We describe a novel approach to analytic rational interpolation problems of the Hermite-Fejér type, based on the fast generalized Schur algorithm for the recursive triangular factorization of structured matrices. We use the interpolation data to construct a convenient so-called generator for the factorization algorithm. The recursive algorithm then leads to a transmission-line cascade of first-order sections that makes evident the interpolation property. We also give state-space descriptions for each section and for the entire cascade.