Which functions can be called a distribution or a generalized function

Question

I am not a student of mathematics, but an engineer who has studied mathematics. So, I did not have distribution theory in my educational background, but I did study it on my own because I need to understand a certain problem. I learnt that test functions, on which distributions act, need to be the intersection set of compactly supported function such that ($\int_{-\infty}^{+\infty}\phi\left(x\right)\mathrm{d}x \leq M\quad \forall M\in\mathbb{R}$) and the set of infinitely differentiable functions ($\mathbb{C}^{\infty}$). Also I know that the test function needs to vanish for $|x|\geq a \quad\forall a\in \mathbb{R}$. But, is there any test for a distribution or the generalized function itself? Like how would I say that a certain function I have would behave as a distribution for lets say $N\to\infty$?

Answer 1

Usually it is said that a distribution is a function in the sense that $L^1_{loc}(\Omega) \subset \mathcal{D}'(\Omega)$ because every locally integrable function $f$ defines a distribution $u_f \in \mathcal{D}'(\Omega)$ by $$ u_f(\varphi)=\int_{\Omega} f(x) \varphi(x) dx $$ for each test function $\varphi$. In particular, since $u(\varphi)=\int_{\Omega} f(x) \varphi(x) dx=0$ $\forall \varphi \in \mathcal{D}(\Omega)$ imply that $f(x)=0$ a.e. in $\Omega$ we have that the operator $f \in L^1_{loc}(\Omega) \rightarrow u_f \in \mathcal{D}'(\Omega)$ is linear and one-to-one, and functions of $L^1_{loc}(\Omega)$ are uniquely determined by the distributions $u_f, u_g \in \mathcal{D}'(\Omega)$ since if $u_f(\varphi)=u_g(\varphi)$ $\forall \varphi \in \mathcal{D}(\Omega)$ we have that $f(x)=g(x)$ a.e. in $\Omega$.