Say we are given a well-behaved function $f(t)$ (either discrete or continuous) and are able to compute its spectrum $F(k)$ using the (discrete) Fourier transform. Then say we lose $f(t)$ and know nothing about it anymore.
We still have $F(k)$. What can we say about $f(t)$? Say for the sake of argument that the spectrum contains billions of nonzero values and thus we cannot compute it. We need a priori arguments.
As an example of the kind of insight I'm looking for, we can say whether the function $f(t)$ is even or odd just by looking at the spectrum. (If the spectrum is all real $f(t)$ is even, if all imaginary, $f(t)$ is odd.)
I am particularly interested in arguments stating whether or not there exists a $t$ such that $f(t) = x$ for some point $x$. Only given $F(k)$, can we definitively rule out or include some points $x$?
Finally, are there any results or sources in the more general, abstract literature of harmonic analysis, spectral theory, or functional analysis which could help here?