Problem: Suppose we want to perform Fourier transform on a smooth function that is defined on an $m$-dimensional smooth submanifold of $\mathbb{R}^n$ ($1 \le m \le n-1$) i.e., we wish to obtain $\mathcal{F}f$ for $f: M \rightarrow \mathbb{C}$, where $dim(M) = m$. I have only learned about Fourier analysis for functions defined on $\mathbb{R}^n$ and not on manifolds. How do we define the Fourier transform here? By using coordinate functions $\{\phi_i\}$ and computing $\mathcal{F} \phi_i f \phi_i^{-1}$? Can we do this for functions defined on a hyperplane in $\mathbb{R}^n$? What about general manifolds?
This question is related to a previous question of mine: A property of the x-ray transform (see the answer to the question)
Any help is appreciated!