Asymptotic Behaviour of Orthogonal Polynomials on the Unit Circle with Asymptotically Periodic Reflection Coefficients Original Research Article

Let {an}n∈N0withan∈C,an+N=anand |an|<1 for alln∈N0, be a periodic sequence of reflection coefficients and let {Pn}n∈N0be the associated sequence of orthogonal polynomials generated byPn+1=zPn−ānP*n. Furthermore let {bn}n∈N0be an asymptotically periodic sequence of reflection coefficients which arises by a perturbation of the sequence {an}n∈N0and thus satisfies the conditions limν→∞ bj+νN=ajforj=0, …, N−1, and |bn|<1 for alln∈N0. Let {Pn}n∈N0generated byPn+1=zPn−b&nP*nbe the disturbed orthogonal polynomials. Using the “periodic” polynomials {Pn}n∈N0as a comparison system we derive so-called comparative asymptotics for the disturbed polynomials on and off the support of the disturbed orthogonality measure, which consists essentially of several arcs of the unit circle. As a by-product of these results we obtain asymptotically a description of the location of the zeros of {Pn}n∈N0. Finally, a representation for the absolutely continuous part of the disturbed orthogonality measure is derived, and it is shown that there are at most finitely many point measures if thebnʹs converge geometrically fast to theanʹs.