Szegő's theorem

Remarks

The integral of $\log w$ is well defined in $[-\infty, 0]$, because Jensen's inequality gives $\int \log w \, \frac{d\theta}{2\pi} \le \log \int w \, \frac{d\theta}{2\pi} \le 0$. By the norm formula for the monic orthogonal polynomials, the left side equals $\lim_{n \to \infty} \norm{\Phi_n}^2$. And $\norm{\Phi_n}^2 = \min \set{ \norm{p}^2 : p \text{ monic of degree } n }$, so the theorem computes the best mean-square prediction error of $z^n$ from the past. A standard proof treats this minimum as an extremal problem in the Hardy space $H^2$ of the disk. Along the way, the singular part $\mu_s$ turns out not to affect the limit.

Equivalently, in terms of the Toeplitz matrices $T_n = (c_{j-k})_{j,k=0}^{n}$ of the moments $c_k = \int e^{-ik\theta} \, d\mu$, one has $\det T_n / \det T_{n-1} = \norm{\Phi_n}^2$. So the theorem also gives $\lim_{n \to \infty} (\det T_n)^{1/(n+1)} = \exp \int \log w \, \frac{d\theta}{2\pi}$, Szegő's first limit theorem.

Example: for $d\mu = d\theta/2\pi$ one has $w = 1$ and all $\alpha_n = 0$, and both sides equal $1$. If $\mu$ is singular, $w = 0$ almost everywhere, so $\prod (1 - \abs{\alpha_n}^2) = 0$ and $\sum \abs{\alpha_n}^2 = \infty$.

Going deeper

For a student

The condition $\int \log w > -\infty$ is the Szegő condition. The same condition governs the factorization $w = \abs{D}^2$ with an outer function $D$ in the Hardy space. It also governs prediction for stationary sequences: the future is not determined by the past exactly when it holds. The theorem is a sum rule, relating the coefficients to the measure, and it is a starting point for finer results such as the Szegő limit theorems on Toeplitz determinants.