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Re: Generalized Fourier localization theorem?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg102528] Re: Generalized Fourier localization theorem?
*From*: AES <siegman at stanford.edu>
*Date*: Wed, 12 Aug 2009 04:35:53 -0400 (EDT)
*Organization*: Stanford University
*References*: <200908092219.SAA09118@smc.vnet.net> <h5r8fk$l9g$1@smc.vnet.net>
In article <h5r8fk$l9g$1 at smc.vnet.net>, danl at wolfram.com wrote:
> > 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]]
Good point!! -- although I should have realized that "I already knew
that!" This is what Ron Bracewell used to call the "shah function".
Thank you, Daniel.
In case you have any interest in the DFT version of the original
question, appended below is part of the Introduction to a notebook I
wrote some years ago and recently revisited. The key content from a
mathematical viewpoint is in the first three paragraphs. This was done
mostly for curiosity-driven reasons, as an offshoot from some work on
rigorous uncertainty principles for periodic functions. If there is
some more serious mathematics associated with this topic, I'd be glad to
hear about it -- even though if it gets too serious, it will be over my
head.
====================================================
"Brad Osgood once raised the question with me as to whether some close
analog of this localization principle also applies to Discrete Fourier
Transforms (DFTs) specifically if an N-term periodic sequence f_n
has two or more contiguous zero elements, is its DFT g_m barred from
also having two or more contiguous zeros?
This notebook shows that this is not the case: One can find N-term DFT
pairs both of which have a string of contiguous zero values over
approximately half of their widths. To demonstrate this point
experimentally, this notebook presents three methods of obtaining such
sequences:
1) In the first method, a discrete sequence f_n of periodic length N
is assumed to have the uppermost (N/2) - 1 of its contiguous elements
set equal to zero. The remaining (N/2) + 1 elements are then used to
set up N coupled linear equations which express the uppermost N
elements of the DFT sequence g_m = DFT[f_n] as a linear superposition
of the (N/2) + 1 lower elements of f . Setting these N equations to
zero then allows one to solve for the elements of the f array from
f[1] up to f[N/2] in terms of f[0].
Carrying through on this procedure indeed leads to a unique, finite,
more or less gaussian-like "maximum localization" result as shown below,
yielding a gaussian-like sequence f with (N/2) -1 contiguous zero
elements and a gaussian-like sequence g with N/2 contiguous elements
that are both bounded and are DFTs of each other. Ergo, discrete
sequences can be simultaneously bounded, or "localized", at least to the
extent demonstrated here. This calculation method slows down rapidly
for N larger than about 16.
2) Examining the results of this approach shows that they can be
shifted into a simple form, in which the sequence g_m can be made
purely real and symmetric about the "m = -1/2" element of the array, and
the sequence f_n then consists of complex conjugate values centered
about the DC or n=0 value, with a small and apparently totally linear
phase shift associated with the half-period offset of the g_m array.
This leads to a somewhat simpler approach in which the same array
coefficients can be solved by inverting an N/4-th order real matrix,
although this matrix apparently becomes seriously ill-conditioned as the
order N increases.
3) Motivated by the Fox and Li approach to optical resonator modes, I
also implement an ad hoc approach in which an iterative process of
transforming between the sequences and then filtering them is used to
try to force the sequences to converge to minimum localization.
Experimentally this approach also converges, at least for not too large
values of N , although it seems to fail to converge (perhaps as a
result of numerical underflow) at values of N greater than 64 or so.
Each of these approaches is demonstrated below, using a common set of
display modules for the results (For simplicity these are all programmed
only for values of N = integer multiples of 4.). These calculations
were initially carried out by AES during May-June 2003; updated and
extended in June 2004; and cleaned up a bit for the latest version 7 of
Mathematica in August 2009."
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