Periodic coefficients of Fourier series

Question

If we have a continuous function $f$ on $[-\pi,\pi]$ and its complex Fourier coefficients are periodic, i.e. $$c(n) = c(n+k)$$ for some $k\in \mathbb{N}$, can we prove that $f$ is identically the zero function?

Answer 1

Hint For all m you have

$$\sum_{n \in \mathbb Z} |c(m+nk)|^2 \leq \sum_{n \in \mathbb Z} |c(n)|^2 =\frac{1}{2 \pi}\int_{-\pi}^\pi |f(x)|^2 dx$$

Answer 2

The Fourier coefficients $c(m)$ tend to $0$ as $ m \to \pm \infty$. Hence, $c(n)=c(n+mk) \to 0$ as $ m \to \infty$ proving that $c_n=0$ for all $n$.