4 ms·
A bare-bones case of Fourier transform would be the discrete finite FT, that is just multiplying by a well crafted matrix W as in [1]. So here it looks to me th
by jesuslop 3y ago
A bare-bones case of Fourier transform would be the discrete finite FT, that is just multiplying by a well crafted matrix W as in [1]. So here it looks to me that the fractional transform would be a rational exponent of W, W^r, r rational. For instance W^(2/3)=(W^(1/3))^2, with W^(1/3)=X and X^3=W hopefully a well-defined X.
[1] https://en.wikipedia.org/wiki/DFT_matrix https://en.wikipedia.org/wiki/DFT_matrix
- jesuslop 3y agoA bit more techy thing to said is that Plancherel theorem here simply means that W in parent comment is unitary, so we can understand fractional FTs applying Stone's theorem for uniparametric unitary groups (the parameter here the order of the fractional, a scalar). The theorem guarantees that there is an infinitesimal generator H (a matrix) such that W^t = e^(tH).