Numerische Mathematik

, Volume 37, Issue 2, pp 297–320

Rational Chebyshev approximation on the unit disk

  • Lloyd N. Trefethen
A numerical Method for Computing Turning Points. II. Numerical Analysis of a Nonlinear Boundary Value Problem

DOI: 10.1007/BF01398258

Cite this article as:
Trefethen, L.N. Numer. Math. (1981) 37: 297. doi:10.1007/BF01398258


In a recent paper we showed that error curves in polynomial Chebyshev approximation of analytic functions on the unit disk tend to approximate perfect circles about the origin [23]. Making use of a theorem of Carathéodory and Fejér, we derived in the process a method for calculating near-best approximations rapidly by finding the principal singular value and corresponding singular vector of a complex Hankel matrix. This paper extends these developments to the problem of Chebyshev approximation by rational functions, where non-principal singular values and vectors of the same matrix turn out to be required. The theory is based on certain extensions of the Carathéodory-Fejér result which are also currently finding application in the fields of digital signal processing and linear systems theory.

It is shown among other things that iffz) is approximated by a rational function of type (m, n) for ɛ>0, then under weak assumptions the corresponding error curves deviate from perfect circles of winding numberm+n+1 by a relative magnitudeOm + n + 2 as ɛ→0. The “CF approximation” that our method computes approximates the true best approximation to the same high relative order. A numerical procedure for computing such approximations is described and shown to give results that confirm the asymptotic theory. Approximation ofez on the unit disk is taken as a central computational example.

Subject Classifications

AMS(MOS) 30D5030E1041A50

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Lloyd N. Trefethen
    • 1
  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA