Szegő limit theorems


In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő.

Notation

Let be a Fourier series with Fourier coefficients, relating to each other as
such that the Toeplitz matrices are Hermitian, i.e., if then. Then both and eigenvalues are real-valued and the determinant of is given by

Szegő theorem

Under suitable assumptions the Szegő theorem states that
for any function that is continuous on the range of. In particular
such that the arithmetic mean of converges to the integral of.

First Szegő theorem

The first Szegő theorem states that, if right-hand side of holds and, then
holds for and. The RHS of is the geometric mean of .

Second Szegő theorem

Let be the Fourier coefficient of, written as
The second Szegő theorem states that, if, then