Inverse spectral analysis for a class of infinite band symmetric matrices

This work deals with the direct and inverse spectral analysis for a class of infinite band symmetric matrices. This class corresponds to operators arising from difference equations 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.