$u \in C^\infty(\mathbb{R^n})$ with compact support $\implies$ $f(x) = ⨏_{\partial B(0,\vert x \vert)} u(t) \ dt\in C^\infty$ with comp. s.? $\newcommand{\avint}{⨍}$
Let $u \in C^\infty(\mathbb{R^n})$ with compact support.
How can we prove using direct calculations or Fourier transform methods that $$f(x) = \avint_{\partial B(0,\vert x \vert)} u(t) \ dt\in C^\infty(\mathbb{R}^n)$$ with compact support?
We use $\partial B(0,\vert x \vert)$ to denote the boundary of the ball of center $0$ and radius $|x|$ (Euclidean norm).

Let $u \in C_c^\infty(\mathbb R^n)$ and let $\sigma$ be the measure on $S^{n-1}$. Then we set
$$f(x) = \frac{1}{\sigma(\partial B(0,|x|))} \int_{\partial B(0,|x|)} u(t) \, dt$$
It's obvious from the definition that $f(x)$ only depends on $|x|$, i.e. $f(x) = \hat f(|x|)$ where 
$$
\hat f(r) 
= \frac{1}{r^{n-1} \sigma(S^{n-1})} \int_{S^{n-1}} u(rt)  \, r^{n-1} dt
= \frac{1}{\sigma(S^{n-1})} \int_{S^{n-1}} u(rt)  \, dt
$$
Since $u \in C^\infty$ and we integrate over a compact set ($S^{n-1}$), derivatives commute with integration, so
$$
\hat f^{(k)}(r) 
= \frac{1}{\sigma(S^{n-1})} \int_{S^{n-1}} \frac{\partial^k}{\partial r^k} u(rt)  \, dt
= \frac{1}{\sigma(S^{n-1})} \int_{S^{n-1}} t^k u^{(k)}(rt)  \, dt
$$
where the last integral is defined for all $k = 0, 1, \ldots$ and all $r \in [0, \infty)$, so $\hat f \in C^\infty([0, \infty))$.
Since the support of $u$ is compact, and a compact set in $\mathbb R^n$ is bounded, there exists $R>0$ such that $u(x)=0$ whenever $|x|>R$. This implies that $\hat f(r) = 0$ for $r>R$. Thus $\hat f$ has compact support.
Now,
$$\frac{\partial}{\partial x_i} f(x) 
= \frac{\partial r}{\partial x_i} \frac{\partial}{\partial r} f(rt) + \frac{\partial t}{\partial x_i} \frac{\partial}{\partial t} f(rt)
= \frac{x_i}{r} \hat f'(r)
$$
since $\frac{\partial}{\partial t} f(rt) = 0$. So $f$ is derivable for $|x|>0$, and it's clear that higher order derivatives can be taken so $f$ is infinitely derivable at $|x|>0$. But what about derivatives at $x=0$? That's the difficult part and perhaps someone else can give a good answer before I have managed to solve it.
