a unitary operator on $L^2$ and an algebra homomorphism from the convolution algebra in $L^1$ to the product algebra in $L^\infty $.
It is not, however, of widespread use in analysis as far as I know. From the point of view of semiclassical analysis, it amounts roughly speaking to consider Planck's constant $h $ rather than $\hslash=h/2\pi $ as the semiclassical parameter (or as the constant set to one in quantum systems). This is somewhat differing from the common practice in physics.