I have been trying to go from the Fourier transform "back" to the Fourier series, i.e. starting with
\begin{align} f(t) &= \frac{1}{2\pi}\int_{-\infty}^{+\infty}f(\omega)e^{i\omega t}d\omega \\ \quad f(\omega) &= \int_{-\infty}^{+\infty}f(t)e^{-i\omega t}dt \end{align}
get to expressions like
\begin{align} f(t) &= \frac{a_0}{2} + \sum_{r=1}^{\infty}a_r\cos(2\pi rt/T)+b_r\sin(2\pi rt/T) \\ \text{with} \quad a_r &= \frac{1}{T}\int_{t_0}^{t_0+T}f(t)\cos(2\pi rt/T)dt \\ \text{and} \quad b_r &= \frac{1}{T}\int_{t_0}^{t_0+T}f(t)\sin(2\pi rt/T)dt \, . \end{align}
Can I check the following argument?
As I understand it, for the Fourier transform we need $f(t)$ to be integrable over the full domain (i.e. $\int_{-\infty}^{+\infty}f(t)dt$ must be finite), so a periodic function over all time wont satisfy this if there's any offset. Is this the right thing to think about first? So this led me to consider these periodic functions instead as being functions defined in a finite time domain of total length $T$. This nicely explains the limits in the integrals of the Fourier coefficients $a_r$ and $b_r$. I think this then forces your frequencies to no longer be a continuum, i.e. the only frequencies in the domain are multiples of $2\pi/T$. At this point I was tempted to say $f(\omega)$ was being multiplied by a dirac delta comb. But I think I realise that by restricting the time domain we aren't saying that $f(\omega)$ is non-zero only at specific $\omega$, we are saying that $f(\omega)$ is no longer a function of a continuous variable. It's not meaningful to multiply by a delta function as there isn't any frequency value between the allowed values. This means I replace $\int_{-\infty}^{+\infty}...\omega...d\omega$ with $\sum_{r=-\infty}^{+\infty}...\omega_k...\Delta\omega$ where $\Delta\omega=2\pi/T$ and $\omega_k=k\Delta\omega$.
A side question here would be: is the frequency-domain dirac delta function in a finite-length time domain accurately said to be (if located at $\omega_0$):
$\delta(\omega_k - \omega_0) = 1/\Delta\omega$ if $\omega_k=\omega_0$, otherwise it is 0
I think in the limit of infinite time domain, this will converge to the correct definition of a delta function for what will now be a continuum frequency domain, yes?
Is the above all correct, or is there another way to be trying to derive Fourier series from Fourier transforms?
Thanks!