Statements about the Fourier series

Question

$f:\mathbb {R} \to \mathbb {R}$ is $2\pi$-periodic and integratable. Which of these following statements is true?

a) If $f(0)=0$, then $\hat {f}(0)=0$

b) If $f(x)=f(-x)$ for all $x \in \mathbb {R}$, then $\hat {f}(k)=\hat {f}(-k)$ for all $k \in \mathbb {Z}$

c) If $\hat {f}(k) = 0$ for all $k \in \mathbb {Z}$, then $f = 0$

I think that a) is true, b) is wrong and that c) is true, is that correct?

Answer 1

1. The first statement is not true, since $\hat {f}(0)=0$ represents the integral of $f$ over the whole domain. Note that the statement would hold for $f(x)=0$.
2. True, see following previous answer 1.
3. The statement is true, since all modes (including the zero'th one, representing the integral of $f$ over the whole domain) are zero. This directly implies that $f$ is constant zero.