In PDE basic theory by the author Taylor I can't understand two facts:
Prop 1. If $u\in \mathcal{S}'(\mathbb{R}^n)$ is supported by $\left\{0\right\}$, then there exists $k$ and complex numbers $a_\alpha$ such that \begin{align} u=\sum_{|\alpha|\leq k} a_\alpha D^\alpha \delta \end{align}
Prop 2. Suppose $u\in \mathcal{S}'(\mathbb{R}^n)$ satisfies $\Delta u=0$ in $\mathbb{R}^n$. Then $u$ is a polynomial in $(x_1,\ldots, x_n)$. Proof. $|\xi|^2\widehat{u}=0$ in $\mathcal{S}'(\mathbb{R}^n)$ implies that $\text{supp}\widehat{u}\subset \left\{0\right\}$. By prop 1, \begin{align} u=\sum_{|\alpha|\leq k} a_\alpha D^\alpha \delta \end{align}
Question 1. Why $\text{supp} \widehat{u}\subset \left\{0\right\}$?
Question 2. Why $u=\sum_{|\alpha|\leq k} a_\alpha D^\alpha \delta$ implies that $u$ is a polynomial in $(x_1,\ldots, x_n)$?