  This note deals with the direct and inverse spectral analysis for a class of
infinite band symmetric matrices. This class corresponds to operators arising
from difference quations with usual and inner boundary conditions. We give a
characterization of the spectral functions for the operators and provide
necessary and sufficient conditions for a matrix-valued function to be a
spectral function of the operators. Additionally, we give an algorithm for
recovering the matrix from the spectral function. The approach to the inverse
problem is based on the rational interpolation theory.
