I was working around with the fractional Fourier transform(FRFT) when the mathematics undergrad found out, by brute-force computations, that the derivative of the FRFT with respect to the parameter can be written as a closed form formula- explicitly, d/da(F^a(f)(y)) = i(1/8+(π/4-π^2y^2/2)d^2/dy^2)F^a(f)(y). 

Then I remembered that the first definition of the FRFT, using "only" a certain set (Hermite-Gaussian functions) of orthonormal eigenfunctions of the FT, is not actually "unique", especially when attempting to discretise the transform.

So I'm asking the following question instead- do unitarity, index (assumed to be real) continuity, index additivity, reduction to the identity(resp. Fourier transform) operator when the index is 0(resp. 1), and the formula above uniquely define the FRFT, as operators on Schwartz functions?

NB: the version of the FRFT I'm using is the one used by the current revision of [the Wikipedia page for the FRFT][1].


  [1]: https://en.wikipedia.org/wiki/Fractional_Fourier_transform