[SciPy-User] orthogonal polynomials ?

[Charles R Harris]

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.

<snip>

Chuck
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