Near-Circularity of the Error Curve In Complex Chebyshev Approximation

Let f(z) be analytic on the unit disk, and let p*(z) be the best (Chebyshev) polynomial approximation to f(z) on the disk of degree at most n. It is observed that in typical problems the “error curve,” the image of the unit circle under (f p*)(z), often approximates to a startling degree a perfect circle with winding number n + 1. This phenomenon is approached by consideration of related problems whose error curves are exactly circular, making use of a classical theorem of Caratheodory and Fejer. This leads to a technique for calculating approximations in one step that are roughly as close to best as the best approximation error curve is close to circular, and hence to strong theorems on near-circularity as the radius of the domain shrinks to 0 or as n increases to co. As a computational example, very tight bounds are given for approximation of ez on the unit disk. The generality of the near-circularity phenomenon (more general domains, rational approximation) is discussed.