Is the following interpretation correct?
I have a $T-$ periodic function $f(t)$ such that $f(t)=f(t+T)$ according to definition. In the trigonometric Fourier series, it is known that the coefficients for e.g. the cosines are computed as, $$a_n = \dfrac{2}{T}\int_{T}f(t)\cos(n\Omega t)\,dt.$$
If $f(t)$ is odd on $T$, it will yield that $a_n = 0$ according my to textbook. This applies for any interval of length $T$. If understand it correctly then;
If I on some interval of length $T$ are able to show that $a_n =0$, then for any interval I choose, as long as it is of length $T$, $a_n$ will be zero.
Furthermore, If I understand it correctly, if the integral according to $a_n$ is zero, this means that $f(t)$ and $\cos(n\Omega t)$ are orthogonal on any interval of length $T$ (with weight function 1 ?). This theory is from a first year mathematics course on an engineering programme.
I am grateful for every response and many thanks in advance.
Edit: excluding the case of $a_0$ of course.