[SciPy-User] Orthogonal polynomials on the unit circle

[Charles R Harris]

On Sat, Oct 27, 2012 at 11:32 AM, Charles R Harris <
charlesr.harris at gmail.com> wrote:

>
>
> 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.
>
>
See also Szego's book, Orthogonal
Polynomials<http://www.amazon.com/Orthogonal-Polynomials-Colloquium-Publications-Mathematical/dp/0821810235>,
ch 11.

Chuck
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