In Real Analysis and Probability, Th 8.1.3 used a proposition that
Let $g$ be a continous map from $\mathbb{R}$ to $\mathbb{C}$, with compact support, that is for some $T>0$, $g=0$ off $[-T,T]$. since $g(-T)=g(T)=0$,$g$ may be transferred to the unit circle in other words, $g(x)=h(exp\frac{i\pi x}{T}), -T<x<T$, for some continous $h$ from $E=\{z\in \mathbb{C}:\lvert z\rvert =1\}$ to $\mathbb{C}$
And I don't understand how this works ? How to consitruct a $h$ like this？