Assume $G\in L^1(\mathbb{T})$ is any function such that $\widehat G(m)=m$ for all $m$ such that $|m|\leq n\ $. show that for any trigonometry polynomial $P$ of degree at most $n$, $P'=P *G$.
let $P(x)=\sum _{|m|\leq n}a_m \ e^{imx}$. then $P'(x)=\sum _{|m|\leq n}im \ a_m \ e^{imx}$
\begin{align} (P *G)x &=\frac{1}{2\pi}\int_{-\pi}^{\pi}P(y)G(x-y)dy\\ &=\frac{1}{2\pi} \int_{-\pi}^{\pi}\left(\sum _{|m|\leq n}a_m \ e^{imy}\right)G(x-y)dy \end{align}
what can we do after that.i do not understand how idota will appear in this expression.
any hint