Suppose I have a measurable complex function on the circle $f : S^1 \rightarrow \mathbb{C}$ and I find its fourier coefficients $\int f(x) e^{-2 i n \pi x} dx$ are all 0.
Is $f$ a.e. 0? Could it be something else?
Edit: I figured out a proof that $f$ a.e. 0. But my proof only works on the circle. Is the same thing true for $f : \mathbb{R} \rightarrow \mathbb{C}$?
Okay so we know $f$ is $L_1$ at least since the first coefficient is the integral. And we also know that for $L_1$ functions the arithmetic averages of partial sums of the fourier approximations converge in $L_1$ to the actual function. But the actual function must be 0 then.