[SciPy-User] Orthogonal polynomials on the unit circle

[Charles R Harris]

On Sat, Oct 27, 2012 at 9:34 AM, <josef.pktd at gmail.com> wrote:

> On Sat, Oct 27, 2012 at 10:35 AM, Charles R Harris
> <charlesr.harris at gmail.com> wrote:
> >
> >
> > On Fri, Oct 26, 2012 at 7:40 PM, <josef.pktd at gmail.com> wrote:
> >>
> >> http://en.wikipedia.org/wiki/Orthogonal_polynomials_on_the_unit_circle
> >> with link to handbook
> >>
> >> application: goodness of fit for circular data
> >>
> >>
> http://onlinelibrary.wiley.com/doi/10.1111/j.1467-842X.2009.00558.x/abstract
> >>
> >> Are those available anywhere in python land?
> >>
> >
> > Well, we have the trivial case: ϕ_n⁡(z)=z^n for the uniform measure. That
> > reduces to the usual exp(2*pi*i*\theta) in angular coordinates when the
> > weight is normalized. But I think you want more ;-)  I don't know of any
> > collection of such functions for python.
>
> I need to see if I can use this. In general, I would like other weight
> functions
> (Von Mises distribution in the density estimation example (?), like
> hermite polynomials for the normal distribution).
>
> I don't know much about the math of circular statistics and functions,
> I just want to estimate distribution densities on a circle, and I
> discovered that periodic or circular polynomials would be useful for
> estimating seasonal/periodic effects. (the clock as a circle)
> The ends don't match up with chebychev
>
> https://picasaweb.google.com/106983885143680349926/Joepy#5747376116689698434
>
> >
> >> What's the difference between orthogonal polynomials on the unit
> >> circle and periodic polynomials like Fourier series?
> >
> >
> > It looks to be the weight. Also, the usual Fourier series include terms
> in
> > 1/z which allows for real functions. I suspect there is some finagling
> that
> > can be done to make things go back and forth, but I am unfamiliar with
> the
> > topic. Hmm, Laurent polynomials on the unit circle might be more what you
> > are looking for, see the reference at http://dlmf.nist.gov/18.33 .
>
> Might we worth looking into, but this "finagling" usually turns out to
> be very time consuming for me, where I don't have the background and
> no pre-made recipes.
>
> (Might be just finding the right coordinate system, or it might mean I
> would have to look into complex random variables.)
>
>
There seems to be quite a bit of literature out there, but not of the
practical sort, i.e., use this for weights that. I thought this paper,
Orthogonal
Trigonometric Polynomials <http://arxiv.org/abs/0805.2640>, was pretty good
as an introduction to the area and it seems to cover the 'finagle', but I
suspect it isn't what you need. I put it out there in case someone wants to
pursue the subject.

Chuck
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