The space of distributions on $\mathbb{R}^n$ is essentially found by requiring that it should be possible to apply the distribution to any bump function. Similarly, compactly supported distributions can be applied to any smooth function.
Is there some space of distributions which can be applied to all functions of polynomial growth? I'm fairly sure this condition should have been given a name by now but I'm struggling to find a reference.
Since I was unable to find any name for this type of distribution I have given it a name myself and proved that it works well. Here are the relevant notions: