A 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 St

A 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).