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We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to H\"older-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refine previous asymptotic trace formulas by Szeg\"o and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to B\"ottcher and Silbermann.

Higher Order Asymptotic Formulas for Traces of Toeplitz Matrices with Symbols in Hölder-Zygmund Spaces