Polynomials

Favard’s theorem

An interesting question is whether a given set of monic polynomials \(\{p_n\}\) with \(n \in \mathbb{Z}^+\) and leading term \(x^n\) can be made orthogonal through the construction of a linear functional \(\lambda\). The trivial functional, \(\lambda=0\), accomplishes the orthogonality, but is not useful for projecting an arbitrary polynomial on the given set of polynomials, for which we need \(\langle \lambda | p_n^2\rangle \neq 0\).

If the set is orthogonal for some non-trivial functional, it satisfies a three-term recurrence: \(p_{n+1} = (x+a_n)p_n + b_np_{n-1}\). Assuming none of the \(\langle \lambda | p_n^2\rangle \) are zero, the \(b_n=-\langle \lambda | p_n^2\rangle/\langle \lambda | p_{n-1}^2\rangle\) are not zero either.

We can extend the recurrence to \(p_1 = (x+a_0)p_0 + b_0 p_{-1}\) with \(p_0=1\) and \(p_{-1}=0\). If we set \(\langle \lambda | 1\rangle=1\), \(\langle \lambda | p_n \rangle = \langle \lambda | x^n +q \rangle=0\), with \(q\) a polynomial of degree less than \(n\), allows us to construct \(\langle \lambda | x^n \rangle\) from the form of \(p_n\) and \(\{\langle \lambda | x^k \rangle: k \lt n\}\).