Multi-point degenerate interpolation problem for generalized Schur functions: description of all solutions (Q643198)

The generalized Schur class \(S_k\) is defined as the set of meromorphic functions in the unit disk of the form \[ f(z)=\frac{s(z)}{b(z)}, \] where \(s\) is a Schur function and \(b\) is a Blaschke product of degree \(k\). The author studies the general Nevanlinna-Pick type interpolation problem in the class \(S_k\): Given the data set \[ \{z_i,\;n_i,\;c_{ij}: \;j=0,1,\dots,n_i-1; \;i=1,2,\dots,k\}, \] where \(\{z_1,\dots,z_k\}\) are distinct points in the disk, \((n_1,\dots,n_k)\) is a tuple of multiplicities, \(\{c_{ij}\}\) are complex numbers, find all solutions \(f\in S_k\), analytic at \(z_i\) and \[ \frac{f^{(j)}(z_i)}{j!}=c_{ij}. \] The objective of the paper is to present an algorithm providing all solutions to this problem in the degenerate indeterminate case. The algorithm suggested in the paper works equally well for every \(k\) for which the problem has infinitely many solutions.