The Nevanlinna-Pick interpolation problem in multiply connected domains

We simplify and strengthen Abrahamse's result on the Nevanlinna-Pick interpolation problem in a finitely connected planar domain, according to which the problem has a solution if and only if the Pick matrices associated with character-automorphic Hardy spaces are positive semidefinite for all characters in ℝ^n-1/ℤ^n-1, where n is the connectivity of the domain. The main aim of the paper is to reduce the indicated procedure (verification of the positive semidefiniteness) for the entire real (n-1)-torus ℝ^n-1/ℤ^n-1 to a part of it, whose dimension is, possibly, less than n-1.