The Book States the following Theorem:
Suppose that $f$ is an integrable function on the circle with $\hat{f}(n) = 0$ for all n\in $\mathbb{Z}.$ Then $f(\theta_{o})=0$ whenever $f$ is continuous at the point $\theta_{o}$.
I do not understand the statement of the theorem and how to link it with the remark preceding it which states that: " if $f$ and $g$ have the same Fourier Coefficients, then $f$&$g$ are necessarily equal." and the book said that"by taking the difference $f - g,$ this proposition can be reformulated as: if $\hat{f}(n) = 0$ for all $n \in \mathbb{Z}$, then $f = 0.$"
Could anyone explain this for me?
Thanks.