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Generalized Fourier localization theorem?


The following is a math question, not a Mathematica question, but it 
relates to a Mathematica calculation I'm attempting to do, so I hope it 
can be raised in this group.

Suppose a complex-valued function  f[x]  with  x  real, has a region of 
finite width within the range -Infinity < x < +Infinity where the 
function  f[x]  is identically zero.

Does this imply that its Fourier transform  g[s]  with  s  real can 
_not_ have any such region of finite width where  g[s]  is identically 
zero within its similar domain?

Similar theorem for the Fourier series of a periodic function?

Thanks for any pointers.


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