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Re: Generalized Fourier localization theorem?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg102496] Re: [mg102444] Generalized Fourier localization theorem?
*From*: danl at wolfram.com
*Date*: Tue, 11 Aug 2009 04:05:12 -0400 (EDT)
*References*: <200908092219.SAA09118@smc.vnet.net>
> 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.
No to the first. A Dirac comb has infinitely many such regions, and it's
FT is also a Dirac comb.
In[1]:= FourierTransform[DiracComb[x], x, t]
Out[1]= DiracComb[-(t/(2 \[Pi]))]/Sqrt[2 \[Pi]]
Though the comb is a periodic function, this does not comprise a
counterexample for the periodic case. There we have discrete frequencies
(by scaling, can take them as the integers), so it's not obvious to me
what would be the correct analogue of the question. But in any case we now
have all components of the FT nonzero (they all are unity).
In[2]:= Integrate[DiracDelta[x]*Exp[I*x*t], {x, -Pi, Pi},
Assumptions -> Element[t, Reals]]
Out[2]= 1
Daniel Lichtblau
Wolfram Research
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