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 complex function ("symbol") on the unit circle. Consider the  Toeplitz matrices , defined by

where

are the Fourier coefficients of . First Szegő theorem
The first Szegő theorem states that, if 
 and , then

The right-hand side of () is the geometric mean of  (well-defined by the arithmetic-geometric mean inequality). Second Szegő theorem

Denote the right-hand side of () by . The second (or strong) Szegő theorem asserts that if, in addition, the derivative of  is Hölder continuous of order , then

References

Theorems in analysis
Matrices
Linear algebra