THE CF TABLE

Let f be a continuous function on the circle |z|=1. We present a theory of the (untruncated) "Carathéodory-Fejér (CF) table" of best supremumnorm approximants to f in the classes {Mathematical expression} of functions {Mathematical expression}, where the series converges in 1< |z| <∞. (The case m=n is also associated with the names Adamjan, Arov, and Krein.) Our central result is an equioscillation-type characterization: {Mathematical expression} is the unique CF approximant {Mathematical expression} to f if and only if {Mathematical expression} has constant modulus and winding number ω≥ m+ n+1-δ on |z|=1, where δ is the "defect" of {Mathematical expression}. If the Fourier series of f converges absolutely, then {Mathematical expression} is continuous on |z|=1, and ω can be defined in the usual way. For general continuous f, {Mathematical expression} may be discontinuous, and ω 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 {Mathematical expression} is continuous at f if and only if (m, n) lies in the upper-right or lower-left corner of its square block. © 1990 Springer-Verlag New York Inc.