Can you prove this?

Question

Math proof that a limited time domain signal is infinite in the frequency domain & a limited band of frequencys signal that is infinite in freq domain, is unlimited in time domain. I suspect Fourier sereis and transforms are involved. Thax in advance.

Answer 1

Yeah, I would do the fourier transforms.

Show that

x(omega) = 1/sqrt (2pi) * integral from time 1 to time2 of exp (-i omega t) x(t) dt

can't be zero for all values of omega above any arbitrary cutoff

Which is exactly the same as showing that

x(t) = 1/sqrt (2pi) * integral from omega1 to omega2 of exp (+i omega t) x(omega) dt

can't be zero for all values of t above any arbitrary cutoff.

So you need to show that for any cutoff freq/time, there must be some greater freq/time with a non-zero fourier component.

That's a start for you.