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 φ : T→C be a complex function on the unit circle. Consider the n×n Toeplitz matrices Tn, defined by
where
are the Fourier coefficients of φ.

First Szegő theorem

The first Szegő theorem states that, if φ > 0 and φ ∈ L1, then
The right-hand side of is the geometric mean of φ.

Second Szegő theorem

Denote the right-hand side of by G. The second Szegő theorem asserts that if, in addition, the derivative of φ is Hölder continuous of order α > 0, then