The uncertainty principle states that if you have a signal which is very concentrated in time, then its Fourier transform will be rather outspread and vice versa. However, I don't really understand how this implies that in general we can't find a function that gives us the frequency spectrum for a particular point in time.
I don't even know what exactly is meant with determining the frequency spectrum at a specific point in time. If I wanted to localise the frequency better (e.g. at zero), I would multiply by a corresponding indicator function and consider the fourier transform, i.e. consider $$ \begin{aligned} \int_{ \mathbb{R}} f( x) {1}_{ [ - \epsilon , \epsilon ] } ( x) e^{ -2\pi i\xi x}\, dx .\end{aligned} $$ However, with this interpretation the frequency spectrum at $ x = 0$ doesn't really make sense, since for $ \epsilon \to 0$ this integral will vanish. Can someone clarify this a bit?
There is a "best possible distribution function" for the simultaneous frequency-time domain: it's called the Wigner distribution function (WDF) for the signal.
However, you can never have a signal that is perfectly localized in both frequency and time, i.e. its WDF is a delta spike in two dimensions. Think about it counterfactually: suppose we had a signal pulse, instant, that occurred both exactly at time 5 seconds and with a frequency exactly at 50 Hertz. Then we had another instant signal, also occurring at time 5 seconds, with a frequency exactly 500 Hertz.
Both are an instant, point-sized signal i.e. a delta spike. The only thing that can differ between them, then, is their generalized amplitude. How, then, can the 50 vs. 500 Hz frequency difference be meaningful? So there must be a limit on the simultaneous localization, and that is on the order of one square unit in the FT space.
But that doesn't mean you can't come up with the distribution. It just means you can't make that distribution to be of point support for any possible signal.