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,…pn−1. 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 n−1-degree polynomial pn−xpn−1. What are the components of this guy in the basis of p1,…pn−1? Well, writing
pn−xpn−1=∑m<nampm
Then for all m<n,
am=⟨pn−xpn−1,pm⟩⟨pm,pm⟩=−⟨xpn−1,pm⟩⟨pm,pm⟩=−⟨pn−1,xpm⟩⟨pm,pm⟩
Now, xpm has degree m+1. So if m+1<n−1, ⟨pn−1,xpm⟩=0. So the only ms that we need to bother about are n−2 and n−1. Therefore:
pn−xpn−1=an−2pn−2+an−1pn−1
So:
pn=−⟨pn−1,xpn−2⟩⟨pn−2,pn−2⟩pn−2+[x−⟨pn−1,xpn−1⟩⟨pn−1,pn−1⟩]pn−1=−⟨pn−1,xpn−2−pn−1⟩+⟨pn−1,pn−1⟩⟨pn−2,pn−2⟩pn−2+[x−⟨pn−1,xpn−1⟩⟨pn−1,pn−1⟩]pn−1=−⟨pn−1,pn−1⟩⟨pn−2,pn−2⟩pn−2+[x−⟨pn−1,xpn−1⟩⟨pn−1,pn−1⟩]pn−1
Examples:
- Legendre polynomials: w is the indicator for [−1,1]. Sequence: 1,x,x2−13,x3−35x,…
- Chebyshev polynomials: w is (1−x2)−1/2 on [−1,1]. Sequence: cos(narccosx)
- Laguere polynomials: w is e−x on R≥0. Sequence: 1,x−1,x2−4x+2,x3−9x2+18x−6
- Hermite polynomials: w is e−x2 everywhere. Sequence: 1,x,x2−12,x3−32x
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