Weighted H2 rational approximation and consistency

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 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 T, that consistency holds only under rather restricted conditions.