The CF table

Abstract

Letf be a continuous function on the circle ¦z¦=1. We present a theory of the (untruncated) “Caratheodory-Fejer (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, Pade, and formal Chebyshev-Pade 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.