ASYMPTOTIC-BEHAVIOR OF ORTHOGONAL POLYNOMIALS ON THE UNIT-CIRCLE WITHASYMPTOTICALLY PERIODIC REFLECTION COEFFICIENTS

Let {a(n))(n is an element of)N-9 with a(n) is an element of C, a(n+N) =a(n) and /a(n)/<1 for all n is an element of N-0, be the periodic seq uence of reflection coefficients and let {P-n}n<is an element of N0> b e the associated sequence of orthogonal polynomials generated by P-n+1 =zP(n)-<(a)over bar (n) P-n>. Furthermore let {b(n)}(n is an element of N0) be an asymptotically periodic sequence of reflection coefficien ts which arises by a perturbation of the sequence {a}n is an element o f n and thus satisfies the conditions lim(v-->x)b(j+vN)=a(j) for j=0,. .., N-1, and /b(n)/ <1 for all n is an element of N0. Let {<(P (n))(n is an element of N0) generated by <(P)over tilde (n+1)>=<z(P)over tild e (n)-(b) over tilden$ <(P)over tilde n> the disturbed orthogonal pol ynomials. Using the ''periodic'' polynomials {P-n}(n is an element of N0) as 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 asymptotic ally a description of the location of the zeros of {<(P)over tilde (n) }n is an element of N-0. 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 th e b(n)'s converge geometrically fast to the a(n)'s. (C) 1997 Academic Press.