Asymptotic eigenvalue distribution of large Toeplitz matrices

نویسندگان

  • Seung-Yeop Lee
  • Hui Dai
  • Eldad Bettelheim
چکیده

We study the asymptotic eigenvalue distribution of Toeplitz matrices generated by a singular symbol. It has been conjectured by Widom that, for a generic symbol, the eigenvalues converge to the image of the symbol. In this paper we ask how the eigenvalues converge to the image. For a given Toeplitz matrix Tn(a) of size n, we take the standard approach of looking at det(ζ − Tn(a)), of which the asymptotic information is given by the Fisher-Hartwig theorem. For a symbol with single jump, we obtain the distribution of eigenvalues as an expansion involving 1/n and logn/n. To demonstrate the validity of our result we compare our result against the numerics using a pure Fisher-Hartwig symbol. PACS numbers: 02.10.Yn AMS classification scheme numbers: 15A15, 15A18, 15A60, 47B35

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تاریخ انتشار 2007