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CAPACITIES AND JACOBI MATRICES

Published online by Cambridge University Press:  10 December 2003

Ahmed Sebbar  and
Thérèse Falliero
Ahmed Sebbar
Affiliation:
UFR Mathématiques et Informatique, Université Bordeaux I, 33405 Talence Cedex, France (sebbar@math.u-bordeaux.fr)
Thérèse Falliero
Affiliation:
Faculté des Sciences, Université d’Avignon, 84000 Avignon, France (Therese.Falliero@univ-avignon.fr)
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Abstract

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In this paper, we use the theorem of Burchnall and Shaundy to give the capacity of the spectrum $\sigma(A)$ of a periodic tridiagonal and symmetric matrix. A special family of Chebyshev polynomials of $\sigma(A)$ is also given. In addition, the inverse problem is considered: given a finite union $E$ of closed intervals, we study the conditions for a Jacobi matrix $A$ to exist satisfying $\sigma(A)=E$. We relate this question to Carathéodory theorems on conformal mappings.

AMS 2000 Mathematics subject classification: Primary 31B15; 30C20; 39A70

Keywords

Jacobi matricescapacitiesBurchnall–Shaundy TheoremCarathéodory Kernel Theorem
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 3 , October 2003 , pp. 719 - 745
Copyright
Copyright © Edinburgh Mathematical Society 2003