# Space of Bandlimited Functions

Hi,

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!

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$x\mapsto 0 \:$ certainly seems like a bounded (timelimited) function that is also bandlimited. $\hspace{.8 in}$ – Ricky Demer Nov 12 '12 at 8:06
a nonzero function cannot be both timelimited and bandlimited, this is a very known result aka Heisenberg's uncertainty – Ohad Asor Nov 12 '12 at 11:03

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.
No, those are not Hilbert spaces, because the $L^2$ limit of functions with bounded support can have unbounded support. And your third set is not even a vector space. – Alexandre Eremenko Nov 21 '12 at 3:55