ISSN:
1432-0940
Keywords:
Rational approximation
;
CF approximation
;
H ∞ approximation
;
AAK approximation
;
Hankel matrix
;
30E10
;
41A20
;
30D50
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Letf be a continuous function on the circle ¦z¦=1. We present a theory of the (untruncated) “Carathéodory-Fejér (CF) table” of best supremumnorm approximants tof in the classes $$\tilde R_{mn} $$ of functions $${{\tilde r(z) = \sum\limits_{k = - \infty }^m {a_k z^k } } \mathord{\left/ {\vphantom {{\tilde r(z) = \sum\limits_{k = - \infty }^m {a_k z^k } } {\sum\limits_{k = 0}^n {b_k } z^k ,}}} \right. \kern-\nulldelimiterspace} {\sum\limits_{k = 0}^n {b_k } z^k ,}}$$ , where the series converges in 1〈 ¦z¦ 〈∞. (The casem=n is also associated with the names Adamjan, Arov, and Krein.) Our central result is an equioscillation-type characterization: $$\tilde r \in \tilde R_{mn} $$ is the unique CF approximant $$\tilde r^* $$ tof if and only if $$f - \tilde r$$ has constant modulus and winding numberω≥ m+ n+1−δ on ¦z¦=1, whereδ is the “defect” of $$\tilde r$$ . If the Fourier series off converges absolutely, then $$\tilde r^* $$ is continuous on ¦z¦=1, andω can be defined in the usual way. For general continuousf, $$\tilde r^* $$ may be discontinuous, andω is defined by a radial limit. The characterization theorem implies that the CF table breaks into square blocks of repeated entries, just as in Chebyshev, Padé, and formal Chebyshev-Padé approximation. We state a generalization of these results for weighted CF approximation on a Jordan region, and also show that the CF operator $$K:f \mapsto \tilde r^* $$ is continuous atf if and only if (m, n) lies in the upper-right or lower-left corner of its square block.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01889358
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