Finite Blaschke product interpolation on the closed unit disc

We show how to construct all finite Blaschke product solutions and the minimal scaled Blaschke product solution to the Nevanlinna–Pick interpolation problem in the open unit disc by solving eigenvalue problems of the interpolation data. Based on a result of Jones and Ruscheweyh we note that there always exists a finite Blaschke product of degree at most n − 1 that maps n distinct points in the closed unit disc, of which at least one is on the unit circle, into n arbitrary points in the closed unit disc, provided that the points inside the unit circle form a positive semi-definite Pick matrix of full rank. Finally, we discuss a numerical limiting procedure.  2002 Elsevier Science (USA). All rights reserved.