$\phi$ has compact support in $\mathbb{R}^n$ does not imply $\phi (\xi + \eta)$ has compact support in $\mathbb{R}^n\times\mathbb{R}^n$

Question

Let $\phi$ be a $C^\infty$ function with compact support in $\mathbb{R}^n$. Some introductory books on distribution theory I'm reading say that the function $(\xi,\eta)\mapsto \phi (\xi + \eta)$ not necessarily has compact support in $\mathbb{R}^n\times\mathbb{R}^n$, but none of them gave a counter-example of this statement.

Could you please show a counter-example or point a direction on how to find one?

Thanks in advance.

Answer 1

It never has compact support, unless $\phi$ is identically zero. Indeed, if $\phi(a)\ne 0$ then the composition is nonzero on the affine subspace $\{(\xi+a,-\xi):\xi\in\mathbb R^n\}$ which is obviously unbounded.