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 | References | Similar Articles | Additional Information

Abstract: Let be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function, and let be the corresponding sets of eigenvalues of . Define a sequence of measures , and a set in the -plane. It is shown that the weak limit of the measures is unique and possesses at most two atoms, and the function which give rise to atoms are identified.

**[1]**L. Carleson,*Mergelyan's theorem on uniform polynomial approximation*, Math. Scand.**15**(1964), 167-175. MR**33**#6368. MR**0198209 (33:6368)****[2]**K. M. Day,*Toeplitz matrices generated by the Laurent series expansion of an arbitrary rational function*, Trans. Amer. Math. Soc.**206**(1975), 224-245. MR**0379803 (52:708)****[3]**E. Hille,*Analytic function theory*. Vol. II, Introduction to Higher Math., Ginn, Boston, Mass., 1962. MR**34**#1490. MR**0201608 (34:1490)****[4]**I. I. Hirschman, Jr.,*The spectra of certain Toeplitz matrices*, Illinois J. Math.**11**(1967), 145-159. MR**34**#4905. MR**0205070 (34:4905)****[5]**P. Schmidt and F. Spitzer,*The Toeplitz matrices of an arbitrary Laurent polynomial*, Math. Scand.**8**(1960), 15-38. MR**23**#A1977. MR**0124665 (23:A1977)****[6]**J. L. Ullman,*A problem of Schmidt and Spitzer*, Bull. Amer. Math. Soc.**73**(1967), 883-885. MR**36**#3056. MR**0219986 (36:3056)**

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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