Even distribution?

Question

What exactly does it mean for a distribution $u\in \mathcal{D}'(\mathbb{R})$ to be even? Does it mean that for even testfunctions $\varphi $ it holds that $\langle u, \varphi |_\mathbb{R_+} \rangle = \langle u, \varphi |_\mathbb{R_-} \rangle$?

Answer 1

If $\varphi\in\mathcal D(\Bbb R)$, define $\widetilde\varphi(x):=\varphi(-x)$. It's still a test function. A distribution $u$ is even if for all $\varphi$, $$\langle u,\varphi \rangle_{\mathcal D'(\Bbb R),\mathcal D(\Bbb R)}=\langle u,\widetilde\varphi \rangle_{\mathcal D'(\Bbb R),\mathcal D(\Bbb R)}.$$