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Transactions of the American Mathematical Society

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Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions


Author: K. Michael Day
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
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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

DOI: https://doi.org/10.1090/S0002-9947-1975-0383018-7
Keywords: Toeplitz matrices, Laurent series, rational functions, measures, atoms
Article copyright: © Copyright 1975 American Mathematical Society

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