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., , or equivalently . Then both and the eigenvalues of 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
i.e. the arithmetic mean of converges to the integral of , i.e. .
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 (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
References