What can be said about an $L^2$ function whose Fourier series is zero almost everywhere?

Question

let $f \in L^2(\mathbb{T})$ whose Fourier coefficients $c_k=0$ except for multiples of $3$. The Fourier series is given by $f(x)=\sum_{k\in \mathbb{Z}}c_ke^{ikx}$

I know that for every $f \in L^2(\mathbb{T})$, it's associated Fourier series converges almost everywhere

Question: Does this mean that the Fourier series is zero almost everywhere? If so, does that imply $f(x)=0$ almost everywhere? I'm struggling a bit with the intuition. Can someone provide a proof of this, or an appropriate counter-example?

Answer 1

Here is another argument : the Fourier transform is an isometry : $L^2(T) \to L^2(T)$. In particular, $f = 0$ a.e iff $\hat f = 0$ a.e. So if your function is non-zero, its Fourier transform will be non-zero.