Weighted $H^2$ rational approximation and consistency

Abstract

We investigate consistency properties of rational approximation of prescribed type in the weighted Hardy space \(H^2_-(\mu)\) for the exterior of the unit disk, where \(\mu\) is a positive symmetric measure on the unit circle \({\mathbb T}\). The question of consistency, which is especially significant for gradient algorithms that compute local minima, concerns the uniqueness of critical points in the approximation criterion for the case when the approximated function is itself rational. In addition to describing some basic properties of the approximation problem, we prove for measures \(\mu\) having a rational function distribution (weight) with respect to arclength on \({\mathbb T}\), that consistency holds only under rather restricted conditions.