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.