Let $f:\mathbb{R}^{2} \rightarrow \mathbb{C}$ given by $$ f(x_1, x_2)=(x_1+ix_2)^{n}.$$ We have $f \in S'$ because as $f \in L^{1}_{loc} (\mathbb{R}^{2})$ and $|f(x_1,x_2)|^{2} \le (1+|(x_1,x_2)|^{2})^{n}$ it is possible to show that $$ \varphi \in S \rightarrow \int f\varphi$$ defines a tempered distribution.
My problem is
How can I show that $$\mathcal{F}{(x_1+ix_2)^{n}}=\mathcal{F}(x_1+ix_2)*\ldots *\mathcal{F}(x_1+ix_2)?$$
Notation: S is the Schwartz space.
S' is the space of tempered distributions
$\mathcal{F}:S' \rightarrow S'$ is the Fourier transform.
$*$ is the convolution.
Thanks for your help.