[SciPy-User] Orthogonal polynomials on the unit circle

Charles R Harris charlesr.harris@gmail....
Sat Oct 27 09:35:40 CDT 2012

On Fri, Oct 26, 2012 at 7:40 PM, <josef.pktd@gmail.com> wrote:

> http://en.wikipedia.org/wiki/Orthogonal_polynomials_on_the_unit_circle
>
> 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.

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 .

Chuck
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