I was reading this answer: Fourier transform of the indicator of the unit ball and couldn't really follow the computations in the answer. Specifically the equality $$\int_{\|x\| \leq 1} e^{-ix_n\rho} dx_1 \dots dx_n = \int_{-1}^1 (1−x_n^2)^{(n−1)/2} \alpha_{n−1} e^{-ix_n\rho} dx_n$$ where $\rho >0$ and $\alpha_{n-1}$ is the volume of the unit ball in $\mathbb{R}^{n-1}$. The computation looks a lot like the ones in https://en.wikipedia.org/wiki/Volume_of_an_n-ball#The_one-dimension_recursion_formula where a recursion formula for the volume of the unit ball is derived, but I can't really wrap my head around it. Could anyone help me understand what is going on here?
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Can you prove it for small values of $n$? – user170231 Jan 27 '23 at 17:43
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@user170231 Yes I actually was able to do it for n=2 just. I think I'll be able to do the general case now. Thank you – Nora Jan 27 '23 at 18:41