Real lossless functions of continuous- and discrete-type orthogonal polynomials on the real line, and polynomial stability tests

The authors deal with the problem of verifying whether a given antisymmetric function is lossless (in the discrete sense), which is closely related to the polynomial stability problem. The proposed approach is based on some simple correspondences between the class of discrete-type lossless functions and the subclass of continuous-type lossless functions having all their finite poles and zeros in a fixed interval of the imaginary axis. They make it possible to derive a collection of discrete-type losslessness tests by a mere translation of suitable refinements of the classical Cauer criteria. The underlying recurrence relations are shown to have interesting interpretations in the classical framework of real line orthogonal polynomial theory