Past exam of the short options course of the University of Oxford / 2021 / 5 / a

Every $a\in \mathbb{R}$ works. Translation preserves a smooth function, and the support of a function becomes the compact interval $[a-1,a+1]$. Hence $\varphi (\cdot -a)\in C_c^{\infty }(\mathbb{R})$, the space of test functions.

Exactly $a\ne 0$ works. For nonzero $a$, composition with $x\mapsto ax$ preserves smoothness, and the support of a function is $[-1/|a|,1/|a|]$, which is compact. For $a=0$ the result is the strictly positive constant $\varphi (0)$ on all of $\mathbb{R}$, so it lacks compact support and is not a test function.

Exactly $|a|>1$ works. Composition with the hyperbolic tangent is smooth for every $a$. If $|a|>1$, set $R=\mathrm{arctanh}(1/|a|)$. For $|x|\ge R$, $|a\tanh x|\ge 1$, so the result vanishes; it is therefore a test function with support of a function $[-R,R]$.

If $|a|\le 1$, every finite $x$ satisfies $|a\tanh x|<1$. The given positivity makes $\varphi (|a|\tanh x)>0$ for every $x$, so its support is all of $\mathbb{R}$. In particular $|a|=1$ is excluded even though the function tends to zero at infinity: decay is not compact support.

Multiplication of a distribution by a smooth function is defined by moving that function onto the test function. The Dirac delta multiplication identity givesIn fact $x^2\delta$ is the zero distribution, independently of the special positivity assumptions on $\varphi$.