Generalized Fourier localization theorem?

*To*: mathgroup at smc.vnet.net*Subject*: [mg102444] Generalized Fourier localization theorem?*From*: AES <siegman at stanford.edu>*Date*: Sun, 9 Aug 2009 18:19:37 -0400 (EDT)*Organization*: Stanford University

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.

**Follow-Ups**:**Re: Generalized Fourier localization theorem?***From:*danl@wolfram.com