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 (1) holds and , then

()

holds for and . The RHS of (2) is the geometric mean of (well-defined by the arithmetic-geometric mean inequality).

Second Szegő theorem

Let be the Fourier coefficient of , written as

The second (or strong) Szegő theorem states that, if , then

See also

• Trigonometric moment problem
• Verblunsky's theorem