If we evaluate the inverse Fourier transform of X(w) how do we know we get x(t) back?
Link to X(w) and x(t) equations
I know that integrating in the frequency domain results in getting information back in the time domain, and vice versa simply due to the definition of frequency. However, mathematically, how can we integrate an arbitrary function that we don't know the characteristics of? Can we substitute another function, such as the delta function in place of X(w) or the exponential? Do we need to use convolution? Mathematically, I'm stumped. Logically, it's clear to see, I just don't understand the proof.