How to cook up test functions?

Question

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!

Answer 1

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?)

Yes, first apply Whitney extension then smoothly cut-off. For actual constructions and such, much of the work can be traced back to Charlie Fefferman; see this article for some references.

Given a bad function, can we smooth it to be a test function (like smooth away the sharp angles of the tent function)?

Yes, convolve it against the Gaussian (or any integrable smooth function) and then smoothly cutoff (multiply against a test function).

Answer 2

The definition of your test functions implies that they are a proper subset of the schwartz functions. These functions has some nice properties such as the fourier transformation being a linear automorphism.