I have the following functions:
- $e^{2x} $
- $e^{-2x} $
- $e^{2x}H(x)$
- $e^{-2x}H(x)$
- $e^{\sin x}$
- $(x^2-1)^3$,
where $H$ is the Heaviside function. The question is - which of these functions can be considered as distributions? Clearly $e^{\pm 2x}$ grow way too fast as $x$ tends to $\pm \infty$. For the same reason, $e^{2}H(x)$ should be disqualified. It can be shown $e^{-2x}H(x) \in L^1_{\text{loc}}({\Omega})$ since $\int_{K \cap[0,+\infty]}|e^{-2x}| dx \leq \dfrac{1}{2}<+\infty$ for every compact $K \subset \Omega$. Since $|e^{\sin x}| =e^{\sin x}\leq e$ we conclude it is locally integrable and so it makes the list. It is pretty obvious $(x^2-1)^3\in \mathcal{D'}$ since it's in $\mathcal{P_6}$.
Question 1: Are my arguments sound? Why not?
Question 2: Is there a more general test to determine if a function can be considered as a distribution?