Szegő limit theorems

Remarks

Proof idea for the first theorem: the ratio $D_{n+1}(f) / D_n(f)$ is the minimum of $\frac{1}{2\pi} \int_{-\pi}^{\pi} \abs{p(e^{i\theta})}^2 f(\theta) \, d\theta$ over polynomials $p(z) = 1 + a_1 z + \cdots + a_n z^n$. These minima decrease to $G(f)$ by Szegő's theorem on prediction, and then the geometric means $D_n^{1/n}$ have the same limit. The strong theorem gives the next term of the asymptotics, $D_n(f) \sim E(f) \, G(f)^n$ with the constant $E(f)$ above.

Example: let $f(\theta) = \abs{1 - a e^{i\theta}}^2$ with $\abs{a} < 1$. Then $T_n(f)$ is tridiagonal with diagonal $1 + \abs{a}^2$, and $D_n(f) = \frac{1 - \abs{a}^{2(n+1)}}{1 - \abs{a}^2}$. Here $g = \log f$ has $\hat g(0) = 0$ and $\hat g(k) = -a^k / k$ for $k \ge 1$, so $G(f) = 1$ and the strong theorem predicts the limit $\exp\left(\sum_{k \ge 1} \abs{a}^{2k}/k\right) = \frac{1}{1 - \abs{a}^2}$, as the formula confirms.

Szegő proved the first theorem in 1915 and the strong theorem in 1952, motivated by the computation of the spontaneous magnetization of the two-dimensional Ising model. The strong theorem was later proved under weaker hypotheses, essentially only the condition $\sum \abs{k} \abs{\hat g(k)}^2 < \infty$.

Going deeper

For a student

The theorems are the analogue for Toeplitz matrices of the fact that the determinant of a diagonal matrix is the product of its entries. The first theorem is closely tied to orthogonal polynomials on the unit circle and to prediction theory of stationary sequences. The strong theorem has applications in statistical mechanics and in random matrix theory, where such determinants compute averages over the unitary group.