Estimate of a (integral) function

Question

I should show that function $H(w)=\int_{-\pi}^{\pi}f(x) e^{iwx}dx$, where $f(x)\in L^2(-\pi,\pi)$, is such that $H(re^{i\theta})=O(e^{\pi r |sin(\theta)|})$. Any suggestion?

Answer 1

1. Notice that $|e^{ire^{i\theta}x}| = e^{{\rm Re}(ire^{i\theta}x)} = e^{-xr\sin(\theta)}$.

2. Use Cauchy-Schwarz inequality to estimate $\displaystyle \int_{-\pi}^{\pi}f(x)\cdot 1\, dx$.