A function on $\mathcal{S}(\mathbb{R}^n)$ written as a combinations of Schwartz functions

Question

I am having a hard time trying to figure out how to do this exercise:

Let $\psi \in \mathcal{S}(\mathbb{R}^n)$ be such that $\psi(0)=0$. Prove that there are functions $\rho_j \in \mathcal{S}(\mathbb{R}^n)$, $j=1,2,...,n$, such that $$\psi(x)=\sum_{j=1}^nx_j\rho_j(x)$$ for all $x \in \mathbb{R}^n$.

I have no idea how to start it. Can someone help me? Thanks.

Answer 1

For any fixed $x\in\mathbb R^n$, define $f_x\colon t\mapsto \psi\left(tx\right)$, $t\in \mathbb R$. Then $\psi(x)=f_x(1)-f_x(0)=\int_0^1f_x'(s)\mathrm ds$. The computation of the derivative will give the expression for the functions $\rho_j$. Then it will remain to check that these functions belong to the Schwartz space.