Constrained Nevanlinna-Pick interpolation

We are interested in complex interpolation problems that have their origins in the work of Nevanlinna and Pick. In the 90 years since their results first appeared, Nevanlinna-Pick problems have been valuable in the development of areas of pure mathematics such as operator theory, operator algebras, harmonic analysis and complex function theory. The study of interpolation problems has also been closely tied to the development of systems theory and H-infinity control theory.;We describe how these interpolation results extend to a class of subalgebras of the algebra of bounded analytic functions on the open unit disk. The problems we will look at include as special cases interpolation theory on multiply connected domains and interpolation on embedded disks.;Our methods use duality techniques, factorization results, averaging techniques and matrix theory.