The theorems of Stieltjes and Favard (Q643206)

A possible way to develop the theory of orthogonal polynomials is to begin with a sequence of polynomials defined by a three-term recurrence relation of the form \(p_0(x)=1\), \(p_1(x)=x-\alpha _0\), \(p_{n+2}(x)=(x-\alpha _{n+1})p_{n+1}(x)-\lambda _np_n(x)\) for \(n\geq 0\), where each \(\alpha _{n+1}\) and \(\lambda _n\) is real, and \(\lambda _n>0\) for \(n=0,1,2,\dots.\) Under these conditions, Favard's theorem asserts that there is a positive measure \(\mu \) such that the obtained polynomials are orthogonal in the sense \(\int _\mathbb{R}p_n(x)\;p_k(x)\;d\mu =0\) for \(n\neq k\). It is well known that the zeros of \(p_{n-1}\) and \(p_n\) are real, simple and interlacing. Beyond this, Stieltjes showed that if \(m<n-1\) then \(p_m\) and \(p_n\) have interlacing zeros. Here the author gives a modern proof of Favard's theorem and another proof of a result of de Boor and Saff which extends Stieltjes' theorem.