[SciPy-User] Orthogonal polynomials on the unit circle
Charles R Harris
Sat Oct 27 13:33:59 CDT 2012
On Sat, Oct 27, 2012 at 11:32 AM, Charles R Harris <
> On Sat, Oct 27, 2012 at 9:34 AM, <firstname.lastname@example.org> wrote:
>> On Sat, Oct 27, 2012 at 10:35 AM, Charles R Harris
>> <email@example.com> wrote:
>> > On Fri, Oct 26, 2012 at 7:40 PM, <firstname.lastname@example.org> wrote:
>> >> http://en.wikipedia.org/wiki/Orthogonal_polynomials_on_the_unit_circle
>> >> with link to handbook
>> >> application: goodness of fit for circular data
>> >> Are those available anywhere in python land?
>> > Well, we have the trivial case: ϕ_n(z)=z^n for the uniform measure.
>> > 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
>> (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
>> >> 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
>> > 1/z which allows for real functions. I suspect there is some finagling
>> > can be done to make things go back and forth, but I am unfamiliar with
>> > topic. Hmm, Laurent polynomials on the unit circle might be more what
>> > 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.
See also Szego's book, Orthogonal
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