Herglotz's theorem on positive definite sequences

Remarks

Positive definite means that $n \mapsto a_n$ is a positive-definite function on the group $\Z$, or that every Toeplitz matrix $(a_{j - k})_{j,k = 0}^{N}$ is positive semidefinite. The theorem is the case $G = \Z$ of Bochner's theorem, with $\widehat{\Z} = \mathbb{T}$.

Proof idea: if $a_n = \int z^n\, d\mu$, then the sum in the definition is $\int \abs{\sum_j c_j z^{n_j}}^2 \, d\mu(z) \ge 0$. Conversely, choosing $c_j = e^{-ij\theta}$ for $j = 0, \dots, N-1$ shows that $p_N(\theta) = \frac{1}{N}\sum_{j,k=0}^{N-1} a_{j-k} e^{-i(j-k)\theta}$ is nonnegative. The measures $p_N(\theta)\, \frac{d\theta}{2\pi}$ have total mass $a_0$, and their $n$-th moments are $(1 - \abs{n}/N)\, a_n$ for $\abs{n} < N$. A weak-* limit point of these measures is the required $\mu$, and uniqueness holds because trigonometric polynomials are dense in the continuous functions on $\mathbb{T}$.

Example: for $0 \le r < 1$, the sequence $a_n = r^{\abs{n}}$ is positive definite. Its measure is $P_r(\theta)\, \frac{d\theta}{2\pi}$, where $P_r(\theta) = \sum_{n} r^{\abs{n}} e^{in\theta} = \frac{1 - r^2}{1 - 2r\cos\theta + r^2} > 0$ is the Poisson kernel.

Non-example: the sequence with $a_0 = a_1 = a_{-1} = 1$ and $a_n = 0$ otherwise is not positive definite. The Toeplitz matrix $(a_{j-k})_{j,k=0}^{2}$ has determinant $-1$.

Going deeper

For a student

Herglotz's theorem is the discrete forerunner of Bochner's theorem, and its proof by Fejér means and weak-* compactness is the model for the general case. In the theory of stationary sequences, it says that the covariance sequence of a weakly stationary process has a spectral measure on the circle. It is also equivalent to the representation of holomorphic functions on the unit disc with positive real part as Poisson–Stieltjes integrals.