In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices.
[1][2][3] They were first proved by Gábor Szegő.
be a Fourier series with Fourier coefficients
, relating to each other as such that the
Toeplitz matrices
{\displaystyle T_{n}(w)=\left(c_{k-l}\right)_{0\leq k,l\leq n-1}}
are Hermitian, i.e., if
and eigenvalues
are real-valued and the determinant of
is given by 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
[4] The first Szegő theorem[1][3][5] states that, if right-hand side of (1) holds and
The RHS of (2) is the geometric mean of
(well-defined by the arithmetic-geometric mean inequality).
be the Fourier coefficient of
, written as The second (or strong) Szegő theorem[1][6] states that, if