I feel like i don't understand the Fourier transform. I've seen what it does and its properties but even after reviewing various proofs i don't understand why we end up explicitly with a relation between time and frequency.
If the transform is an extension of the series for infinite period then why cant I in principle express it as a relation between time and an approximation in time domain. For a series i can do that, i can write a series as as a sum of sines or cosines or exponentials involving frequency, why cant i do this with the transform.
Also would appreciate an explanation of the uncertainty principle in relation to the Fourier transform. I haven't found anything browsing.