[SciPy-User] orthogonal polynomials ?

[Charles R Harris]

On Mon, May 16, 2011 at 9:11 AM, <josef.pktd at gmail.com> wrote:

> On Sat, May 14, 2011 at 6:04 PM, Charles R Harris
> <charlesr.harris at gmail.com> wrote:
> >
> >
> > On Sat, May 14, 2011 at 2:26 PM, <josef.pktd at gmail.com> wrote:
> >>
> >> On Sat, May 14, 2011 at 4:11 PM, nicky van foreest <
> vanforeest at gmail.com>
> >> wrote:
> >> > Hi,
> >> >
> >> > Might this be what you want:
> >> >
> >> > The first eleven probabilists' Hermite polynomials are:
> >> >
> >> > ...
> >> >
> >> > My chromium browser does not seem to paste pngs. Anyway, check
> >> >
> >> >
> >> > http://en.wikipedia.org/wiki/Hermite_polynomials
> >> >
> >> > and you'll see that the first polynomial is 1, the second x, and so
> >> > forth. From my courses on quantum mechanics I recall that these
> >> > polynomials are, with respect to some weight function, orthogonal.
> >>
> >> Thanks, I haven't looked at that yet, we should add wikipedia to the
> >> scipy.special docs.
> >>
> >> However, I would like to change the last part "with respect to some
> >> weight function"
> >> http://en.wikipedia.org/wiki/Hermite_polynomials#Orthogonality
> >>
> >> Instead of Gaussian weights I would like uniform weights on bounded
> >> support. And I have never seen anything about changing the weight
> >> function for the orthogonal basis of these kind of polynomials.
> >>
> >
> > In numpy 1.6, you can use the Legendre polynomials. They are orthogonal
> on
> > [-1,1] as has been mentioned, but can be mapped to other domains. For
> > example
> >
> > In [1]: from numpy.polynomial import Legendre as L
> >
> > In [2]: for i in range(5): plot(*L([0]*i + [1], domain=[0,1]).linspace())
> >    ...:
> >
> > produces the attached plots.
>
> I'm still on numpy 1.5 so this will have to wait a bit.
>
> > <snip>
> >
> > Chuck
> >
>
>
> as a first application for orthogonal polynomials I was trying to get
> an estimate for a density, but I haven't figured out the weighting
> yet.
>
> Fourier polynomials work better for this.
>
>
You might want to try Chebyshev then, the Cheybyshev polynomialas are
essentially cosines and will handle the ends better. Weighting might also
help, as I expect the distribution of the errors are somewhat Poisson.

Chuck
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