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Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras | Alexei Yu. Karlovich
; | Date: |
26 Mar 2008 | Abstract: | Let $alpha,etain(0,1)$ and [ K^{alpha,eta}:=left{ain L^infty(T):
sum_{k=1}^infty |hat{a}(-k)|^2 k^{2alpha}<infty, sum_{k=1}^infty
|hat{a}(k)|^2 k^{2eta}<infty
ight}. ] Mark Krein proved in 1966 that
$K^{1/2,1/2}$ forms a Banach algebra. He also observed that this algebra is
important in the asymptotic theory of finite Toeplitz matrices. Ten years
later, Harold Widom extended earlier results of Gabor SzegH{o} for scalar
symbols and established the asymptotic trace formula [
operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1) quad ext{as}
n oinfty ] for finite Toeplitz matrices $T_n(a)$ with matrix symbols $ain
K^{1/2,1/2}_{N imes N}$. We show that if $alpha+etage 1$ and $ain
K^{alpha,eta}_{N imes N}$, then the SzegH{o}-Widom asymptotic trace
formula holds with $o(1)$ replaced by $o(n^{1-alpha-eta})$. | Source: | arXiv, 0803.3767 | Services: | Forum | Review | PDF | Favorites |
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