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, …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

$${p_n} - x{p_{n - 1}} = \sum\limits_{m < n} {{a_m}{p_m}} $$

Then for all m < n,

$$\begin{align}   {a_m} &= \frac{{\left\langle {{p_n} - x{p_{n - 1}},{p_m}} \right\rangle }}{{\left\langle {{p_m},{p_m}} \right\rangle }} \\    &=  - \frac{{\left\langle {x{p_{n - 1}},{p_m}} \right\rangle }}{{\left\langle {{p_m},{p_m}} \right\rangle }} \\    &=  - \frac{{\left\langle {{p_{n - 1}},x{p_m}} \right\rangle }}{{\left\langle {{p_m},{p_m}} \right\rangle }} \\ \end{align} $$

Now, xpm has degree m + 1. So if m + 1 < n − 1, ⟨pn − 1,xpm⟩ = 0. So the only $m$s that we need to bother about are n − 2 and n − 1. Therefore:

pn − xpn − 1 = an − 2pn − 2 + an − 1pn − 1

So:

$$\begin{align}   {p_n} &=  - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 2}}} \right\rangle }}{{\left\langle {{p_{n - 2}},{p_{n - 2}}} \right\rangle }}{p_{n - 2}} + \left[ {x - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}} \right]{p_{n - 1}} \\    &=  - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 2}} - {p_{n - 1}}} \right\rangle  + \left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 2}},{p_{n - 2}}} \right\rangle }}{p_{n - 2}} + \left[ {x - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}} \right]{p_{n - 1}} \\    &= - \frac{{\left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 2}},{p_{n - 2}}} \right\rangle }}{p_{n - 2}} + \left[ {x - \frac{{\left\langle {{p_{n - 1}},x{p_{n - 1}}} \right\rangle }}{{\left\langle {{p_{n - 1}},{p_{n - 1}}} \right\rangle }}} \right]{p_{n - 1}} \end{align} $$

Examples: