I know many examples of periodic functions $f(t)$ where the Fourier coefficients don't depend on the period $T$. I was wondering if this is a general property or not. So far all I did was
\begin{align} a_n&=\frac{2}{T}\int_{-T/2}^{T/2} f(t) \cos\Bigl(\frac{2 \pi n}{T}t\Bigr)\, dt \\\\ &=2\int_{-T/2}^{T/2} f\Bigl(T\frac{t}{T}\Bigr) \cos\Bigl(2 \pi n \frac{t}{T}\Bigr)\, d\Bigl(\frac{t}{T}\Bigr) \\\\ &=2\int_{-1/2}^{1/2} f(Tu) \cos(2 \pi n u)\, du \\\\ \end{align}
I do believe that the composition $f(Tu)$ doesn't actually depend on $T$ since every appearance of $t$ in $f(t)$ shows up like $t/T$. But I'm not sure that's true or how I can say it formally.