Involutive fourier transform The writer here states

I am introducing a viewpoint  (the involutive convention)  which makes the Fourier transform its own inverse  (i.e., the Fourier transform so defined is an  involution).

If I am reading the notation correctly, the definition given is:
$$F(f)(s) = \int_{-\infty}^{\infty}\exp(2\pi is x)\overline{f(x)}dx.$$
Under this convention, $F$ fails to be a linear operator; but, I don't think this is too big of a deal, since $F$ ends up being conjugate-linear. In any event, I have never seen this definition before. My question is, firstly, does it have any subtle issues that make it a bad idea? If not, a thoughtful discussion of the benefits of this definition would be appreciated.
 A: The fact that this in an antilinear transformation and not a linear one is a slight drawback. As a consequence, Parseval's formula fails:
$$
\langle \mathcal{F}(f),\mathcal{F}(g)\rangle = \overline{\langle f,g\rangle}
$$
in other words: The transform is not unitary any more. (It still preserves the $L^2$-norm, though.) This is annoying in many situations and I guess it also makes the involutive form unsuitable for many fields where unitarity is important like quantum mechanics (e.g. one had to adapt the momentum operator to keep the dual relationship of position and momentum).
To answer the question for reasons for its unpopularity: From a practical point of view there is no big advantage of the involutive form. Being its own inverse seems like a nice property but it does not really help with the daily use of the Fourier transform. I would even say that it is a slight advantage of the conventional Fourier transform that it is asymmetric - it helps to keep track on which side you are operating at the moment.
Remember that a new notion has to prove that its advantages largely outweigh the drawbacks to have is adapted by the community. In this case I do not see such an advantage and I am happy to stick with the usual Fourier transform…
