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Eigenvalues of Hessenberg Toeplitz matrices generated by symbols with several singularities

Commun. Math. Anal. Conf.
Conference 3 (2011), 23 - 41

Eigenvalues of Hessenberg Toeplitz matrices generated by symbols with several singularities

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Abstract

In a recent paper, we established asymptotic formulas for the eigenvalues of the n£n truncations of certain infinite Hessenberg Toeplitz matrices as n goes to infinity. The symbol of the Toeplitz matrices was of the form a(t)=t¡1(1¡t)a f (t) (t 2 T), where a is a positive real number but not an integer and f is a smooth function in H¥. Thus, a has a single power singularity at the point 1. In the present work we extend the results to symbols with a finite number of power singularities. To be more precise, we consider symbols of the form a(t) = t¡1 f (t)ÕKk =1(1¡t=tk)ak (t 2 T), where tk = eiqk , the arguments qk are all different, and the exponents ak are positive real numbers but not integers.