Orthogonal rational functions and interpolatory product rules on the unit circle. III. Convergence of general sequences

Let be the space of rational functions with poles among with. We consider a sequence nested subspaces with. We continue our investigation of the convergence as of quadrature rules which are exact in. In Part II we have discussed the convergence for a particular nesting of the subspaces. In this part we prove similar convergence results for a more general sequence of subspaces. By similar arguments as in part II, we can also here derive related results about the convergence of multipoint rational approximants to the Riesz-Herglotz transform associated with a complex measure. © 2000, Oldenbourg Wissenschaftsverlag GmbH, Rosenheimer Str. 145, 81671 München. All rights reserved.status: publishe