Support of polynomial distribution

Question

Let $P(x_1,\cdots,x_n)$ be a polynomial in $\mathbb{R}^n.$ What is supp$(\widehat{P})$ when $P$ viewed as a tempered distribution. Can supp$(\widehat{P})$ be the boundary of an sphere?

Answer 1

Here is what I think: Let $P=\sum_{|\alpha|\leq k}a_{\alpha}x^{\alpha},$ then $$\widehat{P}=\widehat{\Big(\sum_{|\alpha|\leq k}a_{\alpha}x^{\alpha}\Big)}=\sum_{|\alpha|\leq k}a_{\alpha}\widehat{x^{\alpha}}=\sum_{|\alpha|\leq k}a_{\alpha}\frac{\partial^{\alpha}\delta_0}{\partial x^{\alpha}}$$ Thus supp$(\widehat{P})\subset$ supp$(\sum_{|\alpha|\leq k}a_{\alpha}\frac{\partial^{\alpha}\delta_0}{\partial x^{\alpha}})\subset\,$ supp$(\delta_0)=\{0\}$