Schur coefficients in several variables

In the paper we study weakly continuous Schur-class-valued maps and their associated Schur coefficient families, that we call functional Schur coefficients. A case of special interest is the family of the “slices” through the polytorus of an n-variable function in the unit ball of H∞(Dn), which is shown to be a weakly continuous map from the polytorus into the Schur class. The continuity properties of its functional Schur coefficients are used to characterize the rational inner functions in the polydisk algebra. As a consequence we obtain extensions in several variables of the Schur–Cohn test on zeroes of polynomials. This provides in particular a necessary and sufficient condition of stability for multi-dimensional AR filters.