DOI: 10.5802/ambp.444 ISSN: 1259-1734
Asymptotic behavior of the eigenvalues of Toeplitz matrices with even symbol
Philippe Rambour
In this paper we consider an interval
[
θ
1
,
θ
2
]
⊂
]
0
,
π
[
and
f
a periodic and even function in
C
4
[
0
,
2
π
]
such that
f
(
θ
)
∈
[
f
(
θ
1
)
,
f
(
θ
2
)
]
⇔
θ
∈
[
θ
1
,
θ
2
]
and
f
∈
𝒜
+
[
θ
1
,
θ
2
]
(resp.
f
∈
𝒜
-
[
θ
1
,
θ
2
]
, that means
f
′
(
θ
)
>
0
(resp.
f
′
(
θ
)
<
0
) for all
θ
∈
[
θ
1
,
θ
2
]
. Then we obtain a higher order asymptotic formula for all the eigenvalues of the Toeplitz matrix
T
N
(
f
)
as
N
→
+
∞
which belong to
[
f
(
θ
1
)
,
f
(
θ
2
)
]
(resp.
[
f
(
θ
2
)
,
f
(
θ
1
)
]
).