, Volume 37, Issue 2, pp 297-320

Rational Chebyshev approximation on the unit disk

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Summary

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 magnitudeO m + 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 ofe z on the unit disk is taken as a central computational example.