Connections between the path integral formulation and the Fourier transform

I am just learning abut the Path integral formulation and it seems to me that there is a connection, at least conceptually, to the Fourier transform.

In the Path integral formulation we sum over all possible paths a particle can take between two points. Each path contributes with a phase factor determined by the action. While in theory, all paths contribute, in practice, the paths near the classical path (the one that minimizes the action) have constructive interference, while others largely cancel out due to destructive interference.

$${\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}{\mathcal {D}}\mathbf {x} \,e^{iS[\mathbf {x} ,{\dot {\mathbf {x} }}]}\psi _{0}(\mathbf {x} (t))\,}$$

When doing a Fourier transform we get that as the unit vector rotates, it cancels out for all frequencies, destructive interference, except the frequencies that are present in our function of interest, constructive interference, and we get as the result only the frequencies that are present in the starting function.

$${\displaystyle {\widehat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)\ e^{-i2\pi \xi x}\,dx,\quad \forall \xi \in \mathbb {R} .}$$

You can even see a similarity in the equations, namely there is something being multiplied by $e^{i...}$ and than integrated.

Am I onto something or am I exaggerating in this connection?

For what it's worth, the path integral$^1$ $$ Z[J]~=~ \int \! {\cal D}\phi~e^{\frac{i}{\hbar}\left(S[\phi]+J_k\phi^k\right)}~=~\widehat{e^{\frac{i}{\hbar}S}} \tag{1}$$ can be viewed as an infinite-dimensional Fourier transform of the Boltzmann factor $$ e^{\frac{i}{\hbar}S}\tag{2}$$ from the field variable $\phi^k$ to the source variable $J_k$.

$^1$ Here DeWitt's condensed notation is implicitly used.

You are extremely far away from the actual connection.

The constructive interference part, with a lot of links to renormalisation, JWKB, perturbation theory, Stokes's phenomenon, Fresnel integrals, and so forth, is coming from stationary phase approximation and not Fourier transform.

Another way to make progress would be the Lie group machinery, operator factorisation, symplectic methods, etc.

If you wanted a connection between path integrals and Fourier transform, the obvious point of closest contact is to start with classical trajectories on phase space, and then convert the leftover path integral into infinitely many integrals over the Fourier coefficients. Needless to say, few people care about this. However, enough people care about this that we understand that even the great Feynman himself got it wrong; if you do not start with the classical trajectories, then the convergence of the path integral might fail, especially if you approach it naïvely. In that case, I cannot remember, neither the books and articles that covered the topic (check Schulman at least, if you want, but my own copy is across an ocean), nor if there are additional conditions you can impose to salvage the direct approach.

Fourier transform is expansion in Hilbert space, e.g., in momentum eigenfunctions. It is far more basic than path integrals.

Method of steepest descent is probably closer to what you have in mind. It is also worth looking in the non-path-integral equivalent of the procedure described - WKB approximation.