Appl Math Optim 44:67–85 (2001)
2001 Springer-Verlag New York Inc.
The Potential Theory of Several Intervals and Its Applications
and A. J. Wathen
School of Mathematics, University of Minnesota,
Minneapolis, MN 55455, USA
Department of Mathematics, MIT,
Cambridge, MA 02139, USA
Oxford University Computing Laboratory,
Parks Road, Oxford OX1 3QD, England
Abstract. Motivated both by digital ﬁlter design and polynomial-based matrix
iteration methods, we study Green’s function for the complement of a union of
disjoint closed intervals. The key tool is the Schwarz–Christoffel map. Asymptotic
analysis produces simple and useful leading terms for Green’s function and the
associated equilibrium distribution. Our results are applied to optimal lowpass ﬁlters
and matrix iterations.
Key Words. Optimal polynomials, Digital ﬁlters, Matrix iterations, Green’s func-
tion, Equilibrium distribution, Schwarz–Christoffel mapping.
AMS Classiﬁcation. Primary 41A10, 65E05, Secondary 65D99, 65F10.
Approximation of f (x) by polynomials on a single interval [a, b] is a fundamental and
much studied problem. The key to its analysis is a family of ellipses in the complex
plane, with the points a and b on the real line as foci. These ellipses are the level
curves of Green’s function for the complement of [a, b]. The accuracy of polynomial
approximation is determined by the ﬁrst ellipse to touch a singularity of f (z). The (real)
approximation problem, when f is the restriction of a holomorphic function, is best
understood in the complex plane.
We study the corresponding problem on a union of disjoint closed intervals. This
problem arises naturally in at least two applications:
1. The optimal design of digital ﬁlters ( and  are closest to the present