Help with some Fourier series questions

Question

I need some help with this question about Fourier Series.

1) If $f\in{L_{1}(T)}$ (that's $f$ periodic with period $2\pi$ and $|f|\in{L_{1}([-\pi,\pi]}$)) with Fourier series $\frac{a_0}{2}+\sum_{k=1}^{\infty}a_k cos(kt)+b_k sin(kt)$, then $f$ is even if and only if $b_k=0$ for all $k$, and $f$ is odd if and only if $a_k=0$ for all k.

I have no problem proving the "left to right" implications, just with simple integration. But i can't prove the "right to left" ones. ¿Can anyone help me at this?

Answer 1

Just note that

$$ a_k = \int_{-\pi}^{\pi} f(x)\cos(kx)dx. $$

Now, if $f(x)$ is an odd function then the integrand is an odd function and this implies that the value of the above integral is zero. The same with the other case.