Fourier coefficients are exponentially bounded for a real analytic, periodic function

Question

This question popped up in my exam today, I am curious for a solution.
Let $f:\mathbb{R}\to\mathbb{C}$ a real analytic, $2\pi$-periodic function that is integrable on $[-\pi,\pi]$. Its Fourier coefficients $c_k$ for $k\in\mathbb{Z}$ are given by $$
c_k=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{ikx}dx
$$
Then there exists $\Gamma>0$ and $\eta>0$ such that $|c_k|\leq \Gamma e^{-\frac{1}{2}|k|\eta}$ for all $k\in\mathbb{Z}$.

Answer 1

Since $f$ is periodic and real analytic, it admits an analytic extension to a small neighbourhood of the real line $U(\eta)=\{ x+iy \in \mathbb C: |y|<\eta \}$. Choose $$\Gamma = \max_{|y|\le \eta/2}\int_{-\pi}^{\pi} |f(x+iy)|dx < \infty .$$ Suppose $k\ge 0$. Then as $f(x)e^{ikx}$ is analytic and periodic in the real direction on $U(\eta)$, by Cauchy's theorem (integrating along the boundary of the box $[-\pi,\pi]\times [0,\eta/2]$)

$$ \left| \int_{-\pi}^\pi f(x)e^{ikx} dx\right| = \left|\int_{-\pi}^\pi f(x+i\eta/2)e^{ik(x+i\eta/2)} dx\right|=e^{-k\eta/2}\left|\int_{-\pi}^{\pi} f(x+i\eta)e^{ikx} dx\right| \le e^{-k \eta/2}\Gamma.$$ You can do a similar thing if $k<0$.