Let $\Omega\subset \mathbb{R}$ be open. A test function is a $C^\infty$ function with compact support. This is a rather strong restriction, for instance, no analytic function is a test function. But $\exp(-1/x^2)\exp(-1/(x-1)^2)$ is a test function.
In general I found it difficult to construct test functions with certain properties, but this seems important in proving theorems in distribution theory. Thus I wonder whether there are good ways to cook up certain test functions.
For instance, can we have a test function that pass through finite many fixed points? Can we find a test function whose derivatives have specified values at certain points? (for instance, can we construct a test function whose first $N$ derivatives vanish but the $N+1$ derivative is large?) Given a bad function, can we smooth it to be a test function (like smooth away the sharp angles of the tent function)?
The above are some of the properties I have encountered but I am also interested in other properties. So any suggestion is helpful.
Thanks!