# Is the inverse Fourier transform always defined?

The Fourier transform of a continuous-time function is defined if the function is absolutely integrable, otherwise it does not exist. What about the inverse Fourier transform? If I make up any function of $$\omega$$, can I always invert it to some time-domain counterpart?

Would that mean that every function in the frequency domain will have a corresponding function in the time domain but not vice versa?

Well, you can't just make up any function of $$\omega.$$ Just as for the Fourier transform itself, the function has to be nice enough, e.g. $$L^1$$ or $$L^2.$$ Since the Fourier transform can be extended to tempered distributions the functions can actually grow polynomially at the infinities.