I was asking myself about some necessary and/or sufficient conditions for a function to be bandlimited (i.e. its Fourier transform is zero t residing out of $[-B,B]$ for some $B>0$).
Of course, if a function is bounded (timelimited), it cannot be bandlimited.
But for a non-bounded function, how can we tell if it's bandlimited or not?
Or, how can we know if a function is both time- and band- unlimited? Any sufficient/necessary conditions.
I'll be glad if you can share with me some known results.


Thanks!



Space of Bandlimited Functions
There is a necessary and sufficient condition for a function $f$ of the real variable
to be "band-limited". It is called the Wiener-Paley theorem. $f$ must be a restriction on the real
line of an entire function of exponential type. Of course, here one implicitly assumes that
$f$ belongs to an appropriate space which permits to interpret it as a "signal".
For example, $L^2$ (finite energy), or $L^\infty$ (bounded amplitude), or Schwarz temperate
distribution etc.


P.S. This is a mathematical website, and the engineering terminology of the problem may
sound strange to some mathematicians. So let me try to translate:
We are taking about a function of a real variable for which Fourier transform is defined
in some sense. Band limited means that Fourier transform has bounded support, and 
"time-limited" means that the function itself has bounded support.
Of course the function cannot be simultaneously time- and band- limited, unless it is zero.
This is a (very crude) form of the "indeterminacy principle", and also follows
from the Wiener-Paley theorem.


