Fancy polynomials

Orthogonal polynomials

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* Given a dot product on functions of the form \(\langle f, g \rangle = \iint_{\mathbb{R}} w(x)f(x)g(x)\, dx\), we can consider orthogonal polynomials, i.e. of the form \(\langle f, g\rangle = 0\). A question is if we can generate sequences \(p_n\) 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 \(p_n\) must be given \(p_1,\dots p_{n-1}\). What this means is trying to express the “non-leading part” of \(p_n\) 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 \(p_n-xp_{n-1}\). What are the components of this guy in the basis of \(p_1,\dots p_{n-1}\)? Well, writing

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

Then for all \(m

$$\begin{align}

  {a_m} &= \frac{{\left⟨ {{p_n} - x{p_{n - 1}},{p_m}} \right⟩ }}{{\left⟨ {{p_m},{p_m}} \right⟩ }}

   &=  - \frac{{\left⟨ {x{p_{n - 1}},{p_m}} \right⟩ }}{{\left⟨ {{p_m},{p_m}} \right⟩ }}

   &=  - \frac{{\left⟨ {{p_{n - 1}},x{p_m}} \right⟩ }}{{\left⟨ {{p_m},{p_m}} \right⟩ }}

\end{align} $$

Now, \(xp_m\) has degree \(m+1\). So if \(m+1

\[{p_n} - x{p_{n - 1}} = {a_{n - 2}}{p_{n - 2}} + {a_{n - 1}}{p_{n - 1}}\]

So:

$$\begin{align}

  {p_n} &=  - \frac{{\left⟨ {{p_{n - 1}},x{p_{n - 2}}} \right⟩ }}{{\left⟨ {{p_{n - 2}},{p_{n - 2}}} \right⟩ }}{p_{n - 2}} + \left[ {x - \frac{{\left⟨ {{p_{n - 1}},x{p_{n - 1}}} \right⟩ }}{{\left⟨ {{p_{n - 1}},{p_{n - 1}}} \right⟩ }}} \right]{p_{n - 1}}

   &=  - \frac{{\left⟨ {{p_{n - 1}},x{p_{n - 2}} - {p_{n - 1}}} \right⟩  + \left⟨ {{p_{n - 1}},{p_{n - 1}}} \right⟩ }}{{\left⟨ {{p_{n - 2}},{p_{n - 2}}} \right⟩ }}{p_{n - 2}} + \left[ {x - \frac{{\left⟨ {{p_{n - 1}},x{p_{n - 1}}} \right⟩ }}{{\left⟨ {{p_{n - 1}},{p_{n - 1}}} \right⟩ }}} \right]{p_{n - 1}}

   &= - \frac{{\left⟨ {{p_{n - 1}},{p_{n - 1}}} \right⟩ }}{{\left⟨ {{p_{n - 2}},{p_{n - 2}}} \right⟩ }}{p_{n - 2}} + \left[ {x - \frac{{\left⟨ {{p_{n - 1}},x{p_{n - 1}}} \right⟩ }}{{\left⟨ {{p_{n - 1}},{p_{n - 1}}} \right⟩ }}} \right]{p_{n - 1}}

\end{align} $$

Examples:

• Legendre polynomials: \(w\) is the indicator for \([-1,1]\). Sequence: \(1, x, x^2-\frac13, x^3-\frac35x,\dots\)
• Chebyshev polynomials: \(w\) is \((1-x^2)^{-1/2}\) on \([-1,1]\). Sequence: \(\cos(n\arccos x)\)
• Laguere polynomials: \(w\) is \(e^{-x}\) on \(\mathbb{R}^{\ge 0}\). Sequence: \(1, x-1, x^2-4x+2, x^3-9x^2+18x-6\)
• Hermite polynomials: \(w\) is \(e^{-x^2}\) everywhere. Sequence: \(1, x, x^2-\frac12, x^3-\frac32 x\)