The Szegö matrix recurrence and its associated linear non-autonomous area-preserving map

Abstract

A change to the Szegö matrix recurrence relation, satisfied by orthonormal polynomials on the unit circle, gives rise to a linear map by the action of matrices belonging to the group SU(1; 1). The companion factorization of such matrices, via 2nd-order linear homogeneous difference equations, provides a compact representation of the orthogonal polynomial on the circle. Moreover, an isomorphism SU(1; 1) ≃ SL(2;ℝ) enables the introduction of a linear non-autonomous area-preserving map. This dynamical system has counterparts in those from the complex Szegö recurrence relation, and some basic results are outlined.