Nevanlinna–Pick meromorphic interpolation: The degenerate case and minimal norm solutions

The Nevanlinna–Pick interpolation problem is studied in the class S κ of meromorphic functions f with κ poles inside the unit disk D and with ‖ f ‖ L ∞ ( T ) ⩽ 1 . In the indeterminate case, the parametrization of all solutions is given in terms of a family of linear fractional transformations with disjoint ranges. A necessary and sufficient condition for the problem being determinate is given in terms of the Pick matrix of the problem. The result is then applied to obtain necessary and sufficient conditions for the existence of a meromorphic function with a given pole multiplicity which satisfies Nevanlinna–Pick interpolation conditions and has the minimal possible L ∞ -norm on the unit circle T .