Section 16.1 Brief recap of orthogonal polynomials
Last time we studied Legendre polynomials \(P_n\) and Laguerre polynomials \(L_n\text{.}\) They have degree \(n\text{,}\) \(n=0,1,2\dots\) and their main property is orthogonality, which means different things for different families of orthogonal polynomials.
- \(\int_{-1}^1 Q(x) P_n(x)\, dx = 0\) where \(P_n\) is the Legendre polynomial of degree \(n\) and \(Q\) is any polynomial of degree \(<n\text{.}\)
- \(\int_{0}^\infty Q(x) L_n(x) e^{-x}\, dx = 0\) where \(L_n\) is the Laguerre polynomial of degree \(n\) and \(Q\) is any polynomial of degree \(<n\text{.}\)
A general fact about orthogonal polynomials is that all their roots are real and are contained in the interval over which they are orthogonal: \([-1, 1]\) for Legendre polynomials and \([0, \infty)\) for Laguerre polynomials.
As an aside, there are many other important families of orthogonal polynomials which we do not study, for example Hermite polynomials \(H_n\) which are orthogonal on the entire line \((-\infty, \infty)\) with the weight \(\exp(-x^2)\text{.}\) The theory of orthogonal polynomials is a vast subject.