Re: Finding simplest Fourier series between two given Fourier series
- To: mathgroup at smc.vnet.net
- Subject: [mg105671] Re: Finding simplest Fourier series between two given Fourier series
- From: AES <siegman at stanford.edu>
- Date: Tue, 15 Dec 2009 07:24:59 -0500 (EST)
- Organization: Stanford University
- References: <hg4h4h$gib$1@smc.vnet.net>
In article <hg4h4h$gib$1 at smc.vnet.net>,
Kelly Jones <kelly.terry.jones at gmail.com> wrote:
> Given two Fourier series f1[x] and f2[x], where f1[x]<=f2[x] for all
> x, I want Mathematica to find the "simplest" Fourier series f3[x] that
> lies between them. More specifically:
>
> I. f1[x] <= f3[x] <= f2[x] for all x
>
> II. f3[x] has the fewest non-zero coefficients of all f3 meeting I.
>
> III. If multiple functions meet I and II, choose the one whose
> highest term is smallest (ie, the "least wiggly" one).
>
> If multiple Fourier series satisfy I, II, and III, I'll settle for any
> of them.
I can't say if this will be of any assistance to you, but maybe some
insights on how to define the "simplicity" of Fourier series or DFTs or
the "distances" between them could be obtained from the references
* Forbes and Alonso, "Measures of spread for periodic distributions and
the associated uncertainty relations," Am. J. Phys., vol. 69, pp.
340--347, (March 2001).
* Forbes and Alonso, "Consistent analogs of the Fourier uncertainty
relation," Am. J. Phys., vol. 69, pp. 1091--1095, (October 2001).
* Forbes, Alonso, and Siegman, "Uncertainty relations and minimum
uncertainty states for the discrete Fourier transform and the Fourier
series," J. Phys. A: Math. Gen., vol. 36, pp. 7027--7047, (2003).
I subsequently made some heuristic explorations into the "localization"
of DFTs which are recorded in
<http://www.stanford.edu/~siegman/DFT localization.nb>
Again, no product warranties as to the suitability of this for any
serious purpose -- but when you mention "fewest non-zero coefficients"
that triggers a mental alert.