Does there exists a distribution on $\mathcal{R}$ with restriction $$u(x)=\sum_{k=1}^\infty\delta_{\frac 1k}^k,$$ to $\mathcal{R}_+$? $\delta_{x}^k$ is the Dirac function defined by $(\delta_{x}^k ,\phi)=\partial^k\phi(x),\forall \phi\in C_c^\infty(I)$.
I guess no, but I cannot find a counterexample yet.
On every compact set, a distribution is of finite order. Also, the order on one set is the supremum of the orders on subsets.
What is the order of $u$ on an interval $[\frac1k-\epsilon, \frac1k+\epsilon]$ where $\epsilon>0$ is very small? What is the supremum of the orders on the subsets of $[-1,1]$?