Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras

TL;DR

This paper investigates the asymptotic behavior of finite Toeplitz matrices with symbols in generalized Krein algebras, extending classical results and providing precise error estimates based on the parameters of the algebra.

Contribution

It extends the Szegő-Widom asymptotic trace formula to symbols in generalized Krein algebras with explicit error bounds depending on the algebra parameters.

Findings

Asymptotic trace formula holds for symbols in $K^{ ext{alpha, beta}}$ with $ ext{alpha}+ ext{beta} extgreater=1$

Error term in the asymptotics is $o(n^{1- ext{alpha}- ext{beta}})$

Generalizes classical results for Toeplitz matrices with broader symbol classes.

Abstract

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 $o(n^{1-\alpha-\beta})$.