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.
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).
Liked this view always (look for Hutter prize rationale), but I think it needs to be accomodated in a general perception/action loop that optimizes a lower level fitness/utility/reward (for instance an inner sense of pleasure/pain).
I always liked this viewpoint a lot. I'm enjoying with abandon some dusty lectures that Vito Volterra gave in Madrid on differential and integrodifferential equations, while helping also to create Functional Analysis (a Functional being the analogue of a dual vector). He is constantly exploiting this analogy method from finite variable constructions to infinite, also uncountable variables. Even up to showing some embarrassment of being too repetitive with the idea! People in teaching should join and take a peek.
Givental uses this viewpoint too in his differential equations class notes. It may soothe students who notice a disjunction between linear independence of functions and that of vectors.
While it downloads, speaking about Lambek, in Lambek-Scott they prove the equivalence of CCC's and simply typed lambda calculus. I suffered some functional programming exposure via Lisp, and came later to see that in the core of lisp, the lambdas, were formalized by (you guess it), lambda calculus. This idea migrates also to Haskell, so if one is pressed to give a formal Haskell semantics, that is a good bootstrap. But that is equivalent to CCCs (cartesian closed categories) as per Lambek-Scott, where the essential feature of CCCs is having functions of functions, higher order functions. And that is not just self-pleasure, look at Conal Elliot's "compiling to categories". Applications I think I've heard of: probabilistic programming and automatic differentiation.
Lambek-Scott: Introduction to Higher-Order Categorical Logic
