Maybe a trivial question but I can't seem to find it treated anywhere.

Consider a smooth manifold $Q$ (configuration space) and its cotangent bundle $T^*Q$ (phase space). Any function $F:Q\to\mathbb{R}$ induces a map $q\mapsto dF_q \in T^*_qQ$ which induces the Legendre involution when $F$ is convex/concave. I think of $(q,dF_q)$ as determining a Lagrangian submanifold $L_F\subset T^*Q$, and the Legendre transformation as lifting functions on $Q$ to $L_F$ and subsequently projecting them to a fibre. The geometric construction shows it is a symplectomorphism. In coordinates, this changes the variables from $q$ to $p=\partial F/\partial q$.

My question is how can I understand the Fourier transform in this picture. By the stationary phase approximation (Wiki) or similarly Laplace's method (also on Wiki) we have something like $\mathcal{F}(\exp(i f)) \approx \exp(i\mathcal{L}f)$ where $\mathcal{L}$ is the Legendre transform.

Furthermore, how does this relate to (quantum) field theory? Path integrals always seem to have the form $\int \exp(\frac{i}{h}S[\phi]) F(\phi) \mathcal{D}\phi$ where $S$ is the action functional and integration is over all paths $\phi$ with given endpoints. This action functional ``is'' the tautological one-form (Wiki yet again), so is path integration just some kind of Fourier transform?

Any reference will help, I am not an expert.