Mathematics – Functional Analysis
Scientific paper
2008-03-26
Mathematics
Functional Analysis
16 pages
Scientific paper
Let $\alpha,\beta\in(0,1)$ and \[ K^{\alpha,\beta}:=\left\{a\in L^\infty(\T): \sum_{k=1}^\infty |\hat{a}(-k)|^2 k^{2\alpha}<\infty, \sum_{k=1}^\infty |\hat{a}(k)|^2 k^{2\beta}<\infty \right\}. \] 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 Szeg\H{o} for scalar symbols and established the asymptotic trace formula \[ \operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1) \quad\text{as}\ n\to\infty \] for finite Toeplitz matrices $T_n(a)$ with matrix symbols $a\in K^{1/2,1/2}_{N\times N}$. We show that if $\alpha+\beta\ge 1$ and $a\in K^{\alpha,\beta}_{N\times N}$, then the Szeg\H{o}-Widom asymptotic trace formula holds with $o(1)$ replaced by $o(n^{1-\alpha-\beta})$.
No associations
LandOfFree
Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-50141