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Finding simplest Fourier series between two given Fourier series

  • To: mathgroup at smc.vnet.net
  • Subject: [mg105662] Finding simplest Fourier series between two given Fourier series
  • From: Kelly Jones <kelly.terry.jones at gmail.com>
  • Date: Mon, 14 Dec 2009 00:06:39 -0500 (EST)

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.

Motivation:

 % I'm Fourier-fitting continuous cyclic data that's measured to the
 nearest integer. IE, a datum of 56 means the value I'm measuring is
 between 55.5 and 56.5.

 % Using Fourier series, I can approximate the measured data to
 arbitrary precision, but this feels silly when the terms are of order
 0.1, 5 times smaller than the measurement precision.

 % I believe the data satisfies a fairly simple Fourier relation
 that's being obscured by rounding/measurement precision.

"Extra credit": The data I'm measuring is cyclic, but I'm not
necessarily feeding Mathematica an integral number or cycles. The list
I give Mathematica may have 57.2 cycles instead of 57 or 58. The ideal
solution would compensate for this, though I'd be happy w/ a solution
that just works for an integral number of cycles.

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