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Fancy polynomials

Orthogonal polynomials

Given a dot product on functions of the form f,g=Rw(x)f(x)g(x)dx, we can consider orthogonal polynomials, i.e. of the form f,g=0. A question is if we can generate sequences pn of these polynomials of consecutive degree -- such a set would then be a basis for polynomials up to that degree (do you see why?).

What we want is to pin down what pn must be given p1,pn1. What this means is trying to express the "non-leading part" of pn in terms of these former terms. We can access a "non-leading part" of the polynomial by normalizing the polynomials to monic and considering the n1-degree polynomial pnxpn1. What are the components of this guy in the basis of p1,pn1? Well, writing

pnxpn1=m<nampm
Then for all m<n,

am=pnxpn1,pmpm,pm=xpn1,pmpm,pm=pn1,xpmpm,pm
Now, xpm has degree m+1. So if m+1<n1pn1,xpm=0. So the only ms that we need to bother about are n2 and n1. Therefore:

pnxpn1=an2pn2+an1pn1
So:

pn=pn1,xpn2pn2,pn2pn2+[xpn1,xpn1pn1,pn1]pn1=pn1,xpn2pn1+pn1,pn1pn2,pn2pn2+[xpn1,xpn1pn1,pn1]pn1=pn1,pn1pn2,pn2pn2+[xpn1,xpn1pn1,pn1]pn1

Examples:
  • Legendre polynomials: w is the indicator for [1,1]. Sequence: 1,x,x213,x335x, 
  • Chebyshev polynomials: w is (1x2)1/2 on [1,1]. Sequence: cos(narccosx)
  • Laguere polynomials: w is ex on R0. Sequence: 1,x1,x24x+2,x39x2+18x6
  • Hermite polynomials: w is ex2 everywhere. Sequence: 1,x,x212,x332x

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