An extension problem in Sobolev spaces

Question

Suppose $u\in H^{-s}(0,1)$ and $0<s<1/2$. Is there $\tilde{u}\in H^{-s}(\mathbb{R})$ such that the restriction of $\tilde{u}$ to $(0,1)$ is $u$? Does this work: $\tilde{u}(\phi)=u(\phi|_{(0,1)})$ if the $supp(\phi)\subset (0,1)$, and $\tilde{u}=0$ otherwise, for $\phi\in H^s(\mathbb{R})$.