Acta Applicandae Mathematicae 62: 155–186, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
155
Extension of the Ahiezer–Kac Determinant
Formula to the Case of Real-Valued Symbols
with Two Real Zeros
SERGIO ALBEVERIO
1
and KONSTANTIN A. MAKAROV
2
1
Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany.
e-mail: Albeverio@uni-bonn.de
2
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.
e-mail: makarov@azure.math.missouri.edu
(Received: 15 June 1998; in final form: 24 January 2000)
Abstract. The Fredholm determinant asymptotics for self-adjoint convolution operators on finite
intervals with real symbols vanishing on the real axis is studied. Explicit formulae are obtained in the
case where the symbol satisfies the generalized zero index condition and has only two simple zeros of
analytic type. These formulae are direct extensions of the Ahiezer–Kac–Szegö limit theorem which,
in particular, take into account the oscillating character of the asymptotics.
Mathematics Subject Classifications (2000): Primary: 45P05, 47B35; Secondary: 47A68, 47G10.
Key words: Ahiezer–Kac determinant, asymptotics, convolution operators, Ahiezer–Kac–Szegö limit
theorem, Wiener–Hopf decomposition, generalized zero index condition.
1. Introduction
1.1. HISTORICAL REMARKS ON SZEGÖ’ S LIMIT THEOREM AND ITS
EXTENSIONS
In his fundamental work [26] Szegö proved a basic theorem on the N →∞ limit
of the N × N determinant
D
N
= det(T(N))
of a Toeplitz matrix T(N) of the form
T(N) =
c
0
c
−1
··· c
−N+1
c
1
c
0
··· c
−N+2
. . . .
. . . .
. . . .
c
N−1
c
N−2
··· c
0
,