First some definitions. $\Omega \subseteq \mathbb R^N$.
- ${\cal D}(\Omega)$ denotes the set of all compactly supported smooth functions, $f \in {\cal C}^\infty_0(\Omega)$, equipped with the following convergence notion:
$ {\cal C}^\infty_0(\Omega) \ni \{ f_n \}_{n \in \mathbb N} \overset {n \to \infty}{\longrightarrow} f \in {\cal C}^\infty_0(\Omega) $ if:
- $\exists K \subset \Omega$ compact $\ |$ ${\rm supp}(f_n) \subseteq K \quad \forall n$
- $\partial^\alpha f_n \rightrightarrows \partial^\alpha f $ uniformly, $\forall \alpha $ multiindex.
- a linear functional $u: {\cal D}(\Omega) \to \mathbb C $ is a distribution if $\forall$ convergent sequence $ \{f_n\}_{n \in \mathbb N} \in {\cal D}(\Omega)$ one has:
$$\lim_n u(f_n) = u (\lim_n f_n)$$
${\cal D}' (\Omega) \equiv \{ u: {\cal D}(\Omega) \to \mathbb C \ | \ u
$ is a distribution$\}$
- given $\Omega_0 \subset \Omega $, $\Omega_0$ is said to be a vanishing open set for $u \in {\cal D}'(\Omega) $ if $\forall f \in {\cal D}(\Omega) $ such that ${\rm supp} f \subseteq \Omega_0$, one has:
$$u(f) =0$$
4.$ \forall u \in {\cal D}'(\Omega) $ we define ${\rm supp } \ u \equiv \Omega \setminus \bigcup\limits^{}_{i} \Omega_0^i$, with $\Omega_0^i$ a vanishing open set for $u$.
Question: is ${\rm supp} \ u $ compact $\forall u \in {\cal D}'(\Omega) $?
Best Answer
No, there are distributions with non-compact support. An example is a pulse train: $$ u = \sum_{k=-\infty}^{\infty} \delta_k, $$ where $\delta_k$ is a Dirac delta supported at $x=k$. Less exotic examples come from non-compactly supported functions in $L^1_{loc}$.