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Consider the Cauchy problem for the wave equation : \begin{cases} u_{tt}= \Delta u \\u(x,0)=f(x) \\ u_t(x,0)= g(x)\end{cases} with $t>0$ and $x\in\mathbb{R^d}$

This is an exercise from Stein's Fourier Analysis an Introduction chapter 6 :

A spherical wave is a solution $u(x; t)$ of the Cauchy problem for the wave equation in $\mathbb{R^d}$, which as a function of $x$ is radial. Prove that $u$ is a spherical wave if and only if the initial data $f; g \in \mathcal S$ are both radial, where $\mathcal S$ denotes the Schwartz space.

I have already done the $\Leftarrow$ direction , my doubt is on the $\Rightarrow$ :

My thought: if $u$ is a solution then $u(x,0) = f(x)$ and because $u(x,0)$ depends only on $x$ and $u$ is radial as a function of $x$ (by the definition of spherical wave above) then $f$ is radial by definition since $h(x)$ is radial if $h(x)= h_0(|x|)$ for some $h_0$ . In an analogous way we obtain the result for $g$.

Is this ok ? Or is there a better way to prove it ?

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    $\begingroup$ I'd look at the contrapositive (if $f$ is not radial....) and use the continuity at $t$ goes to $0$. $\endgroup$ – Paul Nov 26 '18 at 21:11
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    $\begingroup$ The hard part of the "complete solution" is to find the Green function is radial so radial initial conditions is equivalent to radial solution. $\endgroup$ – reuns Nov 27 '18 at 4:12

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