Teoplitz

c 2002 Society for Industrial and Applied Mathematics

SIAM J. MATRIX ANAL. APPL.
Vol. 24, No. 1, pp. 59–67

A FINER ASPECT OF EIGENVALUE DISTRIBUTION OF
SELFADJOINT BAND TOEPLITZ MATRICES∗
PETER ZIZLER† , ROB A. ZUIDWIJK‡ , KEITH F. TAYLOR§ , AND SHIGERU ARIMOTO¶
Abstract. The asymptotics of eigenvalues of Toeplitz operators has received a lot of attention
in the mathematical literatureand has been applied in several disciplines. This paper describes
two such application disciplines and provides refinements of existing asymptotic results using new
methods of proof. The following result is typical: Let T (ϕ) be a selfadjoint band limited Toeplitz
operator with a (real valued) symbol ϕ, which is a nonconstant trigonometric polynomial. Consider
finite truncations Tn (ϕ) of T(ϕ), and a finite union of finite intervals of real numbers E . We prove
a refinement of the Szeg¨ asymptotic formula
o
lim

n→∞

N n (E )
1
=
m(F ).
n
2π

Indeed, we show that
N n (E ) −

1
m(F )n = O (1).
2π

Here m(F ) denotes the measure of F = ϕ−1 (E ) on the unit circle, and Nn (E ) denotes the number
of eigenvalues of Tn (ϕ) inside E . We prove similar results for singularvalues of general Toeplitz
operators involving a refinement of the Avram–Parter theorem.
Key words. Toeplitz matrix, eigenvalue distribution, Szeg¨ formula, Avram–Parter theorem
o
AMS subject classifications. 15A18, 47A10, 47A58, 47B35
PII. S089547989834915X

1. Introduction. The eigenvalue distribution of Toeplitz matrices and operators
has been a fascinating and abundant source of topics ofmathematical inquiries. The
prominent monographs [9] and [10] respectively provide extensive analysis of Toeplitz
matrices and operators. Among key historical papers are [11] (on operators), [19]
(on matrices), and [20] (on block matrices). A comprehensive account on the theory
involved is provided in [12].
From the interdisciplinary point of view, the above field also possesses a considerablepotential, especially in terms of a wide range of applications and connections to
disciplines outside mathematics. In the first part of this section, two application areas
(see (I) and (II) below) which have motivated the authors to study the asymptotics
of Toeplitz eigenvalues are addressed.
In the second part of the introduction, the mathematical contribution of this
paper to theasymptotics of eigenvalues and singular values shall be outlined. We
conclude the introduction with some clarification on notation used in the paper.
∗ Received by the editors December 10, 1998; accepted for publication (in revised form) by
A. Bunse-Gerstner November 9, 2001; published electronically June 12, 2002.
http://www.siam.org/journals/simax/24-1/34915.html
† Department of Mathematics,Engineering and Physics, Mount Royal College, 4825 Richard Road
S.W., Calgary, AB, T3E 6K6, Canada (zizler@mtroyal.ab.ca).
‡ Rotterdam School of Management, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands
(rzuidwijk@fbk.fac.eur.nl).
§ Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road,
Saskatoon, SK, S7N 5E6, Canada (taylor@snoopy.usask.ca).
¶ Department ofChemistry, University of Saskatchewan, 110 Science Place, Saskatoon, SK, S7N
5C9, Canada, and Institute for Fundamental Chemistry, 34-4 Takano-Nishihiraki-cho, Sakyo-ku,
Kyoto 606-8103, Japan (arimoto@duke.usask.ca).

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ZIZLER, ZUIDWIJK, TAYLOR, AND ARIMOTO

(I) Vast uncharted regions lie between mathematics and chemistry on the map
of science. Communication across the border of thesedisciplines is still generally
sporadic and uncoordinated, despite modern trends of cross-disciplinary investigations
in each of these fields. In the present work, we have for the first time formed a linkage
between
(i) the mathematical branch of Toeplitz matrices, and
(ii) the “repeat space theory” (RST) in theoretical chemistry, which originates
in the study of the zero-point vibrational...