The ordinary Fourier transform of a function $\Psi(x,0)$ is value of $\Psi(x,\pi/2)$ of a solution of the wave equation

$$ -{\rm i}\cdot d/dt \Psi = \delta \Psi + |x|^2\cdot \Psi $$

The values of $\Psi$ at intermediate values of $t$ are called fractional FTs.

How is the fFT computed?

For the discrete case, the main difficulty is obtaining

• a suitable Laplace approximation and
• finding the constants.

It helps that there is a tridiagonal matrix commuting with the FT is known (F.A. Gruenbaum) for the 1-dim case. From that the eigenvectors can be computed, and the diagonal matrix containing the natural numbers can be transformed back using the eigenvectors making up a Hamiltonian which is an infinitesimal generator of the fFT semigroup. For 2-dim Cartesian space the Kronecker product can be used as for the Laplacian. This way solves both of the above problems, but the matrix is not sparse at all.

Looking at the 1-dim diagonal, it is a parabola with the max energy term at the border and a minimum of that divided by pi (?). The 1st codiagonal is a parabola with zero at the border and minimum -2/pi^2 of the max energy (?).

The right side of the above PDE is called Hamilton function of $\Psi$. Infinitesimally, the Hamiltonian is tridiagonal for a 1-dim domain. For discrete Laplace operators under periodic boundary conditions this is not exactly the case. Therefore it is worthful to find a tridiagonal matrix commuting with the Fourier matrix, in order to find eigenvectors of the Fourier matrix usable to construct the true Laplacian on those spaces.