Let $ f\in L^1(\mathbb{R}^3,\mathbb{C} ) $ be a function such that for $ x_1,x_2\in \mathbb{R}^3 $ such that $ |x_{1}|=|x_{2}| $ we have $ f\left(x_{1}\right)=f\left(x_{2}\right) $.
I have to prove that also for any $ \omega_{1},\omega_{2}\in\mathbb{R}^{3} $ we have $ \mathcal{F}\left\{ f\right\} \left(\omega_{1}\right)=\mathcal{F}\left\{ f\right\} \left(\omega_{2}\right) $.
Here's what I have tried:
Note that $ \mathcal{F}\left\{ f\right\} \left(\omega\right)=\frac{1}{\left(2\pi\right)^{3}}\intop_{\mathbb{R}^{3}}f\left(x\right)e^{-i\omega\cdot x}dx $
(where $ \omega \cdot x $ is the dot product of 3dimensional vectors).
Now I want to switch to spherical cooredinates, so I'll note
$ \omega=\left(R\sin\left(\theta_{1}\right)\cos\left(\varphi_{1}\right),R\sin\left(\theta_{1}\right)\sin\left(\varphi_{1}\right),R\cos\left(\theta_{1}\right)\right) $
and $ x=\left(r\sin\left(\theta\right)\cos\left(\varphi\right),r\sin\left(\theta\right)\sin\left(\varphi\right),r\cos\left(\theta\right)\right) $
So that $ \mathcal{F}\left\{ f\right\} \left(\omega\right)=\frac{1}{\left(2\pi\right)^{3}}\intop_{0}^{\infty}\intop_{0}^{\pi}\intop_{0}^{2\pi}f\left(r,\theta,\varphi\right)e^{-irR\left(\sin\theta_{1}\cos\varphi_{1}\sin\theta\cos\theta+\sin\theta_{1}\sin\varphi_{1}\sin\theta\sin\varphi+\cos\theta_{1}\cos\theta\right)}r^{2}\sin\theta d\varphi d\theta dr $
and since $ r $ is radial I can just write $ f(r) $. Also I will use trigonometric identities to get:
$ =\frac{1}{\left(2\pi\right)^{3}}\intop_{0}^{\infty}\intop_{0}^{\pi}\intop_{0}^{2\pi}f\left(r\right)e^{-irR\left(\sin\theta_{1}\sin\theta\cos\left(\varphi-\varphi_{1}\right)+\cos\theta_{1}\cos\theta\right)}r^{2}\sin\theta d\varphi d\theta dr $
All I have to do is to show that the integral does not depend on $ \theta_1 , \varphi_1 $
But I got stuck here. Any help would be appreciated.
Thanks in advance.
