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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions
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by K. Michael Day PDF
Trans. Amer. Math. Soc. 209 (1975), 175-183 Request permission

Abstract:

Let ${T_n}(a) = ({a_{i - j}})_{i,j = 0}^n$ be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function, and let ${\sigma _n} = \{ {\lambda _{n0}}, \ldots ,{\lambda _{nn}}\}$ be the corresponding sets of eigenvalues of ${T_n}(f)$. Define a sequence of measures ${\alpha _n},{\alpha _n}(E) = {(n + 1)^{ - 1}}{\Sigma _{{\lambda _{ni}} \in E}}1,{\lambda _{ni}} \in {\sigma _n}$, and $E$ a set in the $\lambda$-plane. It is shown that the weak limit $\alpha$ of the measures ${\alpha _n}$ is unique and possesses at most two atoms, and the function $f$ which give rise to atoms are identified.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 209 (1975), 175-183
  • MSC: Primary 45E10; Secondary 30A06
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0383018-7
  • MathSciNet review: 0383018