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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!

Nen
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  • Related: https://math.stackexchange.com/questions/13902/does-a-fourier-transformation-on-a-pseudo-riemannian-manifold-make-sense – Eclipse Sun Sep 08 '18 at 19:47
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    The Fourier transform can be generalized to groups, but not to arbitrary manifolds as far as I know. For hyperplanes, you can just use any isometry to $\mathbb R^m$ and apply the Fourier transform there. –  Sep 08 '18 at 20:14

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