Rational Chebyshev Approximation on the Unit Disk

In a recent paper we showed that er ror curves in po lynomia l Chebyshev a p p r o x i m a t i o n of ana ly t ic functions on the unit disk tend to a p p r o x i m a t e perfect circles abou t the origin [23]. M a k i n g use of a theorem of Ca ra th6odo ry and Fej6r, we der ived in the process a me thod for calculat ing near-bes t a p p r o x i m a t i o n s rapid ly by finding the pr incipal s ingular value and co r respond ing s ingular vector of a complex Hanke l matr ix. This paper extends these deve lopments to the p rob lem of Chebyshev a pp rox ima t ion by ra t iona l functions, where non-pr inc ipa l s ingular values and vectors of the same matr ix turn out to be required. The theory is based on certain extensions of the Ca ra th6odory -Fe j6 r result which are also current ly finding app l i ca t ion in the fields of digi tal signal process ing and l inear systems theory. It is shown a m o n g o ther things that if f ( e z ) is a p p r o x i m a t e d by a ra t iona l function of type (m, n) for ~>0 , then under weak assumpt ions the co r respond ing error curves deviate from perfect circles of winding number m + n + 1 by a relat ive magni tude O(E "+"+2) as ~ 0 . The " C F app rox imat ion" that our me thod computes app rox ima te s the true best a p p r o x i m a t i o n to the same high relat ive order . A numer ica l p rocedure for comput ing such a p p r o x i m a t i o n s is descr ibed and shown to give results that conf i rm the a sympto t i c theory. A p p r o x i m a t i o n of e z on the unit d isk is t aken as a centra l compu ta t i ona l example .