Let $f \in L^1(\mathbb{D})$ where $\mathbb{D}$ is the unit disc in the complex plane. Let $r_0\in (0,1)$ such that
$$f(r_0e^{i\theta})=\sum_{n=-\infty}^{\infty}\hat{f}(n){r_0}^ne^{in\theta}=0$$
for every $\theta\in (-\pi,\pi)$. Will that imply that $\hat{f}(n)=0$ for every $n\in\mathbb{N}$?
I know that the $\hat{f}(n)=0$ for every $n\in\mathbb{N}$ is the given condition holds true for every $re^{i\theta}\in\mathbb{D}$.
$\sum_{n=-\infty}^{\infty} c_n e^{int}=0$ implies each $c_n=0$. This is immediate from the fact that $\{e^{int}:n\in \mathbb Z\}$ is an orthonormal basis for $L^{2}[0,2\pi]$.