Re: Finding simplest Fourier series between two given
- To: mathgroup at smc.vnet.net
- Subject: [mg105678] Re: [mg105662] Finding simplest Fourier series between two given
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Tue, 15 Dec 2009 07:26:20 -0500 (EST)
- References: <200912140506.AAA16985@smc.vnet.net>
- Reply-to: drmajorbob at yahoo.com
Or better yet:
f3[x_] = Piecewise@
{{f1@x, 0 <= f1@x},
{f2@x, f2@x <= 0}
}
Bobby
On Sun, 13 Dec 2009 23:06:39 -0600, 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.
>
> 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.
>
--
DrMajorBob at yahoo.com
- References:
- Finding simplest Fourier series between two given Fourier series
- From: Kelly Jones <kelly.terry.jones@gmail.com>
- Finding simplest Fourier series between two given Fourier series