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. -- We're just a Bunch Of Regular Guys, a collective group that's trying to understand and assimilate technology. We feel that resistance to new ideas and technology is unwise and ultimately futile.
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