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.
Contents
Notation
Let
w
{\displaystyle w}
be a Fourier series with Fourier coefficients
c
k
{\displaystyle c_{k}}
, relating to each other as
w
(
θ
)
=
∑
k
=
−
∞
∞
c
k
e
i
k
θ
,
θ
∈
[
0
,
2
π
]
,
{\displaystyle w(\theta )=\sum _{k=-\infty }^{\infty }c_{k}e^{ik\theta },\qquad \theta \in [0,2\pi ],}
c
k
=
1
2
π
∫
0
2
π
w
(
θ
)
e
−
i
k
θ
d
θ
,
{\displaystyle c_{k}={\frac {1}{2\pi }}\int _{0}^{2\pi }w(\theta )e^{-ik\theta }\,d\theta ,}
such that the
n
×
n
{\displaystyle n\times n}
Toeplitz matrices
T
n
(
w
)
=
(
c
k
−
l
)
0
≤
k
,
l
≤
n
−
1
{\displaystyle T_{n}(w)=\left(c_{k-l}\right)_{0\leq k,l\leq n-1}}
are Hermitian, i.e.,
T
n
(
w
)
=
T
n
(
w
)
∗
{\displaystyle T_{n}(w)=T_{n}(w)^{\ast }}
, or equivalently
c
−
k
=
c
k
¯
{\displaystyle c_{-k}={\overline {c_{k}}}}
. Then both
w
{\displaystyle w}
and the eigenvalues
(
λ
m
(
n
)
)
0
≤
m
≤
n
−
1
{\displaystyle (\lambda _{m}^{(n)})_{0\leq m\leq n-1}}
of
T
n
(
w
)
{\displaystyle T_{n}(w)}
are real-valued and the determinant of
T
n
(
w
)
{\displaystyle T_{n}(w)}
is given by
det
T
n
(
w
)
=
∏
m
=
1
n
−
1
λ
m
(
n
)
{\displaystyle \det T_{n}(w)=\prod _{m=1}^{n-1}\lambda _{m}^{(n)}}
.
Szegő theorem
Under suitable assumptions the Szegő theorem states that
lim
n
→
∞
1
n
∑
m
=
0
n
−
1
F
(
λ
m
(
n
)
)
=
1
2
π
∫
0
2
π
F
(
w
(
θ
)
)
First Szegő theorem
The first Szegő theorem states that, if right-hand side of (1) holds and
w
≥
0
{\displaystyle w\geq 0}
, then
holds for
w
>
0
{\displaystyle w>0}
and
w
∈
L
1
{\displaystyle w\in L^{1}}
. The RHS of (2) is the geometric mean of
w
{\displaystyle w}
(well-defined by the arithmetic-geometric mean inequality).
Second Szegő theorem
Let
c
^
k
{\displaystyle {\widehat {c}}_{k}}
be the Fourier coefficient of
log
w
∈
L
1
{\displaystyle \log w\in L^{1}}
, written as
c
^
k
=
1
2
π
∫
0
2
π
log
(
w
(
θ
)
)
e
−
i