I am reading the notes provided here: https://link.springer.com/content/pdf/10.1007%2F978-1-4419-6055-9.pdf which I have some questions about.

On page 2 of chapter 1, the Radon transform of a function $f: \mathbb{R}^n \to \mathbb{C}$ on a hyperplane $\xi$ is defined as $\hat{f}(\xi) = \int_{\xi} f(x) dm(x)$, where $dm$ is the Euclidean measure on the hyperplane $\xi$.

Later on on page 4, a relationship between Fourier transforms and Radon transform is noted, that $$\tilde{f}(s \omega) = \int_{-\infty}^{\infty} \int_{\langle x, \omega \rangle = r} f(x) e^{-i s \langle x, \omega \rangle} dm(x) dr $$ where $\tilde{f}$ is the Fourier transform of $f$, $\omega$ is a unit vector and $s \in \mathbb{R}$.

My questions:

  1. What exactly is meant by "Euclidean measure on the hyperplane on $\xi$"? The notes do not define it and Google doesn't seem to turn up anything.

  2. I do not understand how the formula for $\tilde{f}(s \omega)$ is obtained. Here, the Fourier transform is written as $\tilde{f}(\omega) = \int f(x) e^{-i \langle x, \omega \rangle} dx$, but I do not see how it can be transformed into the equation involving the Radon Transform.

  • $\begingroup$ You're writing the integral as an integral over a plane that is a distance of $r$ from the origin with normal $\omega$ and then an integral in $r$. The integral over the plane is with respect to the ordinary Euclidean measure on the plane. $\endgroup$ – DisintegratingByParts Mar 13 at 0:50
  • $\begingroup$ Thanks for the reply. I made a small typo in the formula that's now been fixed. With respect to your answer, I'm not sure I understand what "ordinary Euclidean measure on the plane" means. I do see what the interpretation of right-hand side of the equation, but I still don't really see the relation to the LHS. I think it would be most helpful to me to see what precisely what measure $dm$ refers to. $\endgroup$ – mathmajor55567 Mar 13 at 1:29
  • $\begingroup$ Euclidean measure on a plane $x_1=d$ for $(x_1,x_2,\cdots,x_n)\in\mathbb{R}^n$ would correspond to integration with respect to $dx_2 dx_3 dx_4\cdots dx_n$ on an $\mathbb{R}^{n-1}$ plane. For a more general plane, you can always apply a rotation to put it back into this configuration. The rotation preserves measure. A translation in a variable also preserves measure. $\endgroup$ – DisintegratingByParts Mar 13 at 1:49

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