Rapid decrease of Fourier coefficients

Question

On page 23 of Anton Deitmar's book on automorphic forms, he presents a proof that for a function

$$g\in C^\infty(\mathbb{R}/\mathbb{Z}),$$

the Fourier coefficients

$$c_k(g)=\int_0^1g(t)e^{-2\pi i k t}dt,$$

are of rapid decay, meaning that

$$\forall N\in\mathbb{N}, k^Nc_k(g), \text{is bounded as a function of } k.$$

Using integration by parts, the author manages to obtain the inequality (for $k\neq 0$)

$$|c_k(g)|\leq\frac{1}{(4\pi^2k^2)^n}\left|\int_0^1g^{(2n)}(t)e^{-2\pi i k t}dt\right|,$$

for some $n\in\mathbb{N}$. If I understand correctly the integral is bounded as a function of $k$ because both

$$\int_0^1g^{(2n)}(t)dt, \text{ and } \int_0^1e^{-2\pi i k t}dt,$$

are. Furthermore, multiplying by $k^n$, the coefficient becomes $((4\pi^2)^nk)^{-1}$, which is certainly bounded as a function of (non-zero) $k$, and thus is of rapid decay? (because the choice of $n$ is arbitrary)

Secondly, the author goes on to claim that the rapid decay of the $c_k(g)$ implies that the series

$$\sum_{k\in\mathbb{z}}|c_k(g)|,$$

converges. I'm having trouble seeing how this follows from rapid convergence. I played around with the series a bit, but to no avail.

Answer 1

1. We have by compactness of $\mathbb R/\mathbb Z$, $$g^{(2n)}\in C^\infty(\mathbb R/\mathbb Z)\subset L^\infty(\mathbb R/\mathbb Z)\subset L^1(\mathbb R/\mathbb Z),$$ hence $$\left|\int_0^1g^{(2n)}(t)e^{-2\pi i k t}\,\mathrm dt\right|\le\left\lVert g^{(2n)}\right\rVert_1<\infty.$$ This means that $|c_k(g)|$ is less than some constant $C(n)$ times $k^{-2n}$ for every $n\in\mathbb N$.
2. Setting $n=1$, we get $$\sum_{k\in\mathbb Z\setminus\{0\}}|c_k(g)|\le C(1)\sum_{k\in\mathbb Z\setminus\{0\}}k^{-2}=\frac{C(1) \pi^2}3<\infty. $$