Favard's theorem

Remarks

The proof defines a linear functional $L$ on polynomials by $L(1) = 1$ and $L(p_n) = 0$ for $n \ge 1$, which is possible since $p_n$ is monic of degree $n$. The recurrence gives $L(p_m p_n) = 0$ for $m \ne n$ and $L(p_n^2) = b_1 b_2 \cdots b_n > 0$, so $L$ is positive on nonzero squares of polynomials. Then $L(f) \ge 0$ for every polynomial $f \ge 0$ on $\R$, since such $f$ is a sum of two squares, and the Hamburger moment problem gives a measure $\mu$ with $L(f) = \int f\, d\mu$. The measure need not be unique. It is unique when the moment problem is determinate, for instance when $(a_n)$ and $(b_n)$ are bounded, and then $\mu$ has compact support. Example: $a_n = 0$ and $b_n = \frac{1}{4}$ give the monic Chebyshev polynomials of the second kind, and $\mu$ is the semicircle measure $\frac{2}{\pi} \sqrt{1 - x^2}\, dx$ on $[-1, 1]$. The hypothesis $b_n > 0$ is needed. For orthogonal polynomials of a positive measure with infinite support, $b_n = \frac{\int p_n^2\, d\mu}{\int p_{n-1}^2\, d\mu} > 0$, so a recurrence with some $b_n \le 0$ cannot come from such a measure.

Going deeper

For a student

The theorem identifies orthogonal polynomial sequences on the real line with pairs of sequences $(a_n)$, $(b_n)$ with $b_n > 0$, that is, with Jacobi matrices. In operator language, $\mu$ is a spectral measure of the Jacobi operator with diagonal $a_n$ and off-diagonal $\sqrt{b_n}$. It reverses the three-term recurrence, and its proof reduces to the Hamburger moment problem. It is used to show that families defined by recurrences, such as many in the Askey scheme, are orthogonal before the weight is known.